Can you solve the rogue submarine riddle? - Alex Rosenthal
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- Опубліковано 22 вер 2021
- Practice more problem-solving at brilliant.org/TedEd
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Smuggling yourself aboard the rogue submarine was the easy part. Hacking into the nuclear missile launch override - a little harder. And you’ve got a problem: you don’t have the override code. You know you need the same numbers that were used to authorize the launch, but one wrong answer will lock you out. Can you figure out what numbers to enter to stop nuclear war? Alex Rosenthal shows how.
Lesson by Alex Rosenthal, directed by Igor Coric, Artrake Studio.
A very special thank you to Jon Schneider and Brian Chen, who wrote the original version of this puzzle for the 2021 MIT Mystery Hunt and were inestimably helpful in adapting it to this format. You can find that original puzzle - and many more like it - at: bit.ly/MITHunt
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Well, this is definitely one of our most difficult riddles! Did anyone manage to figure it out? Let us know how you did it ⬇️
These aren't riddles, they are math problems with a story.
@Ted-Ed You just mixed number theory with Cheryl's Birthday Puzzle lol
I havent even started the video yet puhleeze
Ozo.
Oh no riddle can be more difficult than that 3 god riddle
I was scuba diving one time in the Indian Ocean and I saw a submarine with minions and a nuclear weapon being launched. I watched this video and was able to stop it. Thanks Ted Ed
Impressive that you managed to watch the video, diffuse a torpedo that used the same logic as this video and then post this comment all in the space of 10 minutes. Kudos to you man!
This is why we do what we do 🙏
The humour is strong with this one!
@I daignose you with invalid Oh, that’s me.
@@TEDEd You guys are one of the most consistent highest quality channels out there. Like Melodysheep, Kurzgezagt and Veritasium.
That boss should have thought twice before giving the codes to minions who are “expert logicians”
Especially when they don’t have the authority to disobey the boss’s orders and not launch the missile
So what I'm hearing is that at least one of the minions has green eyes?
@@Entias and they know the meaning of Ozo and Ulu?
@@Entias And that the man in the green house stole the fish?
lmao
I think the real question is “Why would you secure 2 nuclear lock codes with a single digit?”
Cos it’s easier to remember
Cause who would be looking for single digit numbers when they're described as a code? It'd be so simple that it just becomes too simple and maybe thrown out. It's still a poor choice but there could be some logic to it.
Because it wouldn't matter either way. The system was programmed to lock when a wrong code was entered so brute forcing it wouldn't work.
Cuz the likelihood that someone randomly guess them on the first try is precisely 1 in 42
Edit:
1 in 30. I misheard the boss when he came back to the minions. And it's actually 1 in 60 if they don't have to be in a correct order.
You may be interested to know that for the longest time during the Cold War the password on US nuke silos was 000000
Normally once they explain how to solve the riddle, I'm able to follow along and go "Oh, that makes sense."
This one, I'm still just as lost as when I pressed play.
Same-
This is always for me
Tbh I knew more about before i started the video
Proof by contradiction. By going through the possible results in reverse and eliminating those that would cause a contradiction in the facts, you could find the answer.
I actually had something going, then I just wanted to know the solution so I watched the rest of the video amd for a while I was more confused.
So the boss didn’t want the minions to know each other’s code, but then nonchalantly tells them how to find it?
I commented something like that before I refresh and see this haha
and what kind of boss creates a Single-digit override key/password to his missile???
The boss was concerned that they'd launch without permission, not that they'd over-ride the launch. If that's the only nuke there's no issue.
Lol
That's after the launch is initiated. It doesn't matter anymore.
"Can you solve the rogue submarine riddle?"
I know I never can. I just enjoy the video.
This one is extra hard
I mean you could try though
Same
Me too
@@NabeelFarooquimy brain couldn't handle this riddle, and the explanation, :v
Hears the question: **confusion**
Hears the solution: **even more confusion**
1:36: "6. The minions are expert logicians."
Then why are they stuck as minions?
Minions have a chance to escape when the bomb goes off. Big Bads get taken down one way or another.
Because the boss is a master logician ehe.
Diplomas aren't worth much lately, they still have to start low
Cuz the boss is 10x their size and can kill them with a single finger if they get out of line. XD
99th like
Difficulty level: master
Ah yes, becouse all of the other riddles are just too easy
I don't know, the elemental crystals riddle was pretty straightforward
@@stomyn yes, table riddles are in the lowest level, as much as Troll Paradox was.
the pearl chests takes my potato brain 10 seconds only
This is definitely one of if not the toughest
Where does it say difficulty level
I got a little confused with the “two or more whole numbers” and went with two or more digits per code. Still had fun!
Same
me too
exactly
Yeah me too, however this way the riddle isn’t solvable
same here
I love that I've watched enough of these to recognize that the minions being able to figure out each other's answer is actually useful information. Still can't solve it, but I'm interested to see how it works in the solution.
YES SAME
I think recognizing that as useful information is kinda easy because it’s basically the only information given.
This whole riddle reminded me of that one harry potter bonus riddle
The real genius here is B, he understood what A's number was in no time. He should get a rise and become the BBEG of the sequel to this video...
Sad he's dead.
@@summertriangle4745 ...Plot twist! He's not, and wants his revenge...
@@destegiovi❗❗❗
But he got exploded by their own missiles. Shouldn't have became expert logicians.
"Can you solve this riddle?"
Of course not, but I'm going to watch this anyways
Same
Hahaha... Same.
Am I the weird one? I literally never finish the video until I solve it
@@flargarbason1740 Did you manage to get the answer on this one
🤣
The big boss is the true definition of *”When the villain talks too much about their plans”*
*Puts gun to head pointblank*
"What's the code? Don't make me ask twice."
*Nuclear unarmed, world saved.*
Not really a logician's way but yeah it still works 😁
@Evelyn 21 y.o - check my vidéó Hey, delete your channel instead.
When Ted-Ed adds the difficulty level to the riddles, I know I am up for the challenge to solve them.
Hi we have same pfp😃🤗
@@aryatejc8067 bruh🤣💀🙏
@@aryatejc8067 ua-cam.com/video/lg5WKsVnEA4/v-deo.html&6s
@@Alkalus noice😎👌
its there to scare us
thank you youtube for recommending this to me based on recent events
This is only the baby version of the riddle. The original one goes like this:
Two perfect logicians, S and P, are told that integers x and y have been chosen such that 1 < x < y and x+y < 100. S is given the value x+y and P is given the value xy. They then have the following conversation.
P: I cannot determine the two numbers.
S: I knew that.
P: Now I can determine them.
S: So can I.
Given that the above statements are true, what are the two numbers?
Is it 9 and 18
I believe the answer is 4 and 13
@@carpediem6841 there are no numbers that add up to 4 and multiply to 13
@@cooperplant8907 4 and 13 are x and y. S is given 17 and P is given 52
I consider this version easier because you know by default there are only two numbers.
Plot twist: the boss was not perfectly logical, did a miscalculation, and now you're locked out.
1:00 just your everyday conversation with your boss
"I don't know if two one-digit launch codes really make logical sense as security."
"Well, remember, B, we know that we're perfect logicians, but the boss might not be."
I mean... the thing is your logic is exactly why it's a perfect code;
Nobody would expect such weak security. Force overthinking.
I love how everyone in Ted-ed's puzzles are expert logicians, who can solve multi step processes within minutes mentally.
Step 1: Confirm you have Green Eyes
Step 2: ask the minions for the 2 pass codes
What is it with the green eyes by the way
@@Alexander-km2jo it's a meme from the Green eyed Kids riddle in this channel.
You'd have to watch that one to get it.
I was looking for this comment 😂😂😂
@@lancebird6675what, no it’s the green eyes riddle on this channel
edit: never mind, i read it wrong
I was confused by what it meant for an integer to have 2 elements, when it actually meant that the set had 2 elements. And later the explanation didn’t clearly distinguish between the reasoning would lead to B knowing, and the reasoning would lead to B not knowing; I thought it was a single line of reasoning, and then suddenly we’re left with 4 options, whoa!
Don't worry guys I solved this and now imma go save those people
*Boss after his integer explanation*
A:that’s oddly specific boss…
B:Yeah really specific…
Boss: at least my name parents didn’t name me A or B!
A+B: *SADNESS*
That’s good stuff :)
their parents named him c
I-i am D😢
LOL this is too literal with the scenario, literally the literal meaning of when "you take things too literally"
being brought back here after a certain something really makes me laugh
Ah yes, now I know not to give minions a one-digit code next time. I'll give them a 2 digit one :D
I'm blessed with a perfectly timed upload of ted ed
i love how it's incredibly complicated and takes A LOT of steps more than you think it would, when you were just playing with SUCH SMALL NUMBERS
I think I solved the riddle in about ten minutes. Wasn't timing it, so we'll never know if I could've saved the world.
Genius
@@mubasshir Thanks very much. I had more trouble interpreting the English and the parameters of the puzzle, than I did the math.
I basically just worked through the first couple of possibilities, same as the video. Can't be 1+2||1x2, since it'd be obvious to both of them, and 1+4||1x4 was the first number which had all the elements needed, so it only took the fourth step of trial and error.
@@vanivanov9571 oh you went that far? I was trying randomly. 2,3 and even 4 digits. Should've been more systematic
@@mubasshir I also started with 2,3 as my first example to try and understand the logic of it, interestingly. After that, I decided to start from the smallest end of things, figuring that all I needed was two possible combinations to make the statements true, and that one should appear quickly in the low end of numbers.
Honestly, even after I worked it out I wasn't sure if I got it, since I didn't properly comprehend their explanation at the start. That's my usual problem with these riddles.
I love how they removed the "Difficulty: Master" part of the title
Yeah I'm no master of these riddles but I was able to solve this one in my head in about 10 minutes without writing anything down. The explanation looks pretty complicated in the video, but if you understand the problem completely you quickly realize there aren't many options to work through and they're all very simple arithmetic.
I got lost a little.
The boss says he chose numbers less than 7, but did he specify that the sum and product must also be less than 7?
Yes exactly. Otherwise 3 and 5 can also work.
Yeah I'm still confused as well. Also, why call them elements and not digits?
I was wondering the same, but no, I think it is A’s logic that excludes possibilities above 1x6
@@nehamotwani6477 That's what i thought at first, but 3x5 = 15, same as 3x5x1, so B wouldn't know A's number. Since the number must have at least 2 factors, it must have exactly 2 to make sure 1 is accounted for. Or at least that's how i explained it to myself.
No, it is A's first sentence that excludes all but 1xP P Prime and 1xN^2
Boss: "I can't let my minions be able to launch the nukes on their own. I must ensure the security of my assets!"
Also boss: *makes the password to launch the nukes 2 one digit numbers."
Also also boss: **tells them how to find out each others numbers**
Hits different in 2023
Ted riddles are like: "so, a duck is falling from the sky into a fantasy lake that will start a chain reaction and set the world of fire, but if you manage to shout out the numeric sequence that god was thinking at the time of creation you can prevent it. Here are some incredibly random clues about said number, can you save the world?"
Its kbviously ektger 34 og 67
and i love it.
Teacher: The test isn't that hard
The test: 1:30
Ah yes 1 digit nuclear code
Thank you Ted-Ed for believing I ACTUALLY have the ability to solve this riddle.
I failed by the way, but I mean it was fun to watch.
I like how the laptop is RGB, the little details
Bet it's a Razor laptop.
There are difficulty levels now?!
I probably still can't solve baby difficulty riddles...
Where does it say the difficulty level??
@@mediocre_guy the title got changed.
Too bad people on the Titan submarine didn't know such override code
💀💀💀💀
These stories are always so outlandish and unrealistic :D It's obvious that someone finds or comes up with a riddle and then the storyline is forcibly shoehorned around it. But it's super funny to me. The riddles and the animations are always superb and the seriousness of the presentation makes the "story" very enjoable. And of course it makes it all easier to grasp too.
Oh hey, a new riddle!
Not that I can solve it on my own, I just like to see the story behind each one lol
Mr Ted, in future I would greatly appreciate if in puzzles like this the characters were named something more specific than "A" and "B", and maybe if they had contrasting character designs. It's easier to follow along with the explanation if I can tell who thinks what
I think this is the first time that I was both unable to solve the riddle, _and_ unable to grasp how the solution was deduced
I'm expecting more and more riddle like this from you, these riddles are so good....
I misunderstood the riddle. I thought the sums and products could be greater than 7, but the numbers comprising the sums/products couldn't be greater than 7. I might have made it harder on myself. Oh well, too late, I looked at the answer.
Same
Actually products and sum can indeed be greater than 7 it's not necessary for them to be less than 7 but here for sum or product to be greater than 6 contradicts statement of B that he must know the sum. But, I'm not sure about this though as few other people also have answer more than 7. Needs deep inspection.
@@ashutoshmahapatra537 18 and 360 satisfy the conditions
Sum and product could be greater than 7, but when A starts by saying B might know his number already we know it has to be 1xP, P Prime or 1xN^2.
@@outdoordannyd A says B could already know the answer with his number, but for 360=6x5x4x3x1=6x5x4x3 so B could not know his number.
Alright so I'm gonna be honest here chief these are usually fun little brain teasers but right now I've heard the thing explained to me about 4 times already and the only thing I'm feeling is my brain dripping out of my ears.
I always enjoy watching these, I get completely left behind as soon as they start talking about how to solve the puzzles. It’s fun being clueless but trying to follow along.
This was probably the most confusing riddle I've ever watched on this channel
This is probably the hardest solution I’ve heard from these riddles. Like I usually understand the solution when explained but I’m even more confused now after hearing it.
I'm pretty sure the wording is just not clear enough. When this many people are commenting about being confused you've gotta think that's a flaw of the video
Everybody's talking about how the boss spelled out the solution for finding out the launch codes for his minions, but can we talk about that both launch codes were single-digit numbers?
I think I've got another solution by thinking about missing numbers from the set of (1,2,3,4,5,6) instead.
- If the boss choose (3,4,5,6), then A has 18 and B has 360.
- A will not be sure if (1,2) or (3) are missing, while B will not be sure if (1,2) or (2) are missing.
- However if only (2) is missing, A would have had 19 and A would immediately know the boss set. (There is only one set that sum is 19.)
- After A stated that 'I don't know', B now knows that (1,2) are missing and A got 18.
- After A knows that B knows, A is now certain (1,2) are missing and B got 360.
- Because if (3) is missing instead, B would have got 240, which means B are not sure if (1,3) or (3) are missing, which means B are not sure if A has 17 or 18, but each number does not have unique set like 19; 17 could be (1,2,3,5,6) or (2,4,5,6) while 18 could be (1,2,4,5,6) or (3,4,5,6).
A gets 18 and knows that B has either 240 (1,2,4,5,6) or 360 (3,4,5,6). In both cases, B doesn't know A's number (you explained the reason already). This means A knows that B doesn't know their number. That contradicts A's Statement.
@@cheshire1 Actually no. If B has 360, he would know A would have either 18 or 19. If A got 19 he would immediately know B's number, because it is the only combination with that sum. So B concluded that A has 18.
@@kaienumino1811 No one said that A doesn't know B's number, so B can't use that to rule out 19.
The piece where the boss comes and tells the information needed by the protagonist reminds me of so many films ;)
Rule 1 REALLY needs to be reworded. I needed to see part of the explanation and then re-read it a few times to link it as a "set with at least 2 elements". Initially, I was trying to figure out how the numbers "had" 2 elements and what those "elements" even were. I was also confused as to what subject the "In other words" was rewording - the whole rule or the "elements" part.
Either keep only the "In other words" or change it to "chose at least 2 distinct positive integers".
True...
I confused the rules, so I arrived to the conclusion of "7" (5+2) and "10" (5x2) for "A" and "B" respectively, which actually would be true if the rules said that the 2 Codes are the Sum and the Multiplication of *only two numbers* (I missed the "at least" part lmao) different each other and lower than "7" for "A" and "B" respectively.
It bothers me because I know I'd have been able to solve it if I had all the rules clear (since I followed the same reasoning as him to solve it), but I now don't have any empirical way to prove it 😔
In my opinion, when B says "I know your number, and now know..." it implies that he knew A's number before A said anything. This makes sense if B has the product 4 or 5 but does not make sense if the product was 8.
If A had the sum 6 and B had product 8, he would be able to infer that A has 6 only after A has spoken and indicated that the sum must be 5 or 6. In this case, B would have said "I now know your number, and...". Therefore, A having the sum of 6 and B having the product 5 is also a valid solution as A will know that the product cannot be 8 based on the wording.
Is that tripped me up as well, If you change the order of the conversation to, I know your number, Oh I wasn't certain if you knew my number or not, You know my number now, then the outcome changes. Still I think the original answer is correct as is.
I did the same thing and got 1 and 5. I have no idea how anyone interpreted it the other way.
Actually, he doesn’t have to clarify whether he knew before or figured out from A’s statement. It’s on A to decide whether it was before or after and if he can’t decide(as in the 6 sum case) then it has to be the other one. B’s statement that A NOW knows indicates that what he just said makes clear without a doubt which one he has.
In my opinion there is no such implication. I agree that this wording would be weird in the case of A=6, B=8. But weird doesn't mean false. It is like "I know your number (and I'm not telling you whether I knew it earlier or not) and I know you know mine (and for this part of this sentence I admit I only have this knowledge after you spoke)"
I was thinking of the same thing as well. If B had a product of 5 he could deduce A's number is 6. While A doesn't know at the start if B's number is 5 (1 x 5 ), 8 (2 x 4), or 6 (1 x 2 x 3), after B says that he found A's number, A could deduce that only 5 could be B's number.
These riddles are so well made, keep up the good work!
I love these riddles TED-Ed. I wouldn't mind if there were lots more of them.
Just imagine how the 100th riddle is going to look like...
This editing is crazy good. Also I love the narration, and the riddle is very hard, not for casuals like me. If only I had green eyes...
Then we could ask to leave...
This did not age well
I love the boss just casually picks them up
It feels like forever ago since one of these, absolutely love them ❤
Bad time to reccomend this UA-cam, bad time
This riddle requires you to know what *"the elements of an integer"* means, which I do not know and this video did not explain.
It took me a while to figure this out, but it's the set that has elements, not the integers. It would have been more clearly stated as "The boss chose a set of at least two distinct positive integers, each less than 7."
I have a much simpler solution:
1. Confirm you have green eyes
2. Get into House Minotaur
3. Escape the 3 alien overlords
4. Confront the 2 minions & challenge them to a wizard duel
4.5. Use the worst wand available & miss
5. Once defeating the minions, ask them to stop the missile for you
This has to be one of the most convoluted riddles on this channel
this follows the same sort of logic as the bonus harry potter "how many times did you watch the titanic" riddle!
why don't they ever show the answers to the bonus riddles without you paying ACTUAL MONEY?
Question: Why would the boss even consider that random fact about the numbers to be worth sharing or considered as “funny story”?
This one was such a toughie, even the explanation confused me for a sec and I had to replay it multiple times.
I love these oddly specific riddles
This was tough, especially the part about decoding "I don't know whether you know my number"!
I’m so happy :))) This is the first Ted riddle that I managed to solve in my head without the use of scratch paper!
Wow! Amazing! You must be a mathemathics and game theory genius to be able to solve a puzzle like this and not the other simpler ones (e.g. bridge riddle). And you're solving it all in your head, without any scratch paper, something that even the most trained logicians couldn't do.
Honestly, the only good riddles on youtube
Step 1: Tell the minions that at least one of them has green eyes
Step 2: Ask the minions to ask the boss to leave
This one is actually hard, one of the few on this channel that I didn't managed to solve right away (but I did get very close). I have a feeling that the number must be very small and there is only 2 number in the set (otherwise there will be way too many cases to consider). Then, I realize that the set must have 1, because otherwise B wouldn't possibly know the answer even if it can be factored (i.e, if B's number = A x B, the set could be (1, A, B) or (A, B), which result it different sums). After this, I only tried found (1, 5), and it seems to be working.
After watching the solution, I know (1, 4) works, but I still don't understand why (1, 5) does not work. A's statement is quite self explanatory: when A knows he has 6, B could have 5 (1x5), in which he knows A's number (even before A say anything). B could also have 8 (2x4, 1x2x4) or 6 (2x3, 1x2x3), in which B cannot tell A's number. So A knows that depending on B's number, B could know the underlying set, which gives A's number, or B could not. So he said "I don't know if you know". B's first part also works if B has 5=1x5. However, B knows A will know the underlying set after he said "I know your numbers". This is because he knows A has 6, which can has product of any of 5, 6, 8, and B has 5 is the only possibility that B would know the set (and thus A's number). This means if we separate B's statement (i.e., suppose B said 2 sentences and A comprehend them sequentially), it actually works.
Exactly, I was looking for this comment
If the sum was 6, a product of 5 (1x5) or 8 (2x4) would also tell B that the sum was 6. The reason B can conclude the sum is 6 (when his product is 8) is because he knows 1x2x4 can't be a combination, since their sum is 7. If their sum is 7, A would know for sure that B wouldn't be able to figure out A's number, since all the 7 combinations (1+2+4, 5+2, 4+3 --->> 1x2x4=8, 5x2=10, 4x3=12) wouldn't allow B to narrow down and figure out A's number. Thus B knows that a sum of 7 would contradict A believing B could figure out his sum.
I followed a similar logical path and narrowed it down to (1,4) and (1,5) myself... thought about it for another 10 minutes and couldn't really get anywhere.
I was quite close, but tripped up by 1:20 "B thinks this over."
If B's product is 4, then the information A presents is actually irrelevant -- B immediately knows A's number, as 1x4 is the only possible way for this to be the product, and 5 must therefore be the sum. But B "thinking this over" got me into the headspace that A's information was NECESSARY for B to know A's number.
I came down to 5 & 4 or 6 & 8, but went with 6 & 8 because of this trip-up. Argh...
Great Ted ed. You uploaded it just right when I'm going to sleep. Tonight gonna be another sleepless night for me.
You should do more riddles!
Is it just me or their boss wanted to be a mathematician, but ended up in the criminal world?
You mean Moriarty?
5:55 wait... did i just kill minions A and B? rip
Congratulations! You solved the riddle! You monster.
Its very difficult puzzle ever i heard or seen..
One who came idea of this puzzle is truly genius..
Thanks ted ed for wonderfull lesson..
Me failing this riddle is the worst failure ever involving a submarine.
this aged like fine cheese
It is worrying that 2 perfect logicians agreed that launching nukes to cause chaos was the correct thing to do
Obligatory.
Step 1: Confirm you have green eyes
Step 2: Ask the minions to leave
I love how the big boss was just rambling x2 speed about math-related stuff and I couldn’t understand a single thing, but minion B was like: “Oh now I know your number and you know mine too!”
The rules for this one did not mention that the numbers the boss provided were under 7, they only mentioned that the elements of the numbers were under 7, not the actual product or sum. This led me to see that the only way they could each know which number they had was 21 and 720. Because in the scenario described you can have at least 2 elements. So the only way to be certain is to have all of the elements under 7 non repeating. Something like 123456.
3 (1+2) and 2 (1x2) would also work for this.
It is exactly my point. Initial information aren't complete
I solved this using excel (to be quicker).
Listed all the possible combinations of numbers 1to 6 without repeating digits (there's 57 combination), then calculated the sum and product of the digits. Looked for the non duplicate values in the product column that had a duplicate value on the sum column, and voila the sum is 5, the product is 4 and the seed digits are 1 and 4.
I think you got lucky. The sum also needs to be connected to a duplicate product (else A would know that B knows their number) and it can't be connected to a second non-duplicate product (else A wouldn't know which of them was B's number after B's statement).
I’m kinda proud of myself for nearly solving this problem. I used a different approach but narrowed it down to 6 answers. And then I chose 1 x 6 like a dunderhead. It was 5 in the morning and I was doing it in my head, and I was getting frustrated with the riddle. I had 1 x 5 and 1 x 4 on the list but I just ignored them for some reason.
someone: how did you stop a nuclear missile launch?
me: I watch TED-ed.
When you don’t understand the problem. Watched it. And had yet to understand it anyway.
Step 1 : Confirm you have green eyes
Step 2 : Ask the green minions to give you the code
I'm Sorry
This isn't funny anymore.
I see the MIT Mystery Hunt is now responsible for one of the hardest TED-ED riddles.
I didn’t absorb a single word of that, but I enjoyed every second.
i didn't read "difficulty level: master" and i started the video so confident in myself... now i have headache
Where did you read the difficulty level??
To be fair, if all of the minions are expert logicians and normally a boss is smarter than its goons, i don´t see them dominating the world as a bad thing.
Yes but they were trying to nuke it... Also, it sure does look like gru's brother (13 feet)
It's very simple:
1. Confirm that all minions have green eyes
2. Unplug the computer
3. Use the time you got with this tactic to guide the missiles with a gps spoofer into the Nevada desert
This is probably one of TedEd's hardest puzzles, and yet I figured it out in about a single minute (and lying in bed before sleep), yet I had failed to figure out dozens of much easier puzzles before this. Funny how our brain works! The clue for me was that the boss's two original numbers were distinct, so 1 and 4 are good numbers because their product (4) can only be made in a single way (since 2 and 2 are duplicates). 1 and 5 would add up to 6 which could, however, also be a factor of 2 and 3 so the other minion wouldn't have been able to narrow it down like he did. So the originals must have been 1 and 4.