Child's homework puzzle is baffling adults
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- Опубліковано 12 вер 2024
- Can you solve this number pyramid puzzle?
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To be fair, most adults are baffled by splitting a restaurant bill.
Lmao 😂
Women.. 🍵
Haha so true
2☆ = 40 - 9 - 7
This is easier than splitting most restaurant bills! 😂
@@mickdavies5647 yeah but coming up with the idea of the 2⭐is the hard part for these adults.
The way I did this was notice that the way the pyramid was built, if we call the 3 numbers on the bottom x, y, and z, the number shown at the top is (x + 2y + z). I could then use that to solve for star immediately. Once we know that, we can solve for heart by adding the 9, then add the two together. I got 33.
This was my method. I realised that the sum of 40 was obtained by adding the bottom left/right values once, but the middle number is added twice as you go up the triangle. Once you’ve worked out the bottom number the rest is trivial.
This was my approach as well.
Same instinct here
Me too
Same here :)
*Alternative approach*
The whole pyramid can be written simply as:
40 = (7 + ★) + (★ + 9)
And then you take it from there.
That's how I saw (and solved) it.
33
I took the approach the LHS is 2 less than the RHS : the heart is 21. Given the heart is star + 9, the star is 12. Heart + star is 33.
why are you making this so difficult? It is far easier just to write 40 = (7 + star) + heart. You can go straight to that one, simple equation.
@@dflayneBecause you can't define the value of ❤ without knowing the value ⭐ since (❤=⭐+9). Simply writing 40 = (7+⭐)+❤ is the same, but a step back. Just replace ❤ with ⭐+9.
I get that the question only asks what is ⭐+❤, but finding out each value seems to be realiable.
My first instinct was "40 = blank box + heart" and "blank box = 7 + star" so ... "40 = 7 + star + heart" and then "33 = star + heart" which is the answer. Apparently this was the "clever" answer.
Yeah I did the same thing
Everyone did it that way bro...nothing clever about that solution🤣🤣
@@colors6692 right. The video waited until the end to cover the obvious path. I'm not sure if there's research suggesting that the other paths are more likely to be found in grade school math homework or something else. I didn't feel it was clever - often the videos actually do show at least one path I didn't think of, this one felt like it was stretching to fill time.
@@colors6692yeah he put clever in quotation to show he doesnt think thay but thats what they said
@CharlesBallowe How about "40 = blank box + ♡"
Therefore ♡ = 40 and ☆ = 31.
Isn't this logical too? There's no reason to assume that the blank box isn't really blank.
At a coffee shop my bill was $5.25.
I gave the barista a $10 bill and a quarter.
It was like watching the Hindenburg 2.0
Uhh they don't want the coins either 😂 maybe they were hoping to give you a bunch of change that other people keep giving them 😂
That happens to me all the time. The cashier looks at you and usually replies with something along the line of that’s too much money. Bill comes to 16.95 and you give them a 20 and a two dollar coin (I’m in Canada, we don’t have dollar bills, we have one and two dollar coins), instant deer in headlights look. Just give me back a 5 dollar bill and a nickel. I don’t think they actually know how to use the cash register, they just hit the Cash key like giving exact change and they can’t do the math in their head. Just enter $22.00 before you hit the Cash key and the machine will actually show you how much to give back.
the barista didn't know how much the tip was and his/her brain shortcircuited
I gave a teenage worker at an amusement park $20.75 for a $5.75 Icee. Me and the whole line stared at her doing mental math. I got my Icee and $25 back in change.
@@joepiazza3756 Sorry, I've got to say it: that's not okay. Maybe help someone instead of taking advantage of them.
My logic was that 7 and 9 are two apart, and both 7 and 9 are being added to the same number to create the middle numbers and therefore the middle numbers must also be two apart, and the only way to do that to get 40 is 19+21 so the middle row is 19|21 and the rest of the puzzle solves itself from there.
Same, but I am way worse at explaining my thought process.
Ywp this is simple way, no algebra required
True. Only sum, and subtraction are needed.
That’s how I did it.
This is also how I achieved 33.
I just used the fact that blank = 7 + star, to get to 7 + star + heart = 40.
Meaning that star + heart = 33
I didn't bother working out the individual values at first but when I did I just used the 7 and the 9 to know that the heart and blank have a range of 2, meaning that they are 21 and 19 respectively. Then it's pretty simple to work out that the star is 12
@TravisGn02 Even easier, you don't need to know the 9 at all, it is pointless in this problem because you only need to look at the others.
The kind of problem that leaves you wondering if there's anything more to it?
@@maxhagenauer24 The 9 isn't completely pointless...just mostly. ;) While 7 + star + heart = 40 gives you star + heart = 33, without that 9 there you'd never be able to determine the individual values of star and heart.
@gswcooper7162 Yes but the question was the sum of star and heart, you could find it by finding each individual values, in which case you would need the 9, but you don't have to. That's what I mean by pointless, it's not needed.
@@maxhagenauer24 Me working out the individual values was just an extra bit that I did for fun, I thought that was fairly obvious to be honest. If you don't care about the individual values then just don't read my final paragraph lol
I used a different method: I noticed immediately that the triangle is Pascal’s Triangle-so the top sum is equal to the bottom left plus the bottom right plus double the bottom center. That immediately solved the star to be 12. The heart is then 12+9.
The Pascal’s Triangle method works even if additional rows were added to the bottom of the triangle.
Simply 40-7-9= 2 x star. So star= 12 . Rest follows.
This was also my first idea...
yeah, but what is 12+9?
not so easy anymore, hm?
@@spitalhelles338021, and 7+12 is 19 and 19 + 21 = 40
@@spitalhelles3380holy that’s hard
@@spitalhelles3380 129, ez!
Easiest MYD problem by far
don't take this w from me
I agree, got it straight away 😮
True, I did it in my mind easily
Good for us or more particularly, me! I’d like such an easy MYD problem more often than… hm, how many years has it been?😅
@andrewhooper7603 It's all good. We have to take the W's where we can get them.😂😂
this one is quite easy ngl
It's for 9 year olds
@@NabeelFarooqui lmao
true, solved it in around 30 secs
Solved it in my head before clicking on the video
@@wasikancb same
Another way to solve for 9 year olds who don't yet know algebra is to notice that in the middle of the pyramid the number on the left will always be 2 less than the number on the right since 9 is 2 greater than 7. Then you need to find two numbers which have a difference of 2 that add up to 40 which leads to 19 and 21. From these numbers you subtract 7 from the left and get 12 or 9 from the right and get 12. So star is 12 and heart is 21 and the sum is 33.
That's exactly my solution.
As others also noted, but I like to phrase it differently:
All values come from the bottom and simply trickle up. However the middle value trickles up twice!
So 40 is the sum of the bottom values, with the middle value added twice. This quickly gives you a middle value of 12.
The fact that you don’t need to know star or heart individually actually makes this really easy. Blank (x) is 7 plus star, which then add to heart to make 40. So heart and star just have to make up the leftover after 7.
I figured it out by realizing the bottom row was essentially: 7+star+star+9=40. And that means 40-(7+9) = 2x star.
I think they're pretending to be baffled just to get views on their posts.
My immediate thought was this is some simple simultaneous equations, which is not to say I would expect that of a 9 year old, but the baffled parents? I think so.
There is some ax^c+by^d+e where x increments on the horizontal axis and y increases on the vertical plane, but it isn’t clean cut in any fashion, and very unlikely would the solution be an integer if you were to find a,b,c,d,&e in this equation that works.
In about 7 seconds from the thumbnail. I'm trying to figure out what was supposed to be so baffling about this.
Possibly if the pyramid rules weren't explained in the problem?
@@michaelsorensen7567The "rules" are obvious by examining the completed triangle though..
The puzzle is solvable with basic algebra, but the parent might have assumed the teacher wouldn't possibly throw an algebra problem at a third-grader, and was looking for something even simpler.
Easy enough to do so in your head. Just a small system of equations
Adult: I never used algebra in real life!
Kids: it shows.
if you call the three base numbers a, b, c
then the next layer is a + b and b+c
so the top layer is a+2b+c
The sum require (heart plus star)
is b plus b+c
giving 2b+c
so top number (a+2b+c) minus bottom left a
always equals 2b+c
If you were really stuck, you could start with star = 1, which is too low, and increase star until star = 12 solves the triangle. What I've noticed when helping people with maths is how reluctant students are just to try things to see what happens. Also, doing this basic process reveals how the triangle works - every time you add 1 to star, you add 2 to the apex. Then, you can work out for yourself that 40 = (7+star) + (star + 9) - even if you didn't see that immediately.
this video did not need to be 8 minutes long
I often wish we could speed videos up more than 2x.
This guy wasted time doing the “throw random numbers to see what works” method.
Lot of problems are like that. You don't need to do whole math or solve all the variables in order to find their sum or value that we are looking for. We need to investigate and trust other methods as well. It's also called talent.
(7+✡)+❤ = 40
✡+❤ = 40 - 7 = 33
exactly. The 9 is completely irrelevant to the problem.
@@otakurocklee
7+✡+9+✡=40
✡=12
9+✡=21=❤
❤+✡=33
Wrong ⭐
@@AndrewBlucher Is this getting political. If it's going to be a maths war, I'm backing the ✡.
@@craigsaunders7037 Math is not political, it's factual.
Once you realize that each box is the sum of the two boxes below it it's not too tough, but expecting a 9-year old to solve it is kinda nuts.
37 secs ago ??
This is literally the earliest i have ever been on any video
Edit : I used the 1st way but in the reverse manner so that I get 21 and 19 faster ... I started with 25, 15 and then found out that ♡ = 21 and ☆ = 12 ... thus 12 + 21 = 33
Same
I once saw a video when it said 2 seconds ago
Simple...
7+* is the free shape, *+9=❤ so we can switch them out
7+*+*+9 = 7+9+2* and this is 40
So 16+2*=40, 2*=24 so *=12
And to prove:
7+12=19 for empty, 12+9=21 for ❤
21+19=40
And sum that up you get *+❤=12+21=33
star is just (40-7-9)/2 = 12
heart is then 9+12=21
so 33
Someone see my solution:
What i did was kept the blank space and heart as 20 and 20 respectively. Now ,star is common in both 7+☆=blank and 9+☆=♡. Therefore the difference between the blank and heart should be 2 as 9-7=2. So i subtracted 1 from 20 and adde 1 to 20 to get the values of ♡=21 and blank=19. Then i got ☆=12. Therefore,☆+♡=21+12=33.
Simple!!!!!!!!!
That was pretty simple honestly, 40-(7+9)=24, so we get 7+12+12+9 = 19+21 = 40
So we get 21+12 = *33*
That's how I did it too. Nice to be able to do one of these in my head.
I tried this before watching.
My thought process:
Left triangle gives rules for calculation:
- bottom right + bottom middle = middle right
- bottom left + bottom middle = middle left
- middle left + middle right = top
Using that:
- 40 = blank + ❤
- blank = 7 + ⭐️
- ❤ = ⭐️ + 9
Therefore 40 = 2⭐️ + 7 + 9
2⭐️ = 40 - (9 + 7)
2⭐️ = 24
⭐️ = 12
If ⭐️ = 12, ❤ = 12 + 9
❤ = 21
Then ❤ + ⭐️ = 33
(Also Blank = 19)
40=blank+heart
blank=7+star
Therefore:
7+star+heart=40
Subtract 7 from both sides
Star + heart = 33
Easy 😂
To be fair, that REALLY is a difficult problem for a 9yo. It requires figuring out a way to solve a problem by substitution.
Star = 12
Heart = 21
Edit: Sum = 33
I also had a gut feeling idea, so I tested it with the first pyramid. I added all the bottom numbers, 2+4+3, getting 9. 9 is 4 less than the top number, 13. The middle number on the bottom row was that difference. Then I applied the same logic to the unknown pyramid. 7+star+9=40-star, meaning 2star=24, making star 12. Then I do the simple addition of 12 and 9 to find heart, 21. Add star, 12, and heart, 21, and I get my solituon of 33. It was a fun challenge and I was glad it worked. Very fun to see the other ways to come at this problem though.
Huh? Why such a long video for such a simple problem? 7 + star + heart = 40. star + heart = 33. The 9 plays no role.
I like your solution, but. Clean.
That’s, not how algebra works
@@engi_games What? He’s doing a UA-cam comment, not algebra. Understanding basic algebra is only one requirement for understanding the comment.
@@engi_games The blank space is clearly (7 + star). Therefore ((7 + star) + heart) = 40. Subtract 7 from both sides, star + heart = 33. How is that not how algebra works?
Just saying star + heart = 33 is not the final solution. You still need the 9 to solve for them individually.
Another approach could be that the numbers on the middle row are going to be always with a difference of 2. Then we can ask what two numbers, with a difference of 2 between them, add up to 40? Divide 40 by two, and add and subtract 1 to each side. The answer gives you "heart" and "blank". Having those, you can easily get "star".
But I liked your third approach more.
Way too long for a trivial problem
Maybe because MYD was exploring other solutions?
@@stigmontgomery7901 oh I always appreciate the “here’s three ways” as sometimes seeing a different solution approach helps for a completely different problem, or just understanding how people can see the world differently. But as commented elsewhere the top number is left + 2*middle + right of the bottom row. In this case I’m not sure other approaches added much insight.
There is no need to mess with variable substitution. The simplest way to solve this is to follow the rule given in the instructions and write "7+★" in the blank area; this immediately gives 40 = 7 + ★ + ♥.
Having a blank cell is misleading. We assume that there is a number there, but we shouldn't be forced to assume anything. The question needed to put another symbol there just to indicate that it is not blank.
Well the question is meaningless and unanswerable without a number there.
I don't see why we're not allowed to assume that the question is capable of being answered.
The rules were explained - each number in the top and middle rows is the sum of the two numbers below it.
Therefore for the box to be blank (i.e. have a value of zero) the bottom middle box (⭐️) would have to contain -7 (negative 7).
Which is fine, since negative numbers have not been excluded by our rules.
However, you must then follow that through, and applying those rules, 🤍 would have to be 2.
If "blank" and 2 are added, we naturally get 2, not 40, so "blank = 0" is not the solution.
In other words, there is no "assumption" being made here and nothing is misleading. The rules are clear and if you hypothesise that the blank box really is blank, and test it, it fails the test.
That's entirely consistent with the methods in the video, and other methods in the comments.
Maybe the blank space could be (sum=15)+8 so the pyramids can have the same rules with the first row 1+.
Star can be 8 and heart is 17. When the space is truly blank, anything is allowed, which is my thought.
I noticed that pascal's triangle gives the coefficient of how many times a given block of the pyramid will contribute to the sum at the top, so 40 = 7 + 9 + 2(star)
I didn't watch the video but i did it like this.
2 + 3 + (4*2) = 13
By the same logic.
7 + 9 + (⭐*2)=40
⭐= 12
❤= 21 [12+9]
Used the same logic 🙂
I think you meant 3 instead of that 5 in the first equation
@@clemente3966 yes, thank you
spent the first half of the year learning dynamics and it really helped solve this
I just brute forced it lol
I think I solved this is a pretty simple way: since we know that the middle number is the sum of the 2 numbers below it, I noticed that the top number would be the sum of of the outer numbers on the bottom row + 2 times the middle number. In reverse you just subtract the 2 outer numbers from the top (40-7-9) to get 24 which is 2*12. Thus the bottom middle number is 12 and can be used to calculate the remaining values
71 is the answer. Every any other answer would mean that you either changed the rule or just assumed that a box of nothing could mean whatever.
why tho?
@@milanstevic8424 Because the mid-left box holds no value. So you can’t just arbitrarily put things in it. Whereas the other boxes (and top triangle) have values. Three contain known numbers. And two hold a symbols as place holders of a numbers yet unknown, which we know because of the example, the rule, and the question.
@@eds1942 That makes no sense. The first pyramid is intended as the clear set up for the actual problem. It shows exactly how the problem should be interpreted, and there are no special rules about the empty box, apart from the classical assumption that all empty boxes have to be filled, just like crosswords or sudoku puzzles. In fact all three boxes likely would've been empty, it's just that the question *labels* the right and bottom ones with symbols so that it can ask for their sum. How else would you know which boxes to sum.
Your mistake lies in the assumption that these symbolic labels are always exclusive with the values (as symbolic substitutes), without any evidence to back this up. Whereas a triangle, cross, and seashell (i.e.) could've been labels to valued boxes as well, if only those were needed. Thus, if the question asked for cross + star + heart (with the cross printed in the top box), the solution would have been 73.
In short, these symbols do not stand in place of values (like variables), they're just labels to easily identify the specific boxes.
Let's assume that the mid left box really does contain no value. You still have 2+4+7=13 in the left hand triangle and 7+star+heart=40 in the equivalent boxes in the right hand triangle.
You don't need to even consider the mid left box to get star+heart=33 here.
@@gavindeane3670 True. In this case it doesn't really change the outcome. But there is no reason to assume it should remain blank. Usually these kinds of problems are accompanied with a prompt to "complete the diagram". Regarding the symbols these could've been short line indicators pointing at the boxes, with the symbols printed on the pyramid's outside, clearly denoting they're meant as labels. All of this falls under the categories of notation or visual presentation of the problem, not the problem itself. Imagine if every single crossword puzzle had a full explanation of crossword rules so that the solver won't be mislead into thinking that the blank boxes are supposed to be left blank. Also we should make sure people understand what is meant by "box", not to mention that digits and the meaning of numbers could also be a problem if left undefined... What if I consider the values as written in a hexadecimal base? So many possibilities and quite ridiculous.
To be honest it is absolutely obvious that the middle base entry enters the total sum twice so this is something you solve in less than a minute as 40-16=24.
I'm baffled by where the confusion was.
I did this in 10-15 seconds without watching the video (I stopped at 0:35). I thought it would be harder. But then I saw some fantastic comments overcomplicating this or confidently coming to a solution that is not 33. Now I'm baffled.
@@milanstevic8424I was around 12 when I learned system of equations, and on top of that I was ahead of my peers. This question definitely isn’t for 9 year olds.
@@bobross7473 Yes I can agree with that. Maybe for naturally gifted or trained 8-9 year olds, but not in the general case. There is a lot of knowledge I'm taking for granted, I'm in my 40s.
@@milanstevic8424 I did it about that quickly just from the thumbnail. I'm curious whether the rule of each number being the sum of the two numbers below it was actually given as part of the problem, because it seems trivial to work that out from the first given triangle.
@@Jivvi I like to imagine this being published as a newspapers puzzle, like sudoku, but I'm probably wrong and it's surrounded by all kinds of unfun equations that follow the same kind of semi=textual format, in some school almanac.
My dad can't even figure out a 15% tip
The problem with this problem is the blank space. There should have been a variable assigned to said space. To not do so is poor communication. You had to assign it "x" inorder for algebraic method to work.
The way I did it was the 2 numbers that add to 40 must have a 2 digit difference (because of 7 and 9) therefor must be 19 and 21. The rest is pretty straight forward
Start at 20+20 in the middle row and work in opposite directions along the number line.
It's in pattern recognition and algibra. 13=6+7, 6=2+4, 7=4+3, now apply the same pattern to the other set 40=_+hrt, _=7+str, hrt=str+9. First we need str, 40=(7+str)+(str+9), combine like terms 40=2str+16, subrtact 16 from both sides, 40-16=2str+16-16, 24=2str, devide both sides by 2 24/2=2str/2, 12=str. Next we need hrt, hrt=str+9, hrt=(12)+9 hrt=21. Now we need to test _=7+12, _=19, 40=19+21, 40=40. So str(12)+hrt(21), 12+21=33
In the first triangle, a number is equal to the two adjacent numbers in the row below, or 13=6+7, 6=2+4, and 7=4+3. With substitutions from the bottom row, you can also get 13=(2+4)+7 or 13=6+(4+3).
Now, make analogous equations for the second triangle, disregarding any that would include the blank position. This results in 40=7+☆+♡ and ♡=☆+9.
Next, substitute ♡ and solve for ☆. This gives 40=7+☆+(☆+9), which reduces to 40-(7+9)=2☆, or ☆=12. Finally, ♡=12+9=21.
I figured it was a pascal's triangle of a sort, this made it incredibly easy. Once you figure that out. Simply subtract 7 from 40, and you get the answer.
Since the top number is 7+2star+9=40
And its asking for 2star+9, simply subtract 7 from both sides to answer the question.
Fascinating how the sum heart + star = 33 is not depending on 9. They can put other values different from 9, heart and star will have different values but their sum will remain equal to 33
Isn't it obvious?
All that matters is the value of star, u see, if u change the value from 9 to any other value, the value of star would change accordingly, i.e, star is dependent on 9 itself,
For example, if it was 10, then the heart must be, star + 10... And, the blank area should be star + 7, but we need to find 2 x star + 10
Or in a general solution where the heart is h, star is s, and the value 9 as a variable can be termed as x and blank area be b
We get our h = s+x, and b = s+7
We need to find s+h, or, 2s + x, but we also know 40 = 2s+x+7
And hence 2s+x = 33
No matter whats the value of x is, it will always be 33
I saw the thumbnail and thought, "This will be fun! I may have to use a system of equations to solve this!"
I always start these by writing the problem down (in this case, copying both triangles and their respective values). I quickly realized that I could just derive the value at the top of the triangle with the values at the bottom.
I then wrote my quick proof just to be sure, and it did work out that way. (2+4) + (4+3) = 13 ... Simplified to 2+2(4)+3 = 13... so then I simply wrote out 2*+16 = 40, and it was all downhill from there.
An easier way to think about it is that the top number is always the sum of the bottom row, counting the middle number twice. So 7 + 2x(star) + 9 = 40 so (star) = 12.
My first reaction was to add the two lowest numbers together, then subtract them from the highest number. The number that is generated by this equation would need to be divided by two (because it is part of both numbers in the second row), then substituted for the star. I'm not particularly good at math, but it seemed like a more straightforward approach than guess-and-check.
My approach was to see that 13-(2+3)=8. Since we add the number in the middle of the bottom row to both corners individually, it means we get this number twice. That leaves us to divide 8 by a factor of 2 to get 4. This was the example pyramid.
Now we apply the same trick to our challenge pyramid. 40-(7+9)=24. We now know to divide by 2, so 24/2=12. So star=12. Add 9+12=21. And finally add 21+12=33.
It took me about 5 seconds to come up with this trick.
Another pattern to notice is that the triangle's numbers are approximately 3 times the size as the other one.
Taking 7 and multiplying by 3 gets 21
7*3=21
Then
21-9=12
21+12=33
☆ + ♡ = 33
I made an equation for the star, the heart, and the blank space, which I called A, B, and C. The heart equation I put in terms of A, plugged that into my equation for the blank space (C), and subbed that equation for C into my equation for what added up to 40, so B was my only variable. Solved for B, then plugged that back into the other equations to find A and C, then answered the question asked. C wasn't totally necessary (the blank space in the pyramid), but knowing it helps check your answers.
My initial thought was, since this was for kids, don't overthink it, and just put 8 in the middle position, calculate through, realize that's 8 off the top and added twice, so my guess was 4 off, fix guess and get results.
My solution was to first assume both blocks on the second floor were 20.
Then I took the average of 7 and 9, which is 8, and subtract it to both numbers, 7 and 9 → -1 and +1.
Apply them to the blocks on 2nd floor, as they become 19 and 21 (heart).
Subtract one of them with the already given number below, reveals the star to be 12.
Finally, add heart to it and the result is 33.
I approached it from a logical more than algebraic angle. Although the top number is the sum of the two below it, those numbers are built on the bottom row. Star is included in the sum for both middle row numbers so the top number must be equal to the lower left number plus the lower right number plus 2 star(since it is a component of both middle row numbers. Hence 7+9+2star=40. From there it is simple.
For simplicity of explanation, let's call the star x, the heart y, and the blank space z. So, z=7+x, y=9+x, and 40=y+z, which can also be written 40=(7+x)+(9+x), which simplifies to 40=16+2x, which we can solve to 24=2x and then to 12=x.
Now that we know x is 12, we can find that y is 21 (12+9) and z is 19 (7+12), and thus x+y (or star+heart) equals 33.
I don't even teach math, I'm an English Language teacher.
the two bottom numbers are two apart, so we can assume the two middle numbers will be two apart. The only option that adds up to 40 is 19 and 21. Star is 12, answer is 12 + 21 = 33.
My approach was that the top number=bottom side numbers + 2 times bottom middle number, so 40=7+9+2*, which can be easily solved and gives us *=12, the rest is straight forward.
you can solve for ⭐+❤️ without even knowing what the exact value of either is.
we know that ❤️ must be equal to ⭐+9 based on its position. that means that ⭐+❤️ is equal to ⭐+(⭐+9) or 2⭐+9.
since the top of pyramid is equal to the sum of the two boxes beneath it (we'll call them a+b) and those two boxes are equal to the sum of the two boxes beneath them (a=x+y; b=y+z), we can say that the top of the pyramid is equal to x+2y+z.
x=7, y=⭐, z=9
40=7+2⭐+9
40=7+[our answer]
40-7=33
I used the guess and check method but used Microsoft Excel to speed it up. I put all the possible whole number combinations in a spreadsheet starting with 40 and 0 at the top and 0 and 40 at the bottom. I then subtracted 7 from the left side and subtracted 9 from the right. I got the answer for "star" when both differences were the same, which was 12.
Labelling each box from left to right from top to bottom a to f inclusive, we get the following rule set
Rule 1 a=b+c Rule 2 b=d+e Rule 3 c=e+f.
Substituting rule 2 into rule 1 we get a=c+d+e then deduct d from both sides we get a-d=c+e. Therefore heart (c) + star (e) = a(40) - d(7) = 33.
7 and 9 are separated by 2. Therefore the resulting numbers in the middle row, being the sums of each number, added independently to star will be separated by 2. I divided 40 down the middle and removed a 1 from column 'a' and added it to column 'b'. I then subtracted the corresponding numbers from the bottom row to solve for star.
It was obvious to me looking at the first pyramid that each level was adding bits of the previous level. Then it was simply a matter of expressing those relations algebraically for the second and solving.
Blank box = n, then n = 7 + star => 40 = 7 + star + heart
solving for (star + heart) = 40 - 7 = 33 which give the
required answer. Completing the solution of Pyramid:
Pyramid: top ,, bottom
star + heart = 33
star = 12 and heart = 21
Not sure what a "formula" would be for this, but I am going with (7+X) + (9+Y) = 40. Then I solved by subtracting out the 7 and 9 to get X+Y = 24. I then took 24 and divided it evenly for each variable to get X = 12, and Y =12. Therefore Star = 12. This gives a bottom row of 7+ star + 9. Since Star is 12, Empty Space becomes 19, and heart becomes 12 + 9 = 21. Lastly 19+21 = 40.
33.
My reasoning : following the rules the 40 in the top row is the sum of the sides and twice the sum of the middle of the base.
Thus 7 + 9 + 2 stars = 40 star = 12
Besides, heart = star + 9
Thus star + heart = 2*12 + 9 = 33
This took me about 30 seconds to figure out how to solve. Write all the equations down and you have an easy system of equations. Star=x heart=y blank=z. z+y=40. Both z and y are just x added to a known number. Substitute and you get (x+7)+(x+9)=40, which is easy algebra.
The difference between the numbers in the middle row has to be 2 since the difference between the outer numbers in the lower row is 2. Therefore the numbers in the second row can only be 19 and 21 (heart). Star = 21 - 9 = 12. Star + heart = 33.
Each cell is the sum of the two below it. So 7 + x = y, 9 + x = z, y + z = 40. We only need to know x + z, so sub (1) into (3) for 7 + x + z = 40 and subtract 7 from both sides for x + z = 33. We don't even need to know what x or z are.
You know the "star box" will be counted twice, while the corner boxes will be counted once, so you have 2stars=40-7-9
2stars = 24 star = 12, therefore heart = 21.
More comprehensively, you can start with the top and name box next to heart = 40-heart. And write a system (with heart = star+9)
One only needs the numbers in the top of the triangle and bottom left of the triangle to solve this. The number in the bottom right of the triangle is not necessary. Any other number could be substituted for the bottom right number and the result will still be the same when adding up the middle bottom number and right middle number.
I did it the last way, but forgot that i was just looking for the sum of star and heart, not the individual values of star and heart, so i proceeded to solve them as well.
If the botton row is x y z. Then the middle row is x+y y+z and the top row is x+2y+z
In this case x=7.
We can also guess directly the second row. If we substract the first member with the second, we get : (x+y) - (y+z) =x-z.
Here is how you solve it. The number in the middle on the bottom row is used twice. Now, you add this, 7+9. That equals 16. Now, you subtract 16 from 40, because the higher two numbers equal that. 40-16= 24. Since the middle number will be used twice, you divide by 2. So, that makes the middle number to be 12. 24÷2= 12. Now, add the two left numbers; 7+12= 19. Now, 19 is the left number on the middle row. Add the middle, and right number for the right number of the middle row; 12+9= 21. Add the 2 numbers together to see if that equals 40, and it does; 19+21= 40. Now, the heart plus the star; 12+21= 33. There is your answer.
This was my solution:
Bottom right corners of each triangle is 3 and 9.
3 is one third of 9. Since the triangles are the same proportions, each number in the left triangle is one third of the number in the matching position of the right triangle.
7 matches heart, 7 * 3 = 21
4 matches star, 4 * 3 = 12
21 + 12 = 33
This is a problem that should not be given to kids. All it will do is result in frustration, turn them off to math, and is really just a waste of time.
Your way to approach at the beginning is exactly how I thought to solve it as well: looking for a pattern in the other triangle.
I then did guess and check to verify (starting at 8, then 10, then 12)… mostly because I was solving this in my head while I’m watching on TV and only have my phone available (used for this comment)
From the explanation tower: 2+(2x4)+3=13, bc 13=6+7, so 13=(2+4)+(4+3), with this in mind 40=7+(2x⭐)+9, 24=2⭐, ⭐=12.
⭐+9=🖤, 12+9=🖤, 🖤=21
⭐+🖤=12+21=33
Or: since 7 and 9 are 2 apart, 🖤 and 'blank' also need to be 2 apart, so that means 'blank'=19 and 🖤=21
🖤=7+⭐, 21=7+⭐, ⭐=12
This is trivial. (Top number- bottom left -bottom right)/2 is star. After that just add as described.
This is child's play. Quite literally. I solved the entire puzzle. 40 is equal to the sum of the two unknown variables below it, which each equal the sum of the two numbers below them. This includes 7 and 9, as well as two times ⭐ because the star will be included in both sides of the middle. Simplify that equation and you will get your answer for what each variable is and then you can add the two together. It's a nice little trick and I kinda noticed the pattern on the right, but didn't think about comparing it to the left to make sure it's a pattern.
Star is not needed initially because it contributes to both cells above it. By observation, the empty cell must be Heart - 2.
Attempting to answer this before watching the video.
Clearly, every number in the upper segments is the sum of the two segments beneath it it. Two and four make six, four and three make seven, six and seven make thirteen, etc.
In the opposite pyramid, we are only given three values, and asked to identify the values of the star and the heart so we can finally solve the problem.
If we look at the first pyramid, we can see there is a relationship between the value at the top of the pyramid and the three values at the bottom. The left and right bottom segments each contribute once to the sun at the top, but the center bottom segments contributed twice, since it adds to the sum of both the segments directly above it. To summarize, lower left plus two of lower mid plus lower right equals the value of the top segment.
In the right pyramid, our top segment is forty. If we take away nine and seven, that leaves us 24 remaining. Since the middle segment contributes to this remainder twice, that means our star is twelve. With that value, we can determine that the empty space is 19 (twelve plus seven) and the heart space is 21 (twelve plus nine). We see that our values successfully add up to forty at the top.
The actual question tasks us with determining the sum of the star and heart segments. Twelve plus twenty one is thirty three. The answer is 33.
Now to watch the video and see if I got this right. xD
There is a much easier way to do this without guessing and checking. Because the middle number is used in two factors.
You could write this as 7+2*⭐+9=40
7 + star gives the left number on the second row, and adding it to the number on the right gives 7 + star + love, which is the top number, 40. Star + love is 33 since 40 - 7 is 33
Okay, so there was a much faster way
I could’ve racked my brain more
First thing I noticed was that x is ⭐+7 and 🤍is ⭐+9. Therefore, 🤍is 2 larger than x.
The two numbers that add up to 40, and are 2 apart, are 19 and 21, which makes 🤍=21.
That means⭐=21-9 =12. And, of course, 12+21=33.
Now i will rewrite the heart shape is a, star is b and the empty cell as c.
so we have b+9=a, b+7=c, a+c=40
so that means 2b+9+7=40 -> b=(40-9-7)/2=24/2=12
then we can see a=21 and c=19
we found a,b,c so we can calculate a+b=21+12=33. isn't it easy for adults?
A: star + 9 = heart
B: heart + 7 + star = 40
insert A into B: (star + 9) + 7 + star = 40
--> 2 star = 24
--> star = 12
--> insert in A: 12 + 9 = heart = 21
I solved it first the way you did in the third example (paused the video before watching the solutions). While doing that, the thought occurred to me "Does it scale?" so once I had the solution, I checked back to see. Assuming that a first order approximation would be good enough, I figured 40 and 13 as 3 to 1, and yup. As a first pass it was clearly good enough. Surprisingly good problem.