Child's homework puzzle is baffling adults
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- Опубліковано 6 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.
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
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.😂😂
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!
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..
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.
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.
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.
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.
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 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.
Easy enough to do so in your head. Just a small system of equations
this video did not need to be 8 minutes long
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
40=blank+heart
blank=7+star
Therefore:
7+star+heart=40
Subtract 7 from both sides
Star + heart = 33
Easy 😂
Star = 12
Heart = 21
Edit: Sum = 33
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.
(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.
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.
star is just (40-7-9)/2 = 12
heart is then 9+12=21
so 33
I tried to solve this using a linear system and accidentally found the answer
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
I just brute forced it lol
Quickly spotting the logic, I split into the constituent equations, substituting X, Y and Z for the unknowns.
X + Y = 40, 7 + Z = X, 9 + Z = Y
(7 + Z) + (9 + Z) = 40
16 + 2Z = 40
40 - 16 = 2Z
24 = 2Z
Z = 12
Double check:
(7 + 12) + (9 + 12) = 40
19 + 21 = 40
So Z + Y = 12 + 21 = 33. Variations on this, with a taller pyramid, appear in many local newspapers in the UK, typically advising readers to try it in their head before writing down the numbers.
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
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.
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
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.
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!!!!!!!!!
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.
Elated that I went straight to the algebra solution without thinking too much............ Crushed that I'm not as smart as the 9 year old solution at the end............ Graduate engineer retires to contemplate life, the universe and everything.........
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.
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.
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.
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 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.
Seriously?
7+🌟=⬜
⬜+❤=40
So 7+🌟+❤=40
🌟+❤=33
Longer solution:
7+2x🌟+9=40 (top=sum of bottom with 2x for middle element)
🌟=12
❤=9+🌟=21
So: 🌟+❤=33
Heart is 21, star is 12
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
write as
7+x+9+x=40
x will represent the star
simplify
16+2x=40
subtract 16
2x=24
divide by 2
x=12
the heart is 9 more than the star, so we have
12+21=33
the only reason I couldn’t solve this from the thumbnail was that I didn’t know the rule for how the pyramid is constructed
Solution 1) 40 needs to be the sum of two numbers, easiest place to start is 20/20 and with 7 and 9 as a base it mean the highest star could be is 13 and the lowest is 11, because the base numbers are 2 values apart lowest value comes up one, highest value drops one so 13-1 =12, 11+1=12 star is 12, 12+9=21+12=33.
solution 2) pattern 13-2-3=8/2=4 so 40-7-7-24/2=12 star is =12 solve for heart.
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.
The star is 12, the heart is 21, the blank space is 19, and the answer to the problem is 33.
I got 71 thanks to a twisted brain.
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)
My first reaction was "top box is equal to bottom left plus bottom right plus (bottom middle X 2)". I was able to solve the whole triangle from there and got the final answer.
Start at 20+20 in the middle row and work in opposite directions along the number line.
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 one was actually super easy, barely an inconvenience.
I can't answer any of his puzzles correctly, but this one was child's play.
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.
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)
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.
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.
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)
answer 33
Notice in the second row that the number to the left (6) + the number in the middle of the third row (4) + the number in the right
of the third row (3) = 13
In contrast, the number to the right in the second row (7) + the number in the middle of the third row (4) + the number in the left
of the third row (2) ALSO = 13
The question is similar to the latter (7) + (4) +2 = 13, but two unknowns are in the place of 7 and 4.
Hence, a + b + 7 =40
a + b = 40-7
=33 answer
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
Did this before watching the video. My logic is that the number is the sum of the two numbers under it, which makes ❤=⭐+9, and the blank space 7+⭐.
Based on this logic, we can come up with this equation:
(7+⭐)+(⭐+9)=40
7+2⭐+9=40
16+2⭐=40
2⭐=24
⭐=12
Now that we find ⭐, we can use it to find ❤.
❤=⭐+9
❤=12+9
❤=21
Therefore,
⭐+❤
=12+21
=33
The mid-left box is as vacant as the person's brain who made up this problem.
I thought the question was going to be "why did this baffle adults"; which would have been challenging. 😊
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
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
At first I thought the blank box meant it was zero. Once I realized it was an unknown the problem became simple algebra.
That was super easy . Solved in my head in about 10 seconds.
It took me a moment but once I sat down and filled in the gap, I realized that the actual equation is rather simple.
You see, since we need to add the two numbers below to get the number up top, the empty square had the value 7+⭐, and therefore the 40 was 7+🌟+❤... (because we're looking for ⭐+❤, I'm gonna replace it with an x) so our equation is x+7=40, which you might remember from basic algebra. The answer is 33
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.
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
Took me a moment, but:
Top is left bottom+right bottom+twice middle bottom.
Thus middle has to be 12.
A trivial problem. You can also make a equation 40 = 7 + * + * + 9.
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.
Interesting thing is, the value is independent of 9
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.
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.
The middle row is comprised of two sums, left plus middle and right plus middle. So the 2nd row is left + 2 x middle + right. Since the 2nd row adds up to the top row, 40 = 7 + 9 + 2X, so (40 - 16) / 2 = 24/2 = 12, so X aka star = 12, that makes "heart" = 12 + 9 = 21. So 12 + 21 = 33 is start plus heart. Pretty much the same as @llaffer.
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.
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.
My instincts are always algebra, and my algebra instincts are usually substitution before adding equations together.
I get star = 12 and heart = 21
Rush starts playing
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
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.
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.
7 + star + heart = 40
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
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.
I realized that the two adjacent blank squares had to equal 40 and then it was fairly obvious that the star square had to be 12
I can't believe parents can't do simple algebra
nice to see so many different ways of solving this problem in the comments. Though I left school about 4 years ago, I just used year 8/9 maths (British curriculum) to solve it.
Let ⭐ = a and let ❤ = b. Not knowing the value of the blank box, I let it be 'x'. Looking at the pyramid, the combination of numbers below added together make the number above, so I made these equations:
7 + a = x, a + 9 = b, x + b = 40
from these I further derived the equations for a and x, so that I could substitute them in the first equation:
a = b - 9 and x = 40 - b
So I put these in the first equation and got:
-2 + b = 40 - b
b + b = 40 + 2
2b = 42, so b is 21.
Then a = 21 - 9
a = 12
Just to make sure I wasn't wrong I found x by:
x = 7 + 12 or 40 - 21
x = 19
so then, ⭐ + ❤ is 12 + 21 = 33.
Overall, I don't think it's particularly hard, it just looks hard because of three variables you have to deal with, and the fact that the parents might not have noticed a pattern or two using the first triangle as a guide.