That sounds like an absolute banger of a game show. Two contestants sitting there in complete silence for twenty minutes, until Alice goes "does Bob have 21", and then everyone just sort of goes home. Still better than Deal or No Deal.
It also assumes they both know that the other person has perfect reasoning. It is NOT perfect reasoning to assume the other person will figure it out unless you know they also are very smart.
I got it with less formal reasoning. Their only means of communicating is the clock. One-way and very little information. But it is monotically increasing (in time)... and that is counting. That leads to the same solution (and it doesn't matter if your partner uses induction).
Its easy. If they got 999999999 then they just have to leave that game room and continue their life setting an alarm for 9999999999 minutes. So after 3-4 years they both will get notification that after 1minute their number is coming so they will call game authority and guess other persons number and win 1million.
@@noobgamedev8621 you're probably not noticing how large this number is, its 10^12 -1 and 10^12, suppose it rings every second, it will take 31709 years to guess since every year has 365*24*60*60 = 31536000 seconds that is about 3.2*10^7 only.
Of course, your solution depends on them having been given reasonably small numbers - the game show's producers are likely to lose patience if they're forced to wait more than a few minutes/rings (15? 30? 60?) and either force a guess or send them both home without any winnings.
@@ccanterod68 But probably not 4000 years. As worded, the solution would only work an infinitely small percent of the time, since all positive integers are an infinitely large set, where as the minutes in a human life-span, are not.
I have seen a lot of puzzles that use the same kind of solution, and I can tell you they are flawed and wont work. In the problem, each person is given a number and told the other person has one more or one less. In the solution, you state that the players come to a conclusion based on the possibility that they could have ANY number. You cannot come to a logical conclusion starting off with the statement "if bob/alice had 1" if it is an impossibility for them to have 1. In the example above, Alice had 20 and she is told that Bob has 19 or 21, therefore she would NEVER logically start with the assumption that bob has 1 and start the count from there. There are only 6 possible numbers for each person to wait on. From Alice's point of view, she knows that bob HAS to have either 19 or 21, so she can logically draw the conclusion that if Bob has 19, he knows alice has 18 or 20 and if bob has 21, alice either has 20 or 22. From Bob's point of view, he knows Alice HAS to have either 20 or 22, so he can logically assume that if Alice has 20, she knows Bob has 19 or 21 and if Alice has 22, Bob has either 21 or 23. This means that between the two of them, the lowest number that would be possible between them is 18, so it is a logical impossibility to include the number 1 or 2 or so on.
"From Alice's point of view, she knows that bob HAS to have either 19 or 21, so she can logically draw the conclusion that if Bob has 19, he knows alice has 18 or 20 and if bob has 21, alice either has 20 or 22" Knowing what Bob knows isn't enough, Alice also has to knows what Bob knows about what Alice knows, which in this case, if Bob has 19, he knows that Alice will think Bob has 17 and 19 (in case Alice has 19) and 19 or 21 (the other case) Doing that again and again brings the lowest number possible down to 1
@@randompersondontmindme Except that train of thought is limited by the fact that each person knows their own number. Alice will never think Bob has 17 because Alice's number is 20 The hypothetical ends at allice possibly having 18 because Alice would not have to say "And if bob thinks I have 18 then that means this" because she doesnt have 18. My point is it's not "perfect logic" to start from the number one, it's an arbitrary rule that has been assigned to this problem.
@@wookiebush7449 just bc Alice wouldn't think Bob has 17 doesn't mean Alice should not consider the fact that Bob thinks Alice thinks Bob has 17. And even if that's true, how do they know the lowest number between them to start from? Say the formula is min(Alice-2,Bob-2). That would require Alice to know what Bob's number is to work out the starting number, which is impossible to workout in the beginning since the lowest number between them must be known before the game (which is how they can even get information from the other not knowing).
@@randompersondontmindme the answer is indeed flawed. He builds his answer based on the fact that you gain a piece of info from your partner remaing silent after one minute. But in reality this only applies if participants have 1 and 2 or 2 and 3 as their numbers. If numbers are higher they both know from the begining that the other will remain silent after one minute. There is no information buildup and therefore no way to start counting. They can still figure out the answer though, simply by realizing their only communication device is the clock ticking. They have to both bet that the other will say "I have N and he has N+1" at the Nth minute. The bet is pretty safe because it's the only way to reliably guess the answer. If they both realise that, they win. If there were no fix timing at which they can give their answer the bet is trickier because they also have to guess if their partner will use seconds or minutes as a time frame. The safer bet seems to be minutes because of the time needed to tell the answer being over one second.
It's induction so the other cases are only hypothetical so it's possible for them to have 1 and 2 and this is only used to prove the metod, and the metod works because the first to answer is the one with the lowest number N so the other person must have a number that is N+1.
This is one of those "We silently agree to wait for the right moment" puzzles. Like "Oh, the other person did not guess yet. Then he does not have 1!" But you know this already before the game begins.
No, it's about the difference between private knowledge and public knowledge. After the first moment, it's public knowledge that neither of them have one (they both know that and they both know the other knows). After minute 2, both know neither has two.
@@gordonparks3702 Yes but if one of them was told a random number like 31 then both of them know that it’s not 2, or 3, or 4, etc. so the only chances are one number before or one after. either that or i just did not get a single word from the explanation lol
@@Angel-qi4py I know what you mean, but think of it this way, neither person knows who has the lower number, so they can’t take a shortcut. The only way they can play (even though they know neither has 1, 2, etc) is to say to themselves “a person with 1 would answer here”, then next minute “a person with 2...” etc up until “a person with 20 would answer here”, so Alice answers
@Angel I have the same concern. They are not allowed a strategy, so for me the arbitrary choice to assume the other is eliminating possibilities that are already eliminated is not allowed - it is a strategy. As you say, they know it's not going to be 0 from their own number, not from the positive integer condition. I can't see that not being zero therefore helps in any way. I used their knowledge of each other's uncertainty to have them deduce after three rings that the other has a greater chance of having such-and-such number (the number they in fact have), but couldn't get to 100% that way. I don't think the puzzle is correctly stated if they aren't allowed to strategise.
Raze - T It's not the same method. This one requires both players to agree on a strategy, the real solution was purely logical. Though the outcome may look the same, the methods are very different.
This is the first puzzle of yours that I was actually able to solve! I felt so accomplished, then I realized that I solved it because of another video on your channel with a puzzle a lot like this.
Another way to describe the solution is that they have a perfect system for each of them to discover that they have the lower number. As the clock can be used to count, all that Alice has to do is to count all the way up to her own number and by then she knows that Bob has the higher number, because he's staying silent so far as his number hasn't come up yet. The same applies to Bob from his perspective. They're collectively waiting for the counter to go up to the lower number and whoever has that shouts out the higher number when the lower number comes up. It's really a very simple principle but of course it only works for relatively small numbers in practice.
The logic seems to fall apart when you actually consider the example where Alice gets 20 and Bob gets 21. The problem states that Alice and Bob are aware that their numbers are consecutive. So, they don't need to wait for minutes to tick by on the clock to know that the other's number isn't 1 or 2 or 3 etc. Alice knows right away that Bob has either 19 or 21 and Bob knows right away that Alice has either 20 or 22. And of course, both are aware that the other one is aware of this. So, without planning with each other beforehand, there's no reason they would each independently settle on the "each tick of the clock rules out the next highest number" strategy.
Came here to search for this comment. This puzzle and solution can be reworked to say that Alice and Bob know each of them have a *different* positive integer (not necessarily consecutive), and all they have to do to win is to say whose number is larger or smaller.
@@PutMe I'm also at a loss. The logic for the small numbers makes perfect sense, but for N I don't see why waiting the number of minutes gives them any additional information unless they're using the same strategy--which they're not allowed to coordinate on, so you'd have to show why using that strategy is "perfectly logical". Either the problem, or the solution, is poorly worded. The logic of waiting N minutes wasn't explained at all.
Like some others here, I'm not convinced that the proposed solution is purely logical. An assumption is made that a strategy exists, and that it is the *only* strategy that can be arrived at by reason alone, therefore Alice and Bob will both find it if they reason perfectly. The proposed strategy certainly works, but it's not the only one. Generally the problem is to encode a positive integer N using some positive integer R of rings. The solution R = N is arguably the "simplest" or "most obvious", but those are subjective judgements, not logical conclusions. There are other strategies -- for example, R = 2N would also work. To argue that Alice and Bob are *guaranteed* to both arrive at the R = N strategy, you would somehow have to prove that there is *no other* line of reasoning that would lead to a different strategy. I'm not sure what such a proof would even look like. Given that infinitely many working strategies obviously exist, it seems similar to proving that a theorem is true but can't be proven. Where's Kurt Gödel when you need him...
The solution does happen to "encode a positive integer N using some positive integer R of rings," but that is _not_ the general problem. The game as stated is for either player to determine their partner's value, and to do so with "best" play. A player holding N can say their partner has N+1 after N rings, which follows directly from the insight that a player who holds 1 knows their partner holds 2. There is no way to solve before the first ring, and solving any later wouldn't be "best." Only a player holding 1 can solve at the first ring; only one holding 2 can solve the second; etc. R = N is the only solution that meets the problem requirements.
Great! Just stick with the problems you try to solve. The reason why most ppl fail to solve logic puzzles is that they don't like thinking and quit easily.
How does this work with larger numbers? I get the logic with 2 and 3. If someone has 1, they'll know automatically that the person would guess correctly since 0 isn't positive. But how does this work if one person has 4 and the other has 5? What's the logical process behind this?
Think of it as a waiting game in which the clock is the ONLY communicative tool they have. They merely infer that at the end of the first ringing, they would know both their own number and their opponent's, IF their numbers were 1 and 2. It's irrelevant that their numbers are higher. They simply know that that is the ONLY information they could have possibly gathered by turn 1. Suppose B has a 2. They cannot answer by turn 1. But A can if they have a 1. If A does NOT answer by turn 1, this means A has a 3. B will then answer by turn 2. If B has a number higher than 2, they must wait a third turn, and so on and so on. The burden of answering switches sides each turn. With both participants knowing for a logical FACT, that ONLY the person with the same number as the number of turns of the clock, KNOWS the answer. So suppose you have 32, meaning your partner either has 31 or 33. You have waited 30 minutes. Judging from inference and abiding by the rule of thumb we have stated above (and the video has as well), by the 30 minute ringing mark, you would know NOBODY could answer. You wait for turn 31. Nobody answers. Your partner therefore has 33. If somebody does answer, you know they have 31. It's not so much about inferring information from the information you have received from the excercise before the turns have started. It's about gleaning information from the fact the ONLY realistic way to communicate, in this excercise, is through clock ringing turns. Without ANY communication, this excercise would have been impossible. But counting turns IS a communicative tool, and therefore the ONLY logical one they could use. Because they both know the both of them will have made the logical inference of "If clock=N-1, and nobody answers by turn N-1, then If I have N, my partner must have N+1" Without the clock ringing, the excercise falls apart. But that's what they DO have. And so the excercise is possible.
@@RedFloyd469 Ah, you've brought me back to this video after a year away lol. I was actually able to figure the answer out relatively quickly compared to my 1 yr younger self lol. Thanks for the detailed explanation though!
Going to be a long wait. That's for sure. I figure at that point, you consider the value of time spent waiting vs the amount of money you'd win. In this case of it not being worth it, you'd just guess at the first tick and take your 50/50 chance. 100000 minutes is only 2ish months, so I'd consider doing it since I can't make half a mill in that amount of time normally.
That's not in the rules, but yes, if you were TRAPPED in a room for some reason, you'd obviously only go for a few days or risk dehydration. But maybe there is food and water. You don't know. It isn't stated. Edit: perfect logicians don't lose count.
The solution given here doesn't have a relationship with logic. It is about a strategy: waiting until N top of the clock and saying the other has N+1. For instance, if Alice has 10 and Bob has 9, the space of solutions for both of them is limited to 8, 9, 10, and 11. Therefore, it doesn't make sense to wait until the 9th top of the clock to give the right answer unless it is a strategy (commonly shared strategy). Both of them know that solutions 1, 2, 3, 4, 5, 6, and 7 are impossible.
The thing about not communicating with each other to discuss the algorithm is avoided because they can both deduce this is the one true strategy to use. The rule is not that they cannot have the same strategy ... the rule was that they cannot discuss the strategy to use. Actually it doesn't matter if they discuss the strategy before they know their number ... what does matter is that they don't communicate in the unlimited time, say 25 trillion minutes , it takes to get to the number. They could be discussing if the end of the universe WAS worthwhile staying alive so long for, is it worth the million ? , in eye movement sign language if they can see each others eyes, for example. However the person asking the question in the real world may put an upper limit on the number... 0 to 20 is as good as 0 to infinity, same solution. The realisation is that with only one "bit" of information allowed , the failure to enter a guess is the only information they have..when the minutes gets to my number, and the other person did not know my number at t-1, I know their number, its t+1.. there's no need to divide the minute into parts, you just let them sit silent in the minute of t-1, and if they don't you answer in the minute for t, because if you dont they will incorrectly enter the guess of t+2 during the minute of t+1, possibly in the first second.
This is not logic, it's a strategy. Unless you watch constantly videos like this, a person who doesn't solve this kind of puzzles don't get the idea. I understand how it's solved. But this is mere strategy.
Being perfect logicians they would also see there is a less taxing way to get the answer. They will both realise that the ticking of the clock is a form of communication, counting out their numbers. Then, if the clock gets to your number, you know you have to call out on the next tick. Surely it's that simple.
I fail to realise why your number and the number of the other is dependent on when someone doesn't give an answer. Also, this sounds like something that has to be strategized beforehand.
Because the point of the puzzle is that there is no method of communication other than the clock. So if Alice and Bob have perfect logical reasoning, they would know the only way for either of them to accurately deduce the other's number would be to use the clock. As such, the person with the lower number waits until the clock has rung their number of times - this way they know that the person has the number one higher otherwise they would have made the guess.
@@alexanderdevine4567 But the same can be said for any problem where communication isn't allowed and there is just one solution. "Perfect Logic" should enable anyone when faced with the problem to instantly develop the solution. The problem is, the solution Alice and bob use isn't perfect logic, it is their own perfect logic which enables them to come up with a strategy that makes use of the chiming of the clock but ultimately they are still using a strategy. If there was no chiming of the clock the problem is technically still possible however they have to both independently land on the same timing system instead of the clock which you could still argue has an element of "perfect logic" as there would be an optimal time frequency to count up in to give both people enough time to consider what is going on without wasting time - and since they are both perfectly logical you can only assume that they would choose the same time frame? The only reason it isn't counted as a strategy is because it is their "perfect logic" which independently enabled them to come up with the same idea. However even the concept of the problem assumes that there is such a thing as perfect logic, but it is ridiculous to assume that this is the only way to solve the problem and that it is the best way that they would both settle on. Essentially what I am trying to say is that their strategy has nothing to do with their ability to solve the problem, and instead the mere concept of having perfect logic means that as long as there is something you can do to make the problem solvable, both parties will come up with a strategy independently. Imagine the perfect logical minds of Alice and bob are confronted with a new problem. They both have to write the same number down on a piece of paper independently. Their perfect minds both conclude that there is no logical way to decide on the same number, but if they choose 1 and the other person chooses 1 the win, and this makes sense because they both have perfect logic and therefore would think about the problem in the exact same way, leading to both coming to the same conclusion that 1 is the best number to choose. What these sorts of problems don't consider is what perfectly logical people do when faced with problems that have no "solution". Perfectly logical people basically have telekinesis in the same way that two identical computers will come to the same result when given the same imputs.
@@pwhnckexstflajizdryvombqug9042 The clock's function is not time but _iteration._ Alice and Bob don't decide arbitrarily that they will wait N minutes before guessing N+1. They realize from the rules of the game that someone who had 1 would solve _at the first opportunity._ When is that? After the first ring. If the second ring comes around, the first opportunity was not taken. If someone who had 1 would solve at the first ring, then when the clock rings the second time, everyone knows nobody is holding 1. Because someone who is holding 2 will solve as soon as they know nobody is holding 1, and the second ring tells them exactly that, they will solve at that second ring. And so on. The strategy arises from considering the rules of the game, which each of them can do independently, and because the game's structure is the same for both of them and they each understand it perfectly, they will recognize the same strategy. _Using_ a strategy is not prohibited; they are barred only from conferring with each other. "Write a number down on a piece of paper" doesn't have any structure from which to reason. The rules here conclusively state that if you hold 1, your partner holds 2. Alice and Bob know this, and that there will be structured, iterative opportunities to guess. That's why they can work out what to do. You seem to have a strange idea of what "perfect logic" means. Logic is nothing but rules to manipulate information, particularly values of true and false. Like, in what way is getting the same result from two computers (expected, commonplace) anything like telekinesis (impossible)?
i think it's important to emphasize for these riddles that they are both perfect logicians and they know each other are perfect logicians, it is briefly mentioned in passing but i think it should be made as a more important bullet point
After I saw the tree/cells problem on this channel (which is exactly the same problem btw), the provided solution seemed clearly logic to me. But with this second look at the problem, I think this solution is flawed. My problem with this solution, is that it seems "logic", but it's not more logic than thinking, as Bob : "I'll wait until we are at the (n+1)th minute where n is my number, and say n+1 is the other's number" Because, if one thinks like that, and knows the other thinks the same way, they'll win. It is obviously "a strategy" of some sort, so it isn't considered as a right answer. But the provided solution is not better, because Bob KNOWS that Alice CANNOT have 1 or 2 or 3 since the beginning. It is not logical at all to think "if she had 1, she would have guessed my number" because it's just totally impossible. This reasonning assumes to consider clearly impossible situations (Alice and Bob both know that neither can have the number 1,2,3,4,5...) as possibilities, only to use them as a legitimate way to count the minutes and claim it is logic. This "logical way of thinking" of Bob and Alice is just a strategy like any other, so yes, they'll win if they both think like that. It is probably not the same thing at all, but it makes me think of the unexpected hanging paradox, the reasonning of the prisonner seems perfectly logic, but it's incorrect anyway.
Why should it matter what's possible? I know that sounds glib, but it's a serious question. You haven't justified your assertion, that _"it is not logical to think X because it's impossible."_ Logic doesn't depend on reality! If your premises justify your conclusion, it's logical. Why is it okay to look at your number, let's say 12, and say "what if my partner had 11," but it's not okay to say "what if I had 1"? You don't happen to hold 1, but you can make a sound conclusion about that situation. "If I had 1, I would know my partner has 2. After one minute, we would win." That right there is a 100% airtight true statement. _It's no less true because you are holding 12._ The logic doesn't care what you're holding, only whether the premise and the conclusion match. You don't have 1, but it's nevertheless a fact that you could win the game if you did, and facts are in short supply. Let's not cast away an additional fact so quickly. "My partner would do the same thing if they had 1." Again, 100% justified conclusion. "Therefore, if I had 2, and a minute passed without my partner saying anything, they would have 3." We are just collecting all kinds of facts. So far none of them describe your actual situation, but your situation does not make them untrue or invalid statements. They're all justified by the previous step. Each fact leads to the next fact, and because of how integers work, eventually you will make a statement about what happens if you're holding 12, _which you are,_ and hey presto! you just went from sound conclusions that didn't seem to do you any good to a sound conclusion that wins you a lot of money. After 12 minutes you can say your partner holds 13, and if someone asks how you know, you say, "if she had 11, she would have said I had 12 after 11 minutes." And how do you know that? "Because she would know that if I had 10, I'd have said she had 11 after 10 minutes." And how do you know that? ... etc., etc. "Because if I had 1, I would have said she had 2 after one minute." See, at no point does this depend on an untrue statement, nor on a statement you have no way of knowing. You can know all these things for sure no matter what number you're actually holding.
This seems rather like the hundred monks puzzle, in that they have to follow a huge chain of induction in which at each step they must trust the other to be a perfect reasoner. As always in this sort of puzzle, the whole chain of reasoning is disrupted if, while one is trusting the other reasoner to have conditioned their latest response or non-response on a line of reasoning like, "Ah, since they didn't do thus-and-such on the last turn, they must not know Y", the other person has actually lost track and is thinking about what to have for dinner. But you did say they are both perfect reasoners and know that the other is too. So onward! Originally, both reasoners know that the other does not have an integer less than one. The base case here is when one of the reasoners has the number one. Let's say wlog that it's Alice who was told "one". She knows that Bob can't have zero, because that's not positive, so on the first turn she announces that Bob has two. If that doesn't happen and Bob doesn't announce on the first turn either, then both Alice and Bob know that neither of them has one. Now both reasoners know that the other does not have an integer less than two. The situation is the same as before except the minimum number is one step higher. So each turn is one inductive step, raising the minimum number by 1. Therefore by induction the problem is perfectly solvable. Alice and Bob's correct strategy is, if holding integer N, to wait N-1 turns and, if not pre-empted by the other, to announce that the other holds the next higher number.
first sorry for my english, first of all there's a clock so what if alice has 20 so bob might has 19 or 21 (bob has 21 for example), they just watch @ the clock and they both continuing with "I..." because of "I would say". We know alice has 20, so if bob has 19 she would start saying "I would say" @ 18, because bob has to decide between 18 and 20 if he has 19. BUT bob knows that he has 21 so he stays silent and if alice is the only one talking with "I would say" she knows that he has 21 because she is the only one talking. Of course both must think for this tactic, but still :D
“Each knows the other is perfect at logical reasoning” is not enough. It needs to be what is called “common knowledge”. Everything of the form of “A knows that B knows that ... that A [B] knows that B [A] is perfect at logical reasoning” must be true for the game to work for arbitrarily large numbers.
Yeah, it kind of bothers me because how often do people actually come up with this type of reasoning without having been taught? Alice and Bob need to both know that they both not only have perfect logical reasoning, but that they also will never take the 50% chance, and will immediately come up with a universal strategy to win the game every time, if such a strategy exists. It’s a type of reasoning that does not apply to humans.
@@blackmber there's nothing to teach, it's logical reasoning. And that same ability means they know they can guarantee it, so would not take the 50-50 (unless they reason they can use their time better by guessing because the number is huge).
I stumpled a lot of times on this type of reasoning, and I never agreed to it. IMHO, the fault (just in this case) is that after the two legit cases, the assumption to generalize to the inductive n-case is wrong. let us see. Alice has 1, so no wonder, she tells Bob has 2. Alice has 2, and Bob 3: Alice can't say anything, so Bob after the tick can't say anything either. After the second tick, Alice knows that Bob doesn't have a 1, so she says "3!". But figure Alice has a 3 and Bob has a 4. Alice: nothing - tick, Bob : nothing - tick. Now Alice and Bob have no additional information: they know from the start that both would have said nothing, They can guess that the other person will answer after n-1 tries (or whatever number) but this is not logic, is an emphatic strategy.EDIT Credits to Alessandro Svanascini thst posted before me.
but at the same time as is shown in the video if they use this logic they will indeed win the money, will they not? I can see your standpoint you think that if they have 20 and 21 they didn't get any info after five ticks but they did, they know that 5 minutes had passed and nobody did anything yet try to follow me here: alice has 20. she thinks (if bob has 19, he will think (if alice has 18, she will think (if bob has 17, he will think (if alice has 16, she will think.....)))) there is a concept in logic called "shared knowledge" after each tick their shared knowledge is increased, each now knows that the other person knows something and that the other person knows that the first person knows that the first person knows that the other person knows it take a look at the video again if they follow this "faulty" logic they will win the money. if the strategy works, the logic isn't really faulty, is it? but don't feel bad the concept of shared knowledge is incredibly difficult to grasp for anyone who hadn't specifically studied logic it's not explained in the video at all so it's natural that most people will reject the solution and claim that it doesn't work I bothered explaining this because you bothered to show me your way of thinking
Excellent explanation by Ante Renic! lorenzo, the process generalises to any positive whole number. Each successive minute is informing A and B not about the numbers but about each other's expectations about each other's expectations about each other's expectations about the numbers.
There is no "arbitrary strategy" behind this, it will work with 3 and 4 just as well as with 2 and 3 or with 12412 and 12413. You say they have no extra information after two ticks but that is wrong. Imagine Alice has a 3 and Bob has a 4. Alice is thinking: "Bob has either a 2 or a 4". Alice understands that if Bob had a 2, Bob would be thinking that she had either a 1 or a 3. Now, if Bob really had a 2, after the first tick Bob would realize that Alice has to have a 3. For some reason, however, Bob didn't guess at the second tick. That wouldnt make any sense if Bob really had a 2. Therefore after two ticks Alice knows for 100% that Bob cannot have a 2 and at the third tick she will guess Bob having a 4. This logic continues inductively for all natural numbers.
"but this is not logic, is an emphatic strategy.". Nonsense it is pure logic on each others part. And simply an assumption the other is logical. What does empathy have to do with it. I would know certain of my friends would totally screw this up. Others I know would get it right. Is that empathy?
With these type of puzzles I always have a problem with the inductive part of the solution. For example in this case, yes, it is true that if the other player had been told one then they would have guessed right away that the other player must have been told told two. But, if you are told 19 and you know that the numbers are consecutive, there is no scenario in which the other player was told 1 and therefore isn’t it illogical to look for what they would have done if they had wether they are a perfect logician or not? For these type of riddles I tend to take “your partner is a perfect logician” to mean “you can write your plan to your partner and you can trust that they will understand and execute what you propose perfectly but you cannot communicate in any other way before or after” and then I am more likely to come to the correct solution.
I think you can think of it more simpler in this case. After your number of minutes have passed you guess the one above you. If you had the higher one then your partner would have guessed one minute before, since he didn't it means he has the higher one and you can guess his.
There's nothing wrong with the logic of it. It's true that they don't gain any 'new' information about what the other person's number is on the first ring, but it's still the only thing they can conclude on the first ring. If the other person saw any number other than 1 they would not know with certainty, therefore the only thing you can conclude is that they didn't see the number 1. It doesn't matter whether that's new information or not to you, that's just the only thing you can conclude from it. The reason it changes over multiple iterations isn't that you gain knowledge about what the number actually is, the thing that's changing is that the 'i know that you know that i know that you know....' chain is increasing with each iteration - the important part isn't about what they learn about what the number is, what's actually important is that they gain knowledge about what the other person knows.
@@asdfqwerty9241 I know there is nothing wrong with the logic of it. It’s just that thinking about what would have happened in a scenario that has already been demonstrated to not be the case currently is difficult to think about. At least for me. If you start thinking about things that could have happened you could think about almost anything and very little of it is useful. So I reframe things in the way I described which can then lead me to think about things that are actually helpful in the actual problem. It’s funny because to solve these kind of problems it takes thinking outside the box in an indirect way. But I’m not good at thinking outside the box in that way so I try to think outside the box that’s outside of the original box which makes it easier for me to come up with the plain outside the box thinking that the problem is looking for. I figured that it might help others, specially those who are annoyed by this type of problem as I am. That was the point of my comment.
@@Vendavalez if it helps, this problem is the same as if the setup was "Alice and Bob are given a number each. They know they're positive integers, and they know they're distinct. They win if they correctly announce they have the lower number". The rest of the setup is the same. This way should make it clearer why each ring is important.
The strategy can be described even simplier: wait for clock ring as many times as your number and answer that your mate has N+1 number. If they have N-1 they will answer one turn of clock earlier following the same rule.
LamaLo because he knows that Alice could have 3 and is she has 3 then she would wonder if Bob has 2 and if bob has 2 then he would wonder if Alice had 1. Sorry if its a hard read but its midnight in britain and i really cant be bothered for grammar.
If Bob has 1, then he instantly knows that Alice has 2 and he would guess after the first ring. If Bob has 2, then he knows Alice has either 1 or 3. If Alice has 1, then she would instantly know and guess after the first ring. But if Alice has 3, then she wouldn't know and wouldn't guess after the first ring. That would tell Bob that Alice does not have 1. Thus, Bob would guess after the second ring. If Bob has 3, then he knows that Alice has either 2 or 4. If Alice has 2, then she knows Bob has 1 or 3. Since Bob won't guess after the first ring, she knows he has 3 and so she would guess after the second ring. If Alice has 4, then she wouldn't guess after the second ring, telling Bob that Alice does not have 2. Then Bob would guess after the third ring. If Bob has 4, then he knows that Alice has either 3 or 5. If Alice has 3, then she knows Bob has 2 or 4. Since Bob won't guess after the second ring, she knows Bob doesn't have 2 but instead he has 4. Then she would guess after the third ring. If Alice has 5, then she wouldn't guess after the third ring telling Bob that Alice does not have 3. Then Bob would guess after the fourth ring.
A simpler “explanation” of the solution: they both count the ticks of the clock. Whoever has the smaller number stands up at the appropriate tick. The other then declares the two numbers. Say Alice was given the number 20 and Bob 21. Alice and Bob count the ticks of the clock. Alice stands up at the 20th tick. At which time Bob shouts “20 and 21”. But this explanation breaks the rule of “no communication” between the 2 players! To abide by this rule, Alice simply shouts “We reached 20 ticks and he remained silent. I know I have 20 so he must have 21”. I really enjoy your good work.
I haven't watched the end of the video yet, but I have an idea. The clock must come into play. Alice and Bob aren't allowed to discuss strategy but, if they both have perfect logic, maybe they come up with the same idea as me. I'm thinking each person counts the clock beeps. When the clock beeps the number of times that's the same as their number, they shout out one higher. For example, if Alice has 9 and Bob has 10, she shouts out "ten" after the ninth beep. She knows Bob doesn't have 8 because he would have shouted "nine" after the eighth beep.
I've watched this video three times in a row and still don't get how the clock's ringings relate to their given N, N+1 numbers. Is there any rule that states each ring represents the next integer for them to guess, or maybe a rule forbidding them to take more rings than the smallest of their numbers or something?
I think the reasoning here is to apply the same method used for the numbers 1 & 2 and 2 & 3 to any pair of numbers. If you have 1, you immediately know your partner has 2 (clock has rung zero times). If you have 2, you know after the first ring your partner has 3. Therefore, if both players follow this rule perfectly, they know after the clock rings their number minus one, their partner has a larger number. But I haven't wrapped my head around how thinking like this makes them "perfect logical thinkers"
This puzzle doesn't work for numbers greater than 3. Let's suppose Bob has 4 and Maria 5. After one minute without answer, they'll both know, indeed, that the other one doesn't have a 1. Problem is: they already knew that at the beginning of the game! Bob has a 4, so he knows that Maria has either 3 or 5. He also knows that if Maria had to guess his number, she would choose either 2, 4, or 6 (if she had 3, she'd pick either 2 or 4, and if she had 5, she'd pick either 4 or 6) Same goes for Maria, she knows Bob has either a 4 or a 6, and that if he had to guess Maria's number, he could say either 3, 5, or 7. So, to recap: Bob and Maria both know, that neither of them has a 1, and they also know, that the other one is aware of that fact. Question: why would they wait one minute to confirm that fact, since they both already know it? They both know that they don't have a one, as they know that the other one knows that to. So waiting a turn to eliminate the possibility of one of them having a 1 would be meaningless, ergo it would have to be strategicaly decided, wich is against the rules.
Rafou, Of course it's strategically decided, but it's strategically decided on an individual level, which is not against the rules. Otherwise, there would be no way to play, other than blurting out random guesses at a random time. The rules state that Alice and Bob cannot communicate with eachother, which means that *during the game* , they can not send eachother hints by eyewinks, hand signals or stuff like that to tell the other what number they have; and that they are not allowed to plan a strategy, meaning that *before the game starts* they are not allowed to sit together and consult with eachother on what mutual strategy they should follow.
Bob has 4 and Maria has 5, "thinking perfectly". Bob thinks that maria have 3 or 5, if maria had 3, on the 3nd round she'd say that bob had 4, because if bob had 2, on the second round he'd say maria has 3, so maria cannot have 3. Hence maria has 5 and he says it on the 4th or 5th round
Well, if either was given the number 1, they'd immediately guess and win. If not, a minute goes by and they both know 1 is not either's number. Next minute, 2 or not, etc. So basically you wait till your number of minutes, and then guess. This is a standard inductive reasoning problem. I've heard it a bunch of ways, but the answer is always the same.
I was like, staying silent, what information it gives to the other, it tells that you cannot tell the other's number, so you don't have the number that would enable you to, so you don't have 1, that's how I figured it out, using this kind of reasoning is very effective for logic riddles
While the forward induction solution is correct, it is inefficient. Simpler solution is to wait until n minutes pass where n is my number and then state that the other has n+1. Since the other has not spoken out before me, I must be the one with the lower number.
Not watching the rest of the thing, I can assume it goes by induction. If they're given 10-digit numbers, the game will go on longer than their lifetimes if they want to be sure to win, so they're better off taking a random guess and not devoting their life to it.
I got the same answer but with a different (and less perfect) method. I figured that, since they have no way to communicate, and since they are dealing with whole numbers, and since the clock striking constitues a string of whole numbers, they can use that to communicate. If they both do this, then the clock will strike N times, where N is the number which one of them has. Since the other person has not said anything by that point, then their number must be N+1.
Planning beforehand goes against the rules of the game. An easier way to think about it is that they don't know what the game will be before they are separated.
I got this one right! I don't know why you explained it so complicated though. Just wait until the clock has rung amount of times, and then guess the number above yours
@@stevenloube6784 In that scenario the other player correctly guessed your number one ring before you would have incorrectly guessed theirs. But it does assume both players indepently figure out to do this, since they can;t actually plan to both do it,
The problem with this is that they know the other's number is n-1 or n+1. The point behind this strategy is that you get new information with each ring of the bell (if your partner doesn't say anything on the kth ring, you know their number isnt k). However, you already know that their number is n-1 or n+1. So on the kth ring, if k is not n-1 and k is not n+1, you already know theyre not saying anything, even before the whole thing starts. There is actually no new information there. And logically speaking, why would this be a solution when you get no new information most of the time? Even though they both know they are following the same logic, they are ignoring the fact that they know for most of the initial rings that their partner will not say anything. With no new information, that means there is no logic; it is a predetermined strategy. Which makes this whole thing pointless, because if it is a predetermined strategy, then why not just choose some simpler strategy? If n=20, why not decide to start out at k=15 to save time? Its because they can't communicate beforehand to start at k=15. So that means they can't communicate beforehand to choose to ignore that they know nothing would be said for the first few bell rings
Firstly, separate whoever has the odd and an even numbers. Because the two numbers are consecutive, one will be odd and the other will be even. For our example we'll use Alice to have the odd number. If Alice had 1, she could answer at minute #1, because she knows Bob can't have 0 (not allowed). if they didn't, then if Bob had 2, he could answer at minute #2 because he knows she doesn't have 1. If he doesn't do that, and if Alice had 3, she could answer at minute #3 because she knows he doesn't have 2. etc etc. The numbers are finite, so eventually one of them will reach their number. and they can answer and they both get a fuckton of money easily
they obviously don't follow logic perfectly since the puzzle says they are consecutive numbers, so they should never even be thinking about the possibility of the other person guessing 1 on the first ring.
I think there's not enough information here to solve this puzzle. Your rules say that the two cannot have a plan or strategy worked out before hand. You then state that there is a clock in the room that rings each minute but it is never stated that the clock is integral to the solution. Bob and Alice would have to assume that the clock has nothing to do with the solution unless they had planned to use that strategy from the beginning. I don't see how your logic works here.
The other assumption is that Alice and Bob have perfect logical skills, so they would both figure out that the method is the only way to solve the problem, so they will both act according to the strategy.
Bill Tiemann The clock is important because it is only when the clock rings that they can guess. You don't need a strategy in advance because every time the clock rings you GAIN information about the numbers because the other person stayed silent
Yeah but they may have stayed silent because they were still thinking. I think this isnt really a solution and that the question is actually insolvable
Simple enough. If Alice was told 1, then Bob would have to have the number 2 so she would guess so after the first minute had elapsed. If Alice was told 2, she would know that Bob had either 1 or 3, but if Bob had a 1, he would guess that she has a 2 after the first minute. So Alice would wait, and if Bob didn't say anything, then she would know he has a 3 and says so after the second minute. Thus, this pattern continues, and you could induce that "If Alice has the number n, then Bob has n+1 or n-1. If Bob had n-1, he would determine so after n-2 minutes of silence, and he would guess Alice's number n after the n-1'th minute. If he had n+1, there would be silence after the n-1'th minute, and so Alice would be able to guess his number. Some cool side-effects of this strategy: If Annie has number N, she would only guess Bob's number on the Nth minute. Whoever has the lower number will always make the guess.
Since planning a strategy was not allowed..How would a person know if he/she has to wait N minutes if the number is N and guessing N+1 as others number. Seems conflicting.
They have perfect logical reasoning and both people know this. And because this is the only best strategy, Alice and Bob can assume they are both using that strategy.
ill try to explain it: Let's supose Alice got 37 and Bob 36. Alice knows that Bob's number may be either 36 or 38. Bob knows that Alice's number is either 35 or 37. Bob will wait until 36 minutes happen, while Alice will wait until 37 minutes. When the 35th ring happens and no one makes a guess, Bob will guess on the 36th ring that Alice's number is 37. How? Simple, If Alice's number were 35, she would have known that Bob is 36 because when 34h ring happened and no one made a guess she would have instantly know that Bob's number wasn't 34, and guessed 36 on the 35th ring. But as she couldn't guessed it, that means that Alice's number was Bob's _N_ +1 (37) To summarize: If Alice waits till the _N_ -1 ring, and Bob hasn't guessed, then she knows she is the lower number, so she will guess _N_ +1. Instead, if Bob waits till _N_ -1 rings and Alice couldn't guess, he will guess _N_ +1 on the _Nth_ ring.
The solution would involve some strategizing beforehand to work. Bob can easily contradict Alice. After the clock rings for the first time and neither players guess, Alice may think that Bob has a number greater than her own and answer on the second ring. Similarly, since Alice didn't guess on the first ring, Bob and assume that Alice has a number greater than his. Therefore, when the clock rings a second time, they can still get the answer wrong. They still have a 50% chance of winning (1 of the 2 players would have the correct answer).
Вениамин Феафанов Not exactly, we can say Bob had 4 and Alice had 3. After the clock rings for the first time, Alice assumes Bob has 4 (Which is correct), while Bob assumes Alice has 5 (Which is incorrect). Now, there's a 50% chance of Alice calling out her answer before Bob, therefore, the solution is flawed.
After the clock rings for the first time, Alice does not assume that Bob has 4, she concludes that Bob doesn't have 1. After the second ring she concludes that he doesn't have a 2, and on the third ring she can guess that he has a 4.
Kevin Tong if A = 1 and B = 2, A would know what B is immediately, and if B = 1 and A = 2, B would know what A is immediately. If A=2 and B=3, after the first minute, if nobody guessed, A would know that B cannot be 1. Therefore, A would know that B would be 3. B, being a perfect logician, would not guess on the second minute because B does not know what A is. They have a 100% chance of winning.
You can't have the same number. One of them will always get to their number first and call it out. They are perfect logicsticians. Which is a fancy way of saying you can only have a strategy that requires no planning, just obvious logic.
I considered the case where someone as 1 but I didn't reason up to the solution; I guess I couldn't decide if Alice or Bob could hear each other's guesses. You said they couldn't communicate.
Your answer is paradoxical. If Bob has the number 20, he knows there is a zero percent chance Alice will say her answer on the first minute, so her not saying anything on the first minute doesn't actually provide any information to him since he knew that would happen. Bob has to think there is some chance Alice will say something on the first minute, for her not saying something on the first minute, to give him information. Also, what if their 2 numbers are a quintillion, and a quintillion and one? See you in 2 trillion years, Alice and Bob.
medexamtoolsdotcom You are wrong. Imagine this. Alice has 20, Bob has 21. 19 ticks pass an Bob says nothing. then, Alice knows Bob has 21 because if Bob had 19, then he would have known Alice had 20, since he knew Alice couldn't have 18. He would've knew Alice couldn't have 18 because if Alice had had 18, she wouldve known Bob had 19. She knew Bob had 19 because if Bob had 17, he would've known Alice had 18,...and so on. Eventually we get to where Bob would've had 2, and he would've known Alice couldn't have 1 because then she would've called it out. Do you get how the process staircases down until we get to 1, where we can draw a logical conclusion?
The point is the first 18 minutes are meaningless to either Alice or Bob because Alice has 20 so she knows Bob can't have 1-18 and Bob has 21 so he knows Alice can't have 1-19. There is no information to be gained by having the other person stay silent. Therefore counting off the timer is not a logical move, but a preplanned strategy which is not allowed. The problem is not the logical process staircase, it's that the setup of the question is bad .
chinareds54 you said what I was trying to say here better than I did. It requires the two of them to have this code between them which amounts to a preplanned strategy.
It's a perfectly logical strategy for both contestants to arrive at independently. In fact it's the only way they can guarantee of getting the right answer.
No. That is irrelevant. If they can't plan a strategy in advance, then as long as they don't have 1 and 2, the amount of time that passes doesn't help.
If the smallest number is at least 3, this doesn't work, because this logic works only IF you suppose the smallest can be 1. This sequence of reasoning works only when you cas say for sure that first "if" is true or false.
Alessandro Svanascini if bob and alice both say to themselves that they will guess their number +1 after their number of minutes has past they will always win
This is not true. Let's say person A has a 3 and B has a 4. A knows that B has either a 2 or a 4. But if B had the 2 and A didnt guess on the first ring (which he didnt because A doesnt have a 1), B would guess on the second ring because of the logic described in the video. Therefore A knows, that B doesnt have a smaller number.
You are assuming that the players will guess exactly on the clock ring afterword. Since there is no "price" for waiting, the guess could be made any tine after they "know". If you added to the rules that the 1 million prize was divided by the number of rings when answer was declared, or that the contestants were in a hurry, then maybe this reasoning would be valid.
Actually, the answer assumes that they can only deduce things one minute at a time. In real life, it could take a lot less or a lot more time to figure out that reasoning. By the time the clock got to twenty, the reasoning would be so out of sync that someone would almost definitely guess incorrectly.
Yea I thought the same. But the actual reason has less to do with that inductive logic rather than the fact that they NEED to connect their number with the timer--there is no other way to win this. So given that they need to do this, the only logical/reasonable thing to do is to wait till your number comes up on the timer--there is just no other way to connect the alarm ringing to your number (that would also make sense to the other person). Since that is the only logical thing to do, if your (Bob's) number is 20 and Alice has not spoken till then, you know she does not have 19. So you call out 21. The logic detailed in the video is wrong: that makes it seem like their reasoning takes a minute--that makes no sense. The idea is that you only have to worry about N and N + 1 (given you have to tie your number to the timer and the timer starts from 1). EDIT: I just realized your post was 4 years ago (!!). Wow...this must be a blast from the past for you. Hope you survived Covid, and are having a pleasant life. Wish you well in this transient life of ours.
Alice and Bob are logicians and both know one of them will win $1,000,000 because obviously when they're given the number N, they will have to wait until the clock rings N times. (Of course, it's obvious, everybody knows that!) Bob opens his envelope. His number is 31,541,159. He decided he's not going to waste 60 year of his life counting the clock and goes on a trip with Alice instead.
surprisingly, this is one of the ones that I figured out, it kind of reminded me of the space time riddle (ill explain briefly) basically you and 30 other people are going to be teleported to different planets, and a machine with 2 levers (A and B) will be teleported to one of your planets right next to you. to make the machine teleport you have to switch a lever from on to off, or off to on (they don't affect anything just up or down) to get out of this hell, you have to figure out when the machine has been to every planet at least once you get a few minutes for pregame strategy the solution to this: one of you gets decided as the counter, and your going to be the one who decides when it reaches every planet for everyone else, they follow these simple rules 1) if the machine visits you for the first time you flip A to be on otherwise you flip B either direction 2) if the machine already has A flipped on then you flip B, and don't count it as the machine visiting you for the first time) for the counter, you add one to the count every time you turn the A switch off
but the problem is that if alice gets lets say 30, at the first mins she would not expect bob to say 2, cus bob cant have 1, because bob's number is 29 or 31. so logically, this question make sense in thoery, but never work in reality, this is like the prisoner paradox. where the prisoner was told "he will be executed any day of the week from monday to sunday, and the day of execution will be a surprise", so the prisoner rule out sunday, because if he is still alive on saturday, sunday will not be a surprise to him. but if he cant die on sunday, meaning saturday will not be a surprise (because he know he cant dies on sunday since is been ruled out), so he can not die on saturday. and if he cant die on saturday or sunday, then friday will not be a surprise. and this logic continues until he realize he will not be executed on any day of the week. but at the end, he was executed on a wednesday. and it was indeed a surprise to him.
I dispute the statement that 'zero is not a positive number'. My understanding is that zero is simultaneously both positive and negative, and therefore in the stated example if Alice has 1 then Bob could in fact have zero, as it can be considered positive. Of course if Bob did have zero then he could certainly say immediately that Alice must have 1, as she certainly couldn't have -1, so the logic does still hold, but it means that there is a risk that Alice and Bob could be one off from each other in their counts if they each hold a different position on whether or not zero is positive, which is a mathematical question, and not something that can be concluded solely from being a perfect logician.
plot twist, bob had no idea what was going on and just sat silently afraid to lose. Alice only won because of the luck of always having the lower number.
Intuitively, I would have done it differently! If my partner would have used your method, we would have been screwed! Also, because you said that both don't have the possibility to plan their method beforehand, I guess even your solution is not a real solution either. So here is my naive solution: everyone waits for their number. If the clock reaches their number, the player knows, that the next number is the number of their partner, otherwise the partner would have called the first players number, already. It's exactly the same reasoning you provided, but it is one off. So without each other knowing the strategy the problem is obviously but solvable.
I admit that's better than my solution. My solution is both yell out a number 1 less than their own (or 1 greater than their own, doesn't matter as long as they're consistently doing the same thing). There is 100% chance that one of them will get the right answer (and logically 100% chance that the other person will get the wrong answer). If they both yell out at exactly the same time, say on the first ring, then...they may win? The rule doesn't say anything if both correct and wrong answer are given at the exact same time. Technically (co-)first answer given is correct, hmm?
I managed to figure out the rule of "zeroes are not a possible number, meaning if person A has 1, person B must have 2. Therefore by turn 1, Person A must answer 100% correctly." I didn't quite make the logical inference yet that both of them could act on that information and switch the answering role each ringing of the clock. I did, as a hypothesis outside of logical formulas, consider the possibility that the ONLY way to communicate in this excercise is by using the clock, and the easiest way to do so was through clock ringings. However, I reasoned that strictly using such a hypothesis was a risky move on either participant's side, and that they could assume that a single ringing might mean "N=1" but also "N=60" or "N=1/60 (of an hour)" Using BOTH the idea that they CAN communicate through ringings AND that both of them know the both of them are perfectly logical, and that the rule I stated above in quotation marks MUST have been figured out by the end of ringing 1, we can infer the both of them know they HAVE to use the clock ringings as the definitive communicative tool, and use it in a manner that follows from inference. The only inference we can have are "if A answers , B must have N+1) So the ringing of the clock can not possibly have a meaning other than "Clock turn N= number N" rather than "turn N = 60 or 1/60", as those cannot have a logical inference from any information. In short, all I needed to do was figure out what to do from turn 1 onwards, but I kept gettin sidetracked by the possibility they might have numbers ranging in the dozens or even hundreds. That this is irrelevant was something the video had to show me, and I kicked myself for that.
If the other person stays silent during the first ring of the bell. that conveys the information that the other person does not have a 1. during the second ring, the other person will know that they don't have a 2 because if they had a 2 they would guess 3 since the other person cannot have a 1 (if they follow perfect logic). and you can continue until they come to the number in which Alice understands that Bob does not have a 19. and will then guess correctly.
It’s even simpler than this explanation. Obviously the clock is the only way for them to receive information, and the only way to use it to send meaningful info is to correlate the timing of the guess with the number. But they will not guess their own number, so if the clock gets to their number, they know the other player has a higher one.
I'm not sure if this is really logic - it's more of a trick. They both already know on day 1 that their colleague has at least 19 so their reasoning on the first 18 days is incorrect. There was never a chance of anyone saying yes on day 1 so no new information has really been gained. They eventually get the correct answer because they have effectively shared the rule "wait until the number of minutes passed is equal to your number and then guess your friend has 1 higher". If this is genuine logic it appear to imply that you can gain extra information about the world (your opponents number) by forgetting information you already have (the fact that your opponents number is 1 different from yours.)
why are Alice and Bob always in such crazy situations?
I used to be an adventuror like Alice and Bob, until I got an arrow to the knee...
MistaTwoJeffreyTwenty Yaay I think it's to represent person a and b
Because the evil Eve keeps putting them in these situations.
MistaTwoJeffreyTwenty Yaay at least this time they don't have to die or get trapped forever
Because of the first letters in their names :P
That sounds like an absolute banger of a game show. Two contestants sitting there in complete silence for twenty minutes, until Alice goes "does Bob have 21", and then everyone just sort of goes home.
Still better than Deal or No Deal.
lol
Never mind that...what if the numbers they're given are 849,332,102 and 849,332,101?
Deal or no Deal is my favourite show because the bank determines whether or not you have 200k
@@goodmaro they know they both have perfect logical reasoning, so they’re counting last digits.
HAHAHAHAHHAAJAHAAHHHAHHAAA WHY IS SO FUNNY LMAO
Man,they weren't kidding when they said Alice and Bob would have to have perfect reasoning. Jeez...
It also assumes they both know that the other person has perfect reasoning. It is NOT perfect reasoning to assume the other person will figure it out unless you know they also are very smart.
@@danielhughes3758 It is in the video. "Each knows the other is perfect at logical reasoning."
The hardest part of this game is trusting your partner. It's quite an easy riddle to solve once you trust the parameters that are explained.
I got it with less formal reasoning. Their only means of communicating is the clock. One-way and very little information. But it is monotically increasing (in time)... and that is counting. That leads to the same solution (and it doesn't matter if your partner uses induction).
@@paulmullins3353 they can’t tell each other this strategy so that wouldn’t really work
Host to Alice: the number im giving you is 999999999999
Bob: RIP Alice I guess we aren't winning the $1 Million
lmao fax but then again they are perfect logicians and now it is better to go for the 50 50 then die and wait it out after each ring
Its easy. If they got 999999999 then they just have to leave that game room and continue their life setting an alarm for 9999999999 minutes. So after 3-4 years they both will get notification that after 1minute their number is coming so they will call game authority and guess other persons number and win 1million.
@@noobgamedev8621 you're probably not noticing how large this number is, its 10^12 -1 and 10^12, suppose it rings every second, it will take 31709 years to guess since every year has 365*24*60*60 = 31536000 seconds that is about 3.2*10^7 only.
@@brucelau5359 Yeah but it will work
@@noobgamedev8621 assuming they’re immortal
me watching thumbnail:
thats easy!
me watching video:
ERROR 404: brain.exe not found
BEST COMMENT OF 2017 ^
I got this one, but only because I've seen similar riddles in the past, such as the green eyes riddle and the "how many trees" video you had.
Me too! :-)
Me too! Though in my case, it was from Ted-Ed riddles.
I could'nt be fooled twice!
Øyvind Aanderaa
same。。i saw these riddle on te de d
The Blue-eyed islander puzzle is by far the most famous of these epitomology riddles, but I've never heard of the "tree" one. Do you have a link?
Of course, your solution depends on them having been given reasonably small numbers - the game show's producers are likely to lose patience if they're forced to wait more than a few minutes/rings (15? 30? 60?) and either force a guess or send them both home without any winnings.
Oh, darling, for a million dollars I would wait a week
@@ccanterod68 But probably not 4000 years. As worded, the solution would only work an infinitely small percent of the time, since all positive integers are an infinitely large set, where as the minutes in a human life-span, are not.
@@itzdjyar You're absolutely right. If I got an unbearable number, I might just take the chance.
@@ccanterod68 any number above 5,260,000 is gonna take over 10 years.
The only numbers that are solveable within 24 hours are between 0 and 1440
imagine watching this on live tv
I have seen a lot of puzzles that use the same kind of solution, and I can tell you they are flawed and wont work. In the problem, each person is given a number and told the other person has one more or one less. In the solution, you state that the players come to a conclusion based on the possibility that they could have ANY number. You cannot come to a logical conclusion starting off with the statement "if bob/alice had 1" if it is an impossibility for them to have 1. In the example above, Alice had 20 and she is told that Bob has 19 or 21, therefore she would NEVER logically start with the assumption that bob has 1 and start the count from there. There are only 6 possible numbers for each person to wait on. From Alice's point of view, she knows that bob HAS to have either 19 or 21, so she can logically draw the conclusion that if Bob has 19, he knows alice has 18 or 20 and if bob has 21, alice either has 20 or 22. From Bob's point of view, he knows Alice HAS to have either 20 or 22, so he can logically assume that if Alice has 20, she knows Bob has 19 or 21 and if Alice has 22, Bob has either 21 or 23. This means that between the two of them, the lowest number that would be possible between them is 18, so it is a logical impossibility to include the number 1 or 2 or so on.
"From Alice's point of view, she knows that bob HAS to have either 19 or 21, so she can logically draw the conclusion that if Bob has 19, he knows alice has 18 or 20 and if bob has 21, alice either has 20 or 22"
Knowing what Bob knows isn't enough, Alice also has to knows what Bob knows about what Alice knows, which in this case, if Bob has 19, he knows that Alice will think Bob has 17 and 19 (in case Alice has 19) and 19 or 21 (the other case)
Doing that again and again brings the lowest number possible down to 1
@@randompersondontmindme Except that train of thought is limited by the fact that each person knows their own number. Alice will never think Bob has 17 because Alice's number is 20 The hypothetical ends at allice possibly having 18 because Alice would not have to say "And if bob thinks I have 18 then that means this" because she doesnt have 18. My point is it's not "perfect logic" to start from the number one, it's an arbitrary rule that has been assigned to this problem.
@@wookiebush7449 just bc Alice wouldn't think Bob has 17 doesn't mean Alice should not consider the fact that Bob thinks Alice thinks Bob has 17. And even if that's true, how do they know the lowest number between them to start from? Say the formula is min(Alice-2,Bob-2). That would require Alice to know what Bob's number is to work out the starting number, which is impossible to workout in the beginning since the lowest number between them must be known before the game (which is how they can even get information from the other not knowing).
@@randompersondontmindme the answer is indeed flawed. He builds his answer based on the fact that you gain a piece of info from your partner remaing silent after one minute. But in reality this only applies if participants have 1 and 2 or 2 and 3 as their numbers. If numbers are higher they both know from the begining that the other will remain silent after one minute. There is no information buildup and therefore no way to start counting.
They can still figure out the answer though, simply by realizing their only communication device is the clock ticking. They have to both bet that the other will say "I have N and he has N+1" at the Nth minute. The bet is pretty safe because it's the only way to reliably guess the answer. If they both realise that, they win.
If there were no fix timing at which they can give their answer the bet is trickier because they also have to guess if their partner will use seconds or minutes as a time frame. The safer bet seems to be minutes because of the time needed to tell the answer being over one second.
It's induction so the other cases are only hypothetical so it's possible for them to have 1 and 2 and this is only used to prove the metod, and the metod works because the first to answer is the one with the lowest number N so the other person must have a number that is N+1.
This is one of those "We silently agree to wait for the right moment" puzzles. Like "Oh, the other person did not guess yet. Then he does not have 1!" But you know this already before the game begins.
No, it's about the difference between private knowledge and public knowledge. After the first moment, it's public knowledge that neither of them have one (they both know that and they both know the other knows). After minute 2, both know neither has two.
@@gordonparks3702 Yes but if one of them was told a random number like 31 then both of them know that it’s not 2, or 3, or 4, etc. so the only chances are one number before or one after. either that or i just did not get a single word from the explanation lol
@@Angel-qi4py I know what you mean, but think of it this way, neither person knows who has the lower number, so they can’t take a shortcut. The only way they can play (even though they know neither has 1, 2, etc) is to say to themselves “a person with 1 would answer here”, then next minute “a person with 2...” etc up until “a person with 20 would answer here”, so Alice answers
@Angel I have the same concern. They are not allowed a strategy, so for me the arbitrary choice to assume the other is eliminating possibilities that are already eliminated is not allowed - it is a strategy. As you say, they know it's not going to be 0 from their own number, not from the positive integer condition. I can't see that not being zero therefore helps in any way. I used their knowledge of each other's uncertainty to have them deduce after three rings that the other has a greater chance of having such-and-such number (the number they in fact have), but couldn't get to 100% that way. I don't think the puzzle is correctly stated if they aren't allowed to strategise.
@@xoxb2 I think the rule means they can’t plan a strategy ahead of time with each other
If you have number N, wait N minutes and guess that the other person has N+1.
Uebeltank and that is the same "method" showed here
Raze - T It is.
Raze - T It's not the same method. This one requires both players to agree on a strategy, the real solution was purely logical. Though the outcome may look the same, the methods are very different.
Benjamin Gray I used the same method i just explained it simplified.
Uebeltank I got the same solution but I don't know why.
If Chris Hansen participated in that puzzle, he would have known that the other person "was told 18"
I wants ya and I'm gunna haves ya Chris Handsome. Now we can do this the hard way or the easy way.
This is the first puzzle of yours that I was actually able to solve! I felt so accomplished, then I realized that I solved it because of another video on your channel with a puzzle a lot like this.
Another way to describe the solution is that they have a perfect system for each of them to discover that they have the lower number. As the clock can be used to count, all that Alice has to do is to count all the way up to her own number and by then she knows that Bob has the higher number, because he's staying silent so far as his number hasn't come up yet. The same applies to Bob from his perspective. They're collectively waiting for the counter to go up to the lower number and whoever has that shouts out the higher number when the lower number comes up. It's really a very simple principle but of course it only works for relatively small numbers in practice.
Perfect explanation!!
Yes this is how i got it too. I was surprised by the weird logic in the vid based on the 0 is not positive stuff.
The logic seems to fall apart when you actually consider the example where Alice gets 20 and Bob gets 21. The problem states that Alice and Bob are aware that their numbers are consecutive. So, they don't need to wait for minutes to tick by on the clock to know that the other's number isn't 1 or 2 or 3 etc. Alice knows right away that Bob has either 19 or 21 and Bob knows right away that Alice has either 20 or 22. And of course, both are aware that the other one is aware of this. So, without planning with each other beforehand, there's no reason they would each independently settle on the "each tick of the clock rules out the next highest number" strategy.
Exactly. There's only one way either of them can be 100% certain, and it's if either one of them has 2 or 1.
Came here to search for this comment. This puzzle and solution can be reworked to say that Alice and Bob know each of them have a *different* positive integer (not necessarily consecutive), and all they have to do to win is to say whose number is larger or smaller.
This is exactly my issue with this answer.
You completely misunderstood the solution my dude
@@PutMe I'm also at a loss. The logic for the small numbers makes perfect sense, but for N I don't see why waiting the number of minutes gives them any additional information unless they're using the same strategy--which they're not allowed to coordinate on, so you'd have to show why using that strategy is "perfectly logical". Either the problem, or the solution, is poorly worded. The logic of waiting N minutes wasn't explained at all.
Give them seven digit numbers. Then they have to guess.
XD
Maybe at that point just wait the numbee of the last digit
Like some others here, I'm not convinced that the proposed solution is purely logical. An assumption is made that a strategy exists, and that it is the *only* strategy that can be arrived at by reason alone, therefore Alice and Bob will both find it if they reason perfectly.
The proposed strategy certainly works, but it's not the only one. Generally the problem is to encode a positive integer N using some positive integer R of rings. The solution R = N is arguably the "simplest" or "most obvious", but those are subjective judgements, not logical conclusions. There are other strategies -- for example, R = 2N would also work.
To argue that Alice and Bob are *guaranteed* to both arrive at the R = N strategy, you would somehow have to prove that there is *no other* line of reasoning that would lead to a different strategy. I'm not sure what such a proof would even look like. Given that infinitely many working strategies obviously exist, it seems similar to proving that a theorem is true but can't be proven. Where's Kurt Gödel when you need him...
The solution does happen to "encode a positive integer N using some positive integer R of rings," but that is _not_ the general problem. The game as stated is for either player to determine their partner's value, and to do so with "best" play. A player holding N can say their partner has N+1 after N rings, which follows directly from the insight that a player who holds 1 knows their partner holds 2. There is no way to solve before the first ring, and solving any later wouldn't be "best." Only a player holding 1 can solve at the first ring; only one holding 2 can solve the second; etc. R = N is the only solution that meets the problem requirements.
Game show host: I will give Alice 1 million and Bob 1 million + 1 and let them get on with it for the next two years!
I can't believe it . I've actually figured it out . I was thinking then suddenly the answer came to my mind . This video has given me confidence . 😁😁
Great! Just stick with the problems you try to solve. The reason why most ppl fail to solve logic puzzles is that they don't like thinking and quit easily.
I feel like the 50% thing is better than this time-consuming confusing thing
How does this work with larger numbers? I get the logic with 2 and 3. If someone has 1, they'll know automatically that the person would guess correctly since 0 isn't positive. But how does this work if one person has 4 and the other has 5? What's the logical process behind this?
Think of it as a waiting game in which the clock is the ONLY communicative tool they have. They merely infer that at the end of the first ringing, they would know both their own number and their opponent's, IF their numbers were 1 and 2. It's irrelevant that their numbers are higher. They simply know that that is the ONLY information they could have possibly gathered by turn 1.
Suppose B has a 2. They cannot answer by turn 1. But A can if they have a 1. If A does NOT answer by turn 1, this means A has a 3. B will then answer by turn 2. If B has a number higher than 2, they must wait a third turn, and so on and so on.
The burden of answering switches sides each turn. With both participants knowing for a logical FACT, that ONLY the person with the same number as the number of turns of the clock, KNOWS the answer.
So suppose you have 32, meaning your partner either has 31 or 33. You have waited 30 minutes. Judging from inference and abiding by the rule of thumb we have stated above (and the video has as well), by the 30 minute ringing mark, you would know NOBODY could answer. You wait for turn 31. Nobody answers. Your partner therefore has 33. If somebody does answer, you know they have 31.
It's not so much about inferring information from the information you have received from the excercise before the turns have started. It's about gleaning information from the fact the ONLY realistic way to communicate, in this excercise, is through clock ringing turns. Without ANY communication, this excercise would have been impossible. But counting turns IS a communicative tool, and therefore the ONLY logical one they could use. Because they both know the both of them will have made the logical inference of "If clock=N-1, and nobody answers by turn N-1, then If I have N, my partner must have N+1"
Without the clock ringing, the excercise falls apart. But that's what they DO have. And so the excercise is possible.
@@RedFloyd469 Ah, you've brought me back to this video after a year away lol. I was actually able to figure the answer out relatively quickly compared to my 1 yr younger self lol. Thanks for the detailed explanation though!
What about if the numbers are 100000 and 100001?
Going to be a long wait. That's for sure. I figure at that point, you consider the value of time spent waiting vs the amount of money you'd win.
In this case of it not being worth it, you'd just guess at the first tick and take your 50/50 chance.
100000 minutes is only 2ish months, so I'd consider doing it since I can't make half a mill in that amount of time normally.
Derek Gooding But you can't do anything else during those 2 months.
They would miscount the rings and lose xD
Samvel Matinyan they are perfect logicians so they would realise that it isn't worth their time and say screw it and take the 50/50 guess.
That's not in the rules, but yes, if you were TRAPPED in a room for some reason, you'd obviously only go for a few days or risk dehydration.
But maybe there is food and water. You don't know. It isn't stated.
Edit: perfect logicians don't lose count.
Your logic works only if one of them have 1. The ring of the clock has no relation with the numbers given.
Me: pausing the video again n again & trying to understand 😂😂
"Search your feelings it is two" funny
The solution given here doesn't have a relationship with logic. It is about a strategy: waiting until N top of the clock and saying the other has N+1. For instance, if Alice has 10 and Bob has 9, the space of solutions for both of them is limited to 8, 9, 10, and 11. Therefore, it doesn't make sense to wait until the 9th top of the clock to give the right answer unless it is a strategy (commonly shared strategy). Both of them know that solutions 1, 2, 3, 4, 5, 6, and 7 are impossible.
The thing about not communicating with each other to discuss the algorithm is avoided because they can both deduce this is the one true strategy to use. The rule is not that they cannot have the same strategy ... the rule was that they cannot discuss the strategy to use. Actually it doesn't matter if they discuss the strategy before they know their number ... what does matter is that they don't communicate in the unlimited time, say 25 trillion minutes , it takes to get to the number. They could be discussing if the end of the universe WAS worthwhile staying alive so long for, is it worth the million ? , in eye movement sign language if they can see each others eyes, for example. However the person asking the question in the real world may put an upper limit on the number... 0 to 20 is as good as 0 to infinity, same solution. The realisation is that with only one "bit" of information allowed , the failure to enter a guess is the only information they have..when the minutes gets to my number, and the other person did not know my number at t-1, I know their number, its t+1.. there's no need to divide the minute into parts, you just let them sit silent in the minute of t-1, and if they don't you answer in the minute for t, because if you dont they will incorrectly enter the guess of t+2 during the minute of t+1, possibly in the first second.
What you're missing is that it's a strategy they can come up with independently that is entirely founded in logic.
This is not logic, it's a strategy. Unless you watch constantly videos like this, a person who doesn't solve this kind of puzzles don't get the idea.
I understand how it's solved. But this is mere strategy.
Being perfect logicians they would also see there is a less taxing way to get the answer. They will both realise that the ticking of the clock is a form of communication, counting out their numbers. Then, if the clock gets to your number, you know you have to call out on the next tick. Surely it's that simple.
I fail to realise why your number and the number of the other is dependent on when someone doesn't give an answer. Also, this sounds like something that has to be strategized beforehand.
Because the point of the puzzle is that there is no method of communication other than the clock. So if Alice and Bob have perfect logical reasoning, they would know the only way for either of them to accurately deduce the other's number would be to use the clock. As such, the person with the lower number waits until the clock has rung their number of times - this way they know that the person has the number one higher otherwise they would have made the guess.
@@alexanderdevine4567 But the same can be said for any problem where communication isn't allowed and there is just one solution. "Perfect Logic" should enable anyone when faced with the problem to instantly develop the solution. The problem is, the solution Alice and bob use isn't perfect logic, it is their own perfect logic which enables them to come up with a strategy that makes use of the chiming of the clock but ultimately they are still using a strategy. If there was no chiming of the clock the problem is technically still possible however they have to both independently land on the same timing system instead of the clock which you could still argue has an element of "perfect logic" as there would be an optimal time frequency to count up in to give both people enough time to consider what is going on without wasting time - and since they are both perfectly logical you can only assume that they would choose the same time frame? The only reason it isn't counted as a strategy is because it is their "perfect logic" which independently enabled them to come up with the same idea. However even the concept of the problem assumes that there is such a thing as perfect logic, but it is ridiculous to assume that this is the only way to solve the problem and that it is the best way that they would both settle on.
Essentially what I am trying to say is that their strategy has nothing to do with their ability to solve the problem, and instead the mere concept of having perfect logic means that as long as there is something you can do to make the problem solvable, both parties will come up with a strategy independently.
Imagine the perfect logical minds of Alice and bob are confronted with a new problem. They both have to write the same number down on a piece of paper independently. Their perfect minds both conclude that there is no logical way to decide on the same number, but if they choose 1 and the other person chooses 1 the win, and this makes sense because they both have perfect logic and therefore would think about the problem in the exact same way, leading to both coming to the same conclusion that 1 is the best number to choose.
What these sorts of problems don't consider is what perfectly logical people do when faced with problems that have no "solution". Perfectly logical people basically have telekinesis in the same way that two identical computers will come to the same result when given the same imputs.
@@pwhnckexstflajizdryvombqug9042 The clock's function is not time but _iteration._ Alice and Bob don't decide arbitrarily that they will wait N minutes before guessing N+1. They realize from the rules of the game that someone who had 1 would solve _at the first opportunity._ When is that? After the first ring. If the second ring comes around, the first opportunity was not taken.
If someone who had 1 would solve at the first ring, then when the clock rings the second time, everyone knows nobody is holding 1. Because someone who is holding 2 will solve as soon as they know nobody is holding 1, and the second ring tells them exactly that, they will solve at that second ring. And so on.
The strategy arises from considering the rules of the game, which each of them can do independently, and because the game's structure is the same for both of them and they each understand it perfectly, they will recognize the same strategy. _Using_ a strategy is not prohibited; they are barred only from conferring with each other.
"Write a number down on a piece of paper" doesn't have any structure from which to reason. The rules here conclusively state that if you hold 1, your partner holds 2. Alice and Bob know this, and that there will be structured, iterative opportunities to guess. That's why they can work out what to do.
You seem to have a strange idea of what "perfect logic" means. Logic is nothing but rules to manipulate information, particularly values of true and false. Like, in what way is getting the same result from two computers (expected, commonplace) anything like telekinesis (impossible)?
i think it's important to emphasize for these riddles that they are both perfect logicians and they know each other are perfect logicians, it is briefly mentioned in passing but i think it should be made as a more important bullet point
There's no rule against communicating with the audience soooo...
There is also nothing saying there IS an audience sooooo.....
Derek Gooding it's a game show, so there would be an audience.
You are specifying a LIVE audience. Not all game shows have that. Take survivor. Or big brother.
It isn't stated and it isn't implied.
TheGinginator14 there's no given definition of communication. fg
TheGinginator14 not a necessarily a live audience soooo....
the best part about this is that if the two were allowed to plan before the game started you would probably arrive at the same conclusion
After I saw the tree/cells problem on this channel (which is exactly the same problem btw), the provided solution seemed clearly logic to me. But with this second look at the problem, I think this solution is flawed.
My problem with this solution, is that it seems "logic", but it's not more logic than thinking, as Bob :
"I'll wait until we are at the (n+1)th minute where n is my number, and say n+1 is the other's number"
Because, if one thinks like that, and knows the other thinks the same way, they'll win.
It is obviously "a strategy" of some sort, so it isn't considered as a right answer.
But the provided solution is not better, because Bob KNOWS that Alice CANNOT have 1 or 2 or 3 since the beginning. It is not logical at all to think "if she had 1, she would have guessed my number" because it's just totally impossible. This reasonning assumes to consider clearly impossible situations (Alice and Bob both know that neither can have the number 1,2,3,4,5...) as possibilities, only to use them as a legitimate way to count the minutes and claim it is logic.
This "logical way of thinking" of Bob and Alice is just a strategy like any other, so yes, they'll win if they both think like that.
It is probably not the same thing at all, but it makes me think of the unexpected hanging paradox, the reasonning of the prisonner seems perfectly logic, but it's incorrect anyway.
Why should it matter what's possible? I know that sounds glib, but it's a serious question. You haven't justified your assertion, that _"it is not logical to think X because it's impossible."_ Logic doesn't depend on reality! If your premises justify your conclusion, it's logical.
Why is it okay to look at your number, let's say 12, and say "what if my partner had 11," but it's not okay to say "what if I had 1"? You don't happen to hold 1, but you can make a sound conclusion about that situation. "If I had 1, I would know my partner has 2. After one minute, we would win." That right there is a 100% airtight true statement. _It's no less true because you are holding 12._ The logic doesn't care what you're holding, only whether the premise and the conclusion match. You don't have 1, but it's nevertheless a fact that you could win the game if you did, and facts are in short supply. Let's not cast away an additional fact so quickly.
"My partner would do the same thing if they had 1." Again, 100% justified conclusion. "Therefore, if I had 2, and a minute passed without my partner saying anything, they would have 3." We are just collecting all kinds of facts. So far none of them describe your actual situation, but your situation does not make them untrue or invalid statements. They're all justified by the previous step. Each fact leads to the next fact, and because of how integers work, eventually you will make a statement about what happens if you're holding 12, _which you are,_ and hey presto! you just went from sound conclusions that didn't seem to do you any good to a sound conclusion that wins you a lot of money.
After 12 minutes you can say your partner holds 13, and if someone asks how you know, you say, "if she had 11, she would have said I had 12 after 11 minutes." And how do you know that? "Because she would know that if I had 10, I'd have said she had 11 after 10 minutes." And how do you know that? ... etc., etc. "Because if I had 1, I would have said she had 2 after one minute." See, at no point does this depend on an untrue statement, nor on a statement you have no way of knowing. You can know all these things for sure no matter what number you're actually holding.
Waint till your number of clock sounds and say the number above
Your number is 482,529,496,183,013. Best of luck guessing your friends number.
markus dahle this problem is flawed. Or the solution at least. The number could be above 60 then what?
John Landon Miller after an hour, they would start over at 61. It changes nothing.
Glenn Estrada I thought there is not an hour available
Do you have an intelligent comment?
What about guessing simultaneously the by 1 incremented number? By this one of them has to be right. So, win or not?
Well, we already had that with the cells and the trees.
This seems rather like the hundred monks puzzle, in that they have to follow a huge chain of induction in which at each step they must trust the other to be a perfect reasoner.
As always in this sort of puzzle, the whole chain of reasoning is disrupted if, while one is trusting the other reasoner to have conditioned their latest response or non-response on a line of reasoning like, "Ah, since they didn't do thus-and-such on the last turn, they must not know Y", the other person has actually lost track and is thinking about what to have for dinner. But you did say they are both perfect reasoners and know that the other is too. So onward!
Originally, both reasoners know that the other does not have an integer less than one.
The base case here is when one of the reasoners has the number one. Let's say wlog that it's Alice who was told "one". She knows that Bob can't have zero, because that's not positive, so on the first turn she announces that Bob has two.
If that doesn't happen and Bob doesn't announce on the first turn either, then both Alice and Bob know that neither of them has one. Now both reasoners know that the other does not have an integer less than two. The situation is the same as before except the minimum number is one step higher.
So each turn is one inductive step, raising the minimum number by 1. Therefore by induction the problem is perfectly solvable. Alice and Bob's correct strategy is, if holding integer N, to wait N-1 turns and, if not pre-empted by the other, to announce that the other holds the next higher number.
So to summarise the strategy is: If you are a player with the number N on the Nth ring of the clock you guess the numbers N and N+1.
3:02 That scared the hell out of me!
first sorry for my english,
first of all there's a clock
so what if alice has 20 so bob might has 19 or 21 (bob has 21 for example), they just watch @ the clock and they both continuing with "I..." because of "I would say".
We know alice has 20, so if bob has 19 she would start saying "I would say" @ 18, because bob has to decide between 18 and 20 if he has 19. BUT bob knows that he has 21 so he stays silent and if alice is the only one talking with "I would say" she knows that he has 21 because she is the only one talking.
Of course both must think for this tactic, but still :D
talking about minutes btw, because seconds were way too fast
“Each knows the other is perfect at logical reasoning” is not enough. It needs to be what is called “common knowledge”. Everything of the form of “A knows that B knows that ... that A [B] knows that B [A] is perfect at logical reasoning” must be true for the game to work for arbitrarily large numbers.
Yeah, it kind of bothers me because how often do people actually come up with this type of reasoning without having been taught? Alice and Bob need to both know that they both not only have perfect logical reasoning, but that they also will never take the 50% chance, and will immediately come up with a universal strategy to win the game every time, if such a strategy exists. It’s a type of reasoning that does not apply to humans.
@@blackmber there's nothing to teach, it's logical reasoning. And that same ability means they know they can guarantee it, so would not take the 50-50 (unless they reason they can use their time better by guessing because the number is huge).
Use morphogenetic fields and determine whether Alice has to Ally with or Betray Bob.
This is the only way to do it. Definitely don't jump - reality can be different!
I stumpled a lot of times on this type of reasoning, and I never agreed to it. IMHO, the fault (just in this case) is that after the two legit cases, the assumption to generalize to the inductive n-case is wrong. let us see. Alice has 1, so no wonder, she tells Bob has 2. Alice has 2, and Bob 3: Alice can't say anything, so Bob after the tick can't say anything either. After the second tick, Alice knows that Bob doesn't have a 1, so she says "3!". But figure Alice has a 3 and Bob has a 4. Alice: nothing - tick, Bob : nothing - tick. Now Alice and Bob have no additional information: they know from the start that both would have said nothing, They can guess that the other person will answer after n-1 tries (or whatever number) but this is not logic, is an emphatic strategy.EDIT Credits to Alessandro Svanascini thst posted before me.
but at the same time
as is shown in the video
if they use this logic they will indeed win the money, will they not?
I can see your standpoint
you think that if they have 20 and 21 they didn't get any info after five ticks
but they did, they know that 5 minutes had passed and nobody did anything yet
try to follow me here:
alice has 20. she thinks (if bob has 19, he will think (if alice has 18, she will think (if bob has 17, he will think (if alice has 16, she will think.....))))
there is a concept in logic called "shared knowledge"
after each tick their shared knowledge is increased, each now knows that the other person knows something and that the other person knows that the first person knows that the first person knows that the other person knows it
take a look at the video again
if they follow this "faulty" logic they will win the money. if the strategy works, the logic isn't really faulty, is it?
but don't feel bad
the concept of shared knowledge is incredibly difficult to grasp for anyone who hadn't specifically studied logic
it's not explained in the video at all so it's natural that most people will reject the solution and claim that it doesn't work
I bothered explaining this because you bothered to show me your way of thinking
Excellent explanation by Ante Renic! lorenzo, the process generalises to any positive whole number. Each successive minute is informing A and B not about the numbers but about each other's expectations about each other's expectations about each other's expectations about the numbers.
If Alice's number is 30, then she wouldn't even begin thinking whether Bob's number is 1 because she knows it cannot be 1.
There is no "arbitrary strategy" behind this, it will work with 3 and 4 just as well as with 2 and 3 or with 12412 and 12413. You say they have no extra information after two ticks but that is wrong. Imagine Alice has a 3 and Bob has a 4. Alice is thinking: "Bob has either a 2 or a 4". Alice understands that if Bob had a 2, Bob would be thinking that she had either a 1 or a 3. Now, if Bob really had a 2, after the first tick Bob would realize that Alice has to have a 3. For some reason, however, Bob didn't guess at the second tick. That wouldnt make any sense if Bob really had a 2. Therefore after two ticks Alice knows for 100% that Bob cannot have a 2 and at the third tick she will guess Bob having a 4. This logic continues inductively for all natural numbers.
"but this is not logic, is an emphatic strategy.". Nonsense it is pure logic on each others part. And simply an assumption the other is logical. What does empathy have to do with it. I would know certain of my friends would totally screw this up. Others I know would get it right. Is that empathy?
With these type of puzzles I always have a problem with the inductive part of the solution. For example in this case, yes, it is true that if the other player had been told one then they would have guessed right away that the other player must have been told told two.
But, if you are told 19 and you know that the numbers are consecutive, there is no scenario in which the other player was told 1 and therefore isn’t it illogical to look for what they would have done if they had wether they are a perfect logician or not?
For these type of riddles I tend to take “your partner is a perfect logician” to mean “you can write your plan to your partner and you can trust that they will understand and execute what you propose perfectly but you cannot communicate in any other way before or after” and then I am more likely to come to the correct solution.
I think you can think of it more simpler in this case.
After your number of minutes have passed you guess the one above you.
If you had the higher one then your partner would have guessed one minute before, since he didn't it means he has the higher one and you can guess his.
There's nothing wrong with the logic of it. It's true that they don't gain any 'new' information about what the other person's number is on the first ring, but it's still the only thing they can conclude on the first ring. If the other person saw any number other than 1 they would not know with certainty, therefore the only thing you can conclude is that they didn't see the number 1. It doesn't matter whether that's new information or not to you, that's just the only thing you can conclude from it.
The reason it changes over multiple iterations isn't that you gain knowledge about what the number actually is, the thing that's changing is that the 'i know that you know that i know that you know....' chain is increasing with each iteration - the important part isn't about what they learn about what the number is, what's actually important is that they gain knowledge about what the other person knows.
@@asdfqwerty9241 I know there is nothing wrong with the logic of it. It’s just that thinking about what would have happened in a scenario that has already been demonstrated to not be the case currently is difficult to think about. At least for me. If you start thinking about things that could have happened you could think about almost anything and very little of it is useful.
So I reframe things in the way I described which can then lead me to think about things that are actually helpful in the actual problem.
It’s funny because to solve these kind of problems it takes thinking outside the box in an indirect way. But I’m not good at thinking outside the box in that way so I try to think outside the box that’s outside of the original box which makes it easier for me to come up with the plain outside the box thinking that the problem is looking for.
I figured that it might help others, specially those who are annoyed by this type of problem as I am. That was the point of my comment.
@@Vendavalez if it helps, this problem is the same as if the setup was "Alice and Bob are given a number each. They know they're positive integers, and they know they're distinct. They win if they correctly announce they have the lower number". The rest of the setup is the same.
This way should make it clearer why each ring is important.
@@SleepyHarryZzz no. That’s not the problem at all.
The strategy can be described even simplier: wait for clock ring as many times as your number and answer that your mate has N+1 number. If they have N-1 they will answer one turn of clock earlier following the same rule.
If Bob has 4, why should he ask himself whether Alice has 1 or not? He knows she does'nt have 1
LamaLo because he knows that Alice could have 3 and is she has 3 then she would wonder if Bob has 2 and if bob has 2 then he would wonder if Alice had 1.
Sorry if its a hard read but its midnight in britain and i really cant be bothered for grammar.
If Bob has 1, then he instantly knows that Alice has 2 and he would guess after the first ring.
If Bob has 2, then he knows Alice has either 1 or 3. If Alice has 1, then she would instantly know and guess after the first ring. But if Alice has 3, then she wouldn't know and wouldn't guess after the first ring. That would tell Bob that Alice does not have 1. Thus, Bob would guess after the second ring.
If Bob has 3, then he knows that Alice has either 2 or 4. If Alice has 2, then she knows Bob has 1 or 3. Since Bob won't guess after the first ring, she knows he has 3 and so she would guess after the second ring. If Alice has 4, then she wouldn't guess after the second ring, telling Bob that Alice does not have 2. Then Bob would guess after the third ring.
If Bob has 4, then he knows that Alice has either 3 or 5. If Alice has 3, then she knows Bob has 2 or 4. Since Bob won't guess after the second ring, she knows Bob doesn't have 2 but instead he has 4. Then she would guess after the third ring. If Alice has 5, then she wouldn't guess after the third ring telling Bob that Alice does not have 3. Then Bob would guess after the fourth ring.
thank you. I think I got it now. Don't be sorry :)
Winston Smith c
Math Man I don't get it. Why does the number of time the ring sound even matter? If Alice has 5, why wouldn't she guess after the third ring?
A simpler “explanation” of the solution: they both count the ticks of the clock. Whoever has the smaller number stands up at the appropriate tick. The other then declares the two numbers.
Say Alice was given the number 20 and Bob 21. Alice and Bob count the ticks of the clock. Alice stands up at the 20th tick. At which time Bob shouts “20 and 21”.
But this explanation breaks the rule of “no communication” between the 2 players! To abide by this rule, Alice simply shouts “We reached 20 ticks and he remained silent. I know I have 20 so he must have 21”.
I really enjoy your good work.
it's easy, but at least it's not one of those dumb order of operation "riddle"
I haven't watched the end of the video yet, but I have an idea. The clock must come into play. Alice and Bob aren't allowed to discuss strategy but, if they both have perfect logic, maybe they come up with the same idea as me. I'm thinking each person counts the clock beeps. When the clock beeps the number of times that's the same as their number, they shout out one higher. For example, if Alice has 9 and Bob has 10, she shouts out "ten" after the ninth beep. She knows Bob doesn't have 8 because he would have shouted "nine" after the eighth beep.
These are the kind of logic puzzles we love
I've watched this video three times in a row and still don't get how the clock's ringings relate to their given N, N+1 numbers.
Is there any rule that states each ring represents the next integer for them to guess, or maybe a rule forbidding them to take more rings than the smallest of their numbers or something?
I think the reasoning here is to apply the same method used for the numbers 1 & 2 and 2 & 3 to any pair of numbers.
If you have 1, you immediately know your partner has 2 (clock has rung zero times). If you have 2, you know after the first ring your partner has 3. Therefore, if both players follow this rule perfectly, they know after the clock rings their number minus one, their partner has a larger number.
But I haven't wrapped my head around how thinking like this makes them "perfect logical thinkers"
they must be communicating to figure that out
This is great until the numbers start getting really large and we have to take into account Alice and Bob's mortality as human beings.
They are hypothetical optimal players, not human beings.
@@colgatelampinen2501 We need to rid this world of these hypothetical optimal players that disguise themselves as regular people.
I figured it had something to do with waiting however many rings of the clock as the number you have, but I couldn’t figure out the logic
This puzzle doesn't work for numbers greater than 3.
Let's suppose Bob has 4 and Maria 5. After one minute without answer, they'll both know, indeed, that the other one doesn't have a 1.
Problem is: they already knew that at the beginning of the game!
Bob has a 4, so he knows that Maria has either 3 or 5. He also knows that if Maria had to guess his number, she would choose either 2, 4, or 6 (if she had 3, she'd pick either 2 or 4, and if she had 5, she'd pick either 4 or 6)
Same goes for Maria, she knows Bob has either a 4 or a 6, and that if he had to guess Maria's number, he could say either 3, 5, or 7.
So, to recap: Bob and Maria both know, that neither of them has a 1, and they also know, that the other one is aware of that fact.
Question: why would they wait one minute to confirm that fact, since they both already know it? They both know that they don't have a one, as they know that the other one knows that to.
So waiting a turn to eliminate the possibility of one of them having a 1 would be meaningless, ergo it would have to be strategicaly decided, wich is against the rules.
Red King because waiting is the only way they will win the prize for sure, but by guessing they will have 50% chance.
Rafou, Of course it's strategically decided, but it's strategically decided on an individual level, which is not against the rules. Otherwise, there would be no way to play, other than blurting out random guesses at a random time.
The rules state that Alice and Bob cannot communicate with eachother, which means that *during the game* , they can not send eachother hints by eyewinks, hand signals or stuff like that to tell the other what number they have; and that they are not allowed to plan a strategy, meaning that *before the game starts* they are not allowed to sit together and consult with eachother on what mutual strategy they should follow.
Bob has 4 and Maria has 5, "thinking perfectly".
Bob thinks that maria have 3 or 5, if maria had 3, on the 3nd round she'd say that bob had 4, because if bob had 2, on the second round he'd say maria has 3, so maria cannot have 3.
Hence maria has 5 and he says it on the 4th or 5th round
solution, alice and bob have perfect reasoning, we dont need to figure out how they play it, they know already
Well, if either was given the number 1, they'd immediately guess and win. If not, a minute goes by and they both know 1 is not either's number. Next minute, 2 or not, etc.
So basically you wait till your number of minutes, and then guess. This is a standard inductive reasoning problem. I've heard it a bunch of ways, but the answer is always the same.
Derek Gooding its wrong. Look up the unexpected hanging paradox to know why. Its basically the same idea.
It's similar, but it isn't the same.
Way to watch the video and repeat exactly what he said.... You're a real scholar
Yep that is the point. Both players have to come to the same logic.
I was like, staying silent, what information it gives to the other, it tells that you cannot tell the other's number, so you don't have the number that would enable you to, so you don't have 1, that's how I figured it out, using this kind of reasoning is very effective for logic riddles
Your videos are really interesting, good brain exercise. Thank you and please keep bringing us such a good quality stuff.
While the forward induction solution is correct, it is inefficient. Simpler solution is to wait until n minutes pass where n is my number and then state that the other has n+1. Since the other has not spoken out before me, I must be the one with the lower number.
Not watching the rest of the thing, I can assume it goes by induction. If they're given 10-digit numbers, the game will go on longer than their lifetimes if they want to be sure to win, so they're better off taking a random guess and not devoting their life to it.
am i the only one who feels like you use this method pretty damn often, dude find something better please 😀
Lexically Ambiguityness hahaha
If I was to make a snap choice on who might be an asshole, I'd logic the person cursing is more likely to be.
hold your horses guys, i just think that tricks involving collective knowledge are too often
I don't mind the repetition too much as it means I can watch one video and get stumped but then on the next one go "aha, I know how to do this!"
John Jebunatovic that's because induction is fun
I got the same answer but with a different (and less perfect) method. I figured that, since they have no way to communicate, and since they are dealing with whole numbers, and since the clock striking constitues a string of whole numbers, they can use that to communicate. If they both do this, then the clock will strike N times, where N is the number which one of them has. Since the other person has not said anything by that point, then their number must be N+1.
I came at a similar solution, just assumed that they would plan beforehand. This was unexpected... 🙌🏽
Planning beforehand goes against the rules of the game. An easier way to think about it is that they don't know what the game will be before they are separated.
I got this one right! I don't know why you explained it so complicated though.
Just wait until the clock has rung amount of times, and then guess the number above yours
But what if the number is the one below yours?
@@stevenloube6784 In that scenario the other player correctly guessed your number one ring before you would have incorrectly guessed theirs. But it does assume both players indepently figure out to do this, since they can;t actually plan to both do it,
but people dont calculate simular
this only works if both know the strategy before
me and my partner, getting the numbers 18,391,806 and 18,391,807 thinking perfectly logically:
The problem with this is that they know the other's number is n-1 or n+1. The point behind this strategy is that you get new information with each ring of the bell (if your partner doesn't say anything on the kth ring, you know their number isnt k). However, you already know that their number is n-1 or n+1. So on the kth ring, if k is not n-1 and k is not n+1, you already know theyre not saying anything, even before the whole thing starts. There is actually no new information there. And logically speaking, why would this be a solution when you get no new information most of the time? Even though they both know they are following the same logic, they are ignoring the fact that they know for most of the initial rings that their partner will not say anything. With no new information, that means there is no logic; it is a predetermined strategy. Which makes this whole thing pointless, because if it is a predetermined strategy, then why not just choose some simpler strategy? If n=20, why not decide to start out at k=15 to save time? Its because they can't communicate beforehand to start at k=15. So that means they can't communicate beforehand to choose to ignore that they know nothing would be said for the first few bell rings
Firstly, separate whoever has the odd and an even numbers. Because the two numbers are consecutive, one will be odd and the other will be even. For our example we'll use Alice to have the odd number.
If Alice had 1, she could answer at minute #1, because she knows Bob can't have 0 (not allowed).
if they didn't, then if Bob had 2, he could answer at minute #2 because he knows she doesn't have 1.
If he doesn't do that, and if Alice had 3, she could answer at minute #3 because she knows he doesn't have 2.
etc etc.
The numbers are finite, so eventually one of them will reach their number. and they can answer and they both get a fuckton of money easily
But this is a strategy right? the person who gets the smaller number N always guesses after the clock rings N times.
But strategies aren't allowed
Can't have a strategy being *discuss* but this is just logic by themselves.
they obviously don't follow logic perfectly since the puzzle says they are consecutive numbers, so they should never even be thinking about the possibility of the other person guessing 1 on the first ring.
No, but there’s no logical reasoning why they’d start the count on any other number, so they just have to go through all the numbers.
I think there's not enough information here to solve this puzzle. Your rules say that the two cannot have a plan or strategy worked out before hand. You then state that there is a clock in the room that rings each minute but it is never stated that the clock is integral to the solution. Bob and Alice would have to assume that the clock has nothing to do with the solution unless they had planned to use that strategy from the beginning. I don't see how your logic works here.
The other assumption is that Alice and Bob have perfect logical skills, so they would both figure out that the method is the only way to solve the problem, so they will both act according to the strategy.
Logic gives the answer and we are to understand that both persons have 100 % knowledge of what Logic is.
Bill Tiemann
The clock is important because it is only when the clock rings that they can guess. You don't need a strategy in advance because every time the clock rings you GAIN information about the numbers because the other person stayed silent
Yeah but they may have stayed silent because they were still thinking. I think this isnt really a solution and that the question is actually insolvable
Study Gym they are perfectly smart, however, so we just ignore "thinking"
"Truly you have a dizzying intellect"
Presh: "just wait till I get going!"
Simple enough. If Alice was told 1, then Bob would have to have the number 2 so she would guess so after the first minute had elapsed. If Alice was told 2, she would know that Bob had either 1 or 3, but if Bob had a 1, he would guess that she has a 2 after the first minute. So Alice would wait, and if Bob didn't say anything, then she would know he has a 3 and says so after the second minute.
Thus, this pattern continues, and you could induce that "If Alice has the number n, then Bob has n+1 or n-1. If Bob had n-1, he would determine so after n-2 minutes of silence, and he would guess Alice's number n after the n-1'th minute. If he had n+1, there would be silence after the n-1'th minute, and so Alice would be able to guess his number.
Some cool side-effects of this strategy:
If Annie has number N, she would only guess Bob's number on the Nth minute.
Whoever has the lower number will always make the guess.
Since planning a strategy was not allowed..How would a person know if he/she has to wait N minutes if the number is N and guessing N+1 as others number. Seems conflicting.
They have perfect logical reasoning and both people know this. And because this is the only best strategy, Alice and Bob can assume they are both using that strategy.
I just can't seem to understand how the logic applicable to 1&2 can be used for larger numbers
ill try to explain it: Let's supose Alice got 37 and Bob 36. Alice knows that Bob's number may be either 36 or 38. Bob knows that Alice's number is either 35 or 37. Bob will wait until 36 minutes happen, while Alice will wait until 37 minutes. When the 35th ring happens and no one makes a guess, Bob will guess on the 36th ring that Alice's number is 37.
How? Simple, If Alice's number were 35, she would have known that Bob is 36 because when 34h ring happened and no one made a guess she would have instantly know that Bob's number wasn't 34, and guessed 36 on the 35th ring. But as she couldn't guessed it, that means that Alice's number was Bob's _N_ +1 (37)
To summarize: If Alice waits till the _N_ -1 ring, and Bob hasn't guessed, then she knows she is the lower number, so she will guess _N_ +1. Instead, if Bob waits till _N_ -1 rings and Alice couldn't guess, he will guess _N_ +1 on the _Nth_ ring.
The solution would involve some strategizing beforehand to work. Bob can easily contradict Alice. After the clock rings for the first time and neither players guess, Alice may think that Bob has a number greater than her own and answer on the second ring. Similarly, since Alice didn't guess on the first ring, Bob and assume that Alice has a number greater than his. Therefore, when the clock rings a second time, they can still get the answer wrong. They still have a 50% chance of winning (1 of the 2 players would have the correct answer).
Kevin Tong but that would mean they were both told 2!
Вениамин Феафанов Not exactly, we can say Bob had 4 and Alice had 3. After the clock rings for the first time, Alice assumes Bob has 4 (Which is correct), while Bob assumes Alice has 5 (Which is incorrect). Now, there's a 50% chance of Alice calling out her answer before Bob, therefore, the solution is flawed.
After the clock rings for the first time, Alice does not assume that Bob has 4, she concludes that Bob doesn't have 1. After the second ring she concludes that he doesn't have a 2, and on the third ring she can guess that he has a 4.
Kevin Tong if A = 1 and B = 2, A would know what B is immediately, and if B = 1 and A = 2, B would know what A is immediately. If A=2 and B=3, after the first minute, if nobody guessed, A would know that B cannot be 1. Therefore, A would know that B would be 3. B, being a perfect logician, would not guess on the second minute because B does not know what A is. They have a 100% chance of winning.
You can't have the same number. One of them will always get to their number first and call it out. They are perfect logicsticians. Which is a fancy way of saying you can only have a strategy that requires no planning, just obvious logic.
I considered the case where someone as 1 but I didn't reason up to the solution; I guess I couldn't decide if Alice or Bob could hear each other's guesses. You said they couldn't communicate.
Your answer is paradoxical. If Bob has the number 20, he knows there is a zero percent chance Alice will say her answer on the first minute, so her not saying anything on the first minute doesn't actually provide any information to him since he knew that would happen. Bob has to think there is some chance Alice will say something on the first minute, for her not saying something on the first minute, to give him information.
Also, what if their 2 numbers are a quintillion, and a quintillion and one? See you in 2 trillion years, Alice and Bob.
medexamtoolsdotcom You are wrong. Imagine this. Alice has 20, Bob has 21. 19 ticks pass an Bob says nothing. then, Alice knows Bob has 21 because if Bob had 19, then he would have known Alice had 20, since he knew Alice couldn't have 18. He would've knew Alice couldn't have 18 because if Alice had had 18, she wouldve known Bob had 19. She knew Bob had 19 because if Bob had 17, he would've known Alice had 18,...and so on. Eventually we get to where Bob would've had 2, and he would've known Alice couldn't have 1 because then she would've called it out. Do you get how the process staircases down until we get to 1, where we can draw a logical conclusion?
The point is the first 18 minutes are meaningless to either Alice or Bob because Alice has 20 so she knows Bob can't have 1-18 and Bob has 21 so he knows Alice can't have 1-19. There is no information to be gained by having the other person stay silent. Therefore counting off the timer is not a logical move, but a preplanned strategy which is not allowed.
The problem is not the logical process staircase, it's that the setup of the question is bad .
chinareds54 you said what I was trying to say here better than I did. It requires the two of them to have this code between them which amounts to a preplanned strategy.
It's a perfectly logical strategy for both contestants to arrive at independently. In fact it's the only way they can guarantee of getting the right answer.
It's not logical because the premise can never be "If Bob or Alice's number is 1" if Either Bob or Alice has a number 3 or higher.
I don’t get it. Why is how much time passes related to their number? Why is that a thing?
danielacti because of the number of times they are asked in total
No. That is irrelevant. If they can't plan a strategy in advance, then as long as they don't have 1 and 2, the amount of time that passes doesn't help.
If the smallest number is at least 3, this doesn't work, because this logic works only IF you suppose the smallest can be 1. This sequence of reasoning works only when you cas say for sure that first "if" is true or false.
If the smallest number is three then if one person gets a 3 they know the other has to have 4 and the logic continues from there.
absolutly!! Alessandro Svanascini
! that's what i was thinking....the logic that emplies on 1 or 2 is no good for let's say 20 and 21
Alessandro Svanascini if bob and alice both say to themselves that they will guess their number +1 after their number of minutes has past they will always win
This is not true.
Let's say person A has a 3 and B has a 4.
A knows that B has either a 2 or a 4. But if B had the 2 and A didnt guess on the first ring (which he didnt because A doesnt have a 1), B would guess on the second ring because of the logic described in the video. Therefore A knows, that B doesnt have a smaller number.
Positive whole numbers are +1 and higher.
Read the instructions.
You are assuming that the players will guess exactly on the clock ring afterword. Since there is no "price" for waiting, the guess could be made any tine after they "know". If you added to the rules that the 1 million prize was divided by the number of rings when answer was declared, or that the contestants were in a hurry, then maybe this reasoning would be valid.
i thought of the 'N rings' approach immediately, but it wasn't as clear in my mind as it is now after the explanation. thanks!
Actually, the answer assumes that they can only deduce things one minute at a time. In real life, it could take a lot less or a lot more time to figure out that reasoning. By the time the clock got to twenty, the reasoning would be so out of sync that someone would almost definitely guess incorrectly.
eh em the question clearly states suppose they have both perfect logical reasoning
Yea I thought the same. But the actual reason has less to do with that inductive logic rather than the fact that they NEED to connect their number with the timer--there is no other way to win this.
So given that they need to do this, the only logical/reasonable thing to do is to wait till your number comes up on the timer--there is just no other way to connect the alarm ringing to your number (that would also make sense to the other person).
Since that is the only logical thing to do, if your (Bob's) number is 20 and Alice has not spoken till then, you know she does not have 19. So you call out 21.
The logic detailed in the video is wrong: that makes it seem like their reasoning takes a minute--that makes no sense. The idea is that you only have to worry about N and N + 1 (given you have to tie your number to the timer and the timer starts from 1).
EDIT: I just realized your post was 4 years ago (!!). Wow...this must be a blast from the past for you. Hope you survived Covid, and are having a pleasant life. Wish you well in this transient life of ours.
Alice and Bob are logicians and both know one of them will win $1,000,000 because obviously when they're given the number N, they will have to wait until the clock rings N times. (Of course, it's obvious, everybody knows that!)
Bob opens his envelope. His number is 31,541,159. He decided he's not going to waste 60 year of his life counting the clock and goes on a trip with Alice instead.
This is a very good method, but doesn't that count as strategy that's planned before the game?
surprisingly, this is one of the ones that I figured out, it kind of reminded me of the space time riddle (ill explain briefly)
basically you and 30 other people are going to be teleported to different planets, and a machine with 2 levers (A and B) will be teleported to one of your planets right next to you. to make the machine teleport you have to switch a lever from on to off, or off to on (they don't affect anything just up or down) to get out of this hell, you have to figure out when the machine has been to every planet at least once you get a few minutes for pregame strategy
the solution to this:
one of you gets decided as the counter, and your going to be the one who decides when it reaches every planet
for everyone else, they follow these simple rules
1) if the machine visits you for the first time you flip A to be on otherwise you flip B either direction
2) if the machine already has A flipped on then you flip B, and don't count it as the machine visiting you for the first time)
for the counter, you add one to the count every time you turn the A switch off
but the problem is that if alice gets lets say 30, at the first mins she would not expect bob to say 2, cus bob cant have 1, because bob's number is 29 or 31. so logically, this question make sense in thoery, but never work in reality, this is like the prisoner paradox. where the prisoner was told "he will be executed any day of the week from monday to sunday, and the day of execution will be a surprise", so the prisoner rule out sunday, because if he is still alive on saturday, sunday will not be a surprise to him. but if he cant die on sunday, meaning saturday will not be a surprise (because he know he cant dies on sunday since is been ruled out), so he can not die on saturday. and if he cant die on saturday or sunday, then friday will not be a surprise. and this logic continues until he realize he will not be executed on any day of the week. but at the end, he was executed on a wednesday. and it was indeed a surprise to him.
I dispute the statement that 'zero is not a positive number'. My understanding is that zero is simultaneously both positive and negative, and therefore in the stated example if Alice has 1 then Bob could in fact have zero, as it can be considered positive. Of course if Bob did have zero then he could certainly say immediately that Alice must have 1, as she certainly couldn't have -1, so the logic does still hold, but it means that there is a risk that Alice and Bob could be one off from each other in their counts if they each hold a different position on whether or not zero is positive, which is a mathematical question, and not something that can be concluded solely from being a perfect logician.
plot twist, bob had no idea what was going on and just sat silently afraid to lose. Alice only won because of the luck of always having the lower number.
okay, so you just wait until the minute sounder rings the number of times as your number and then guess your team mate has one more than you
They use the clock to count if bob has 21 then he waits 21 minutes
Incidentally you can always win on the second go because the lower number won't see the larger number make a guess
Intuitively, I would have done it differently! If my partner would have used your method, we would have been screwed!
Also, because you said that both don't have the possibility to plan their method beforehand, I guess even your solution is not a real solution either.
So here is my naive solution: everyone waits for their number. If the clock reaches their number, the player knows, that the next number is the number of their partner, otherwise the partner would have called the first players number, already. It's exactly the same reasoning you provided, but it is one off. So without each other knowing the strategy the problem is obviously but solvable.
I admit that's better than my solution. My solution is both yell out a number 1 less than their own (or 1 greater than their own, doesn't matter as long as they're consistently doing the same thing). There is 100% chance that one of them will get the right answer (and logically 100% chance that the other person will get the wrong answer). If they both yell out at exactly the same time, say on the first ring, then...they may win? The rule doesn't say anything if both correct and wrong answer are given at the exact same time. Technically (co-)first answer given is correct, hmm?
I managed to figure out the rule of "zeroes are not a possible number, meaning if person A has 1, person B must have 2. Therefore by turn 1, Person A must answer 100% correctly."
I didn't quite make the logical inference yet that both of them could act on that information and switch the answering role each ringing of the clock.
I did, as a hypothesis outside of logical formulas, consider the possibility that the ONLY way to communicate in this excercise is by using the clock, and the easiest way to do so was through clock ringings. However, I reasoned that strictly using such a hypothesis was a risky move on either participant's side, and that they could assume that a single ringing might mean "N=1" but also "N=60" or "N=1/60 (of an hour)"
Using BOTH the idea that they CAN communicate through ringings AND that both of them know the both of them are perfectly logical, and that the rule I stated above in quotation marks MUST have been figured out by the end of ringing 1, we can infer the both of them know they HAVE to use the clock ringings as the definitive communicative tool, and use it in a manner that follows from inference. The only inference we can have are "if A answers , B must have N+1) So the ringing of the clock can not possibly have a meaning other than "Clock turn N= number N" rather than "turn N = 60 or 1/60", as those cannot have a logical inference from any information.
In short, all I needed to do was figure out what to do from turn 1 onwards, but I kept gettin sidetracked by the possibility they might have numbers ranging in the dozens or even hundreds. That this is irrelevant was something the video had to show me, and I kicked myself for that.
What does waiting for the rings mean, do they just use the timing as communication? Wouldn’t that be against the rules?
If the other person stays silent during the first ring of the bell. that conveys the information that the other person does not have a 1. during the second ring, the other person will know that they don't have a 2 because if they had a 2 they would guess 3 since the other person cannot have a 1 (if they follow perfect logic). and you can continue until they come to the number in which Alice understands that Bob does not have a 19. and will then guess correctly.
Step 1: Wait for the Clock to ring
Step 2: Say you have green eyes
Step 3: Leave the game show
It’s even simpler than this explanation. Obviously the clock is the only way for them to receive information, and the only way to use it to send meaningful info is to correlate the timing of the guess with the number. But they will not guess their own number, so if the clock gets to their number, they know the other player has a higher one.
I'm not sure if this is really logic - it's more of a trick. They both already know on day 1 that their colleague has at least 19 so their reasoning on the first 18 days is incorrect. There was never a chance of anyone saying yes on day 1 so no new information has really been gained. They eventually get the correct answer because they have effectively shared the rule "wait until the number of minutes passed is equal to your number and then guess your friend has 1 higher". If this is genuine logic it appear to imply that you can gain extra information about the world (your opponents number) by forgetting information you already have (the fact that your opponents number is 1 different from yours.)
_bangs on the wall that separates until it reaches the number_