Math Question for Fun!

[Shadow Wolf]

I did it as I would doing inventory and chunking rows together adding the sums of those chunks (minus the machine).

 

[bobhikes]

Suppose you have this two-dimensional pyramid of cups:



How can you count the number of cups without counting the cups individually?

Bonus question: Assume this pyramid is turned into a pyramid with four faces, with each face looking like the pyramid in the picture. How many cups are there in that tetrahedral pyramid?

Please use spoiler tags for your answers!

I would turn it into a rectangle and then calculate the area. I would still need to count length and width but that is all I would need to count.

 

[SalixIncendium]

I know this is a very non-mathematical answer, but as a former bowler, my eyes automatically cordoned them off into groups of 10. I saw 7 sets of 10, then saw the 5 remaining in the 5th row and the three in an upside-down triangle directly below.

264. 78 x 4 minus the 48 shared at each corner.

I'm probably wrong, though. I didn't double check my work.

 

[LuisDantas]

I know this is a very non-mathematical answer, but as a former bowler, my eyes automatically cordoned them off into groups of 10. I saw 7 sets of 10, then saw the 5 remaining in the 5th row and the three in an upside-down triangle directly below.

264. 78 x 4 minus the 48 shared at each corner.

I'm probably wrong, though. I didn't double check my work.

Hey, this is sharp. I must try this "ispoiler" tag sometime.

 

[Debater Slayer]

Tagging everyone who has given this a shot now that I'm posting the answer!

@Jedster @Subduction Zone @fantome profane @LuisDantas @ChristineM @exchemist @Shadow Wolf @bobhikes @SalixIncendium

1) To count the cups in the two-dimensional pyramid, we can treat each level as a term in an arithmetic sequence with a common difference of one, a first term = 1, and a last term = 12. The sum of an arithmetic series up to the nth term is given by this formula:

This makes the sum until the 12th term = 12[(1 + 12)/2] = 12(6.5) = 78 cups.

2) For the tetrahedral pyramid, we have to account for the four shared columns of cups, which make up the lateral edges:

The problem with multiplying 78 by 4 to get the answer is that it means we'll be counting the lateral edges eight times instead of four. This gives an extra 12 x 4 = 48 cups.

Instead, we will construct the outer structure made up of the four columns, which have 11 x 4 = 44 cups, and then calculate the number of cups in each of the four faces, excluding the columns:

You'll notice that the top cup is shared, which means the four columns, highlighted in red, each have 11 cups, so the outer structure has 4 x 11 = 44 cups.

For the inner triangle making up the four identical faces, we have an arithmetic series with a common difference of one, a first term = 1, and a last term = 10. The sum until the 10th (and last) term is 10[(1 + 10)/2] = 10(5.5) = 55. So the four faces have 55 x 4 = 220 cups.

Adding the cups from the four columns and the shared cup on top, the tetrahedral pyramid has 1 + 44 + 220 = 265 cups.

 

[Debater Slayer]

I know this is a very non-mathematical answer, but as a former bowler, my eyes automatically cordoned them off into groups of 10. I saw 7 sets of 10, then saw the 5 remaining in the 5th row and the three in an upside-down triangle directly below.

264. 78 x 4 minus the 48 shared at each corner.

I'm probably wrong, though. I didn't double check my work.

I find this absolutely fascinating because it highlights how our brains can be trained to approach problems differently and arrive at similar or identical results. The first things I thought of when I looked at the pyramid were

1) whether each level increased by a constant number such that I could form an arithmetic series

2) whether there were any exceptions to that general approach (e.g., the four outer columns and the shared cup on top), and

3) whether I could further simplify my equations to reduce the complexity of the calculations.

Your approach worked just as well, and I'm still trying to wrap my head around how you grouped the cups together into sets of 10—especially the part where you mentioned the fifth row. Could you elaborate on that more? I'd like to understand all of the different ways to solve this.

 

[Debater Slayer]

@anna. You get a mention as well for your kind gesture of bumping the thread!

 

[ChristineM]

Tagging everyone who has given this a shot now that I'm posting the answer!

@Jedster @Subduction Zone @fantome profane @LuisDantas @ChristineM @exchemist @Shadow Wolf @bobhikes @SalixIncendium

1) To count the cups in the two-dimensional pyramid, we can treat each level as a term in an arithmetic sequence with a common difference of one, a first term = 1, and a last term = 12. The sum of an arithmetic series up to the nth term is given by this formula:

This makes the sum until the 12th term = 12[(1 + 12)/2] = 12(6.5) = 78 cups.

2) For the tetrahedral pyramid, we have to account for the four shared columns of cups, which make up the lateral edges:

The problem with multiplying 78 by 4 to get the answer is that it means we'll be counting the lateral edges eight times instead of four. This gives an extra 12 x 4 = 48 cups.

Instead, we will construct the outer structure made up of the four columns, which have 11 x 4 = 44 cups, and then calculate the number of cups in each of the four faces, excluding the columns:

You'll notice that the top cup is shared, which means the four columns, highlighted in red, each have 11 cups, so the outer structure has 4 x 11 = 44 cups.

For the inner triangle making up the four identical faces, we have an arithmetic series with a common difference of one, a first term = 1, and a last term = 10. The sum until the 10th (and last) term is 10[(1 + 10)/2] = 10(5.5) = 55. So the four faces have 55 x 4 = 220 cups.

Adding the cups from the four columns and the shared cup on top, the tetrahedral pyramid has 1 + 44 + 220 = 265 cups.

Damn i forgot the top cup was shared on 2 sides

 

[Debater Slayer]

I'm still trying to work out why it is not 72, i.e. half of 12 x 12.

That is what one gets by making a rhombus of cups, but that is another way to solve it.

I would turn it into a rectangle and then calculate the area. I would still need to count length and width but that is all I would need to count.

The reason that half of 12 x 12 is not the number of cups in this case is that constructing a square around the triangle would give a square with double the area of the triangle, but the area doesn't account for the irregularities in the shape of the cups.

In a triangle formation, the cups can't be arranged to perfectly fill exactly half the area of the square. It's like pouring a given amount of batter into a square pan, where it perfectly takes the shape of the pan, versus pouring it into a circular one with the same area and then trying to fit the cake into the square pan. Although the area is the same, the different shape means that the cake will have to be cut at the corners and not fit perfectly.

 

[SalixIncendium]

Your approach worked just as well, and I'm still trying to wrap my head around how you grouped the cups together into sets of 10—especially the part where you mentioned the fifth row. Could you elaborate on that more? I'd like to understand all of the different ways to solve this.

I didn't physically draw it out yesterday morning as I did just now in my attempt to demonstrate the breakdown, and I swear I was able to mentally map out 7 sets of 10 with 8 left over. But for whatever reason, I'm only to find 6 with 18 left over. But this is basically the way my brain broke it down, though obviously not exactly.

Somehow yesterday, the 5th row of 5 and three of the cups in a triangle below it were the left over cups in the 7 sets of 10, but now I'm unclear how that happened or if it was just an error on my part.

 

[SalixIncendium]

Damn i forgot the top cup was shared on 2 sides

I made a similar error in omitting the top cup and was off by 1.

 

[Debater Slayer]

I didn't physically draw it out yesterday morning as I did just now in my attempt to demonstrate the breakdown, and I swear I was able to mentally map out 7 sets of 10 with 8 left over. But for whatever reason, I'm only to find 6 with 18 left over. But this is basically the way my brain broke it down, though obviously not exactly.

View attachment 80566

Somehow yesterday, the 5th row of 5 and three of the cups in a triangle below it were the left over cups in the 7 sets of 10, but now I'm unclear how that happened or if it was just an error on my part.

Thanks! The thought process behind it is still very interesting to see demonstrated in drawing, nonetheless. I wouldn't have thought of grouping them into smaller triangles.

 

[Subduction Zone]

The reason that half of 12 x 12 is not the number of cups in this case is that constructing a square around the triangle would give a square with double the area of the triangle, but the area doesn't account for the irregularities in the shape of the cups.

In a triangle formation, the cups can't be arranged to perfectly fill exactly half the area of the square. It's like pouring a given amount of batter into a square pan, where it perfectly takes the shape of the pan, versus pouring it into a circular one with the same area and then trying to fit the cake into the square pan. Although the area is the same, the different shape means that the cake will have to be cut at the corners and not fit perfectly.

Dang it, I said rhombus when I meant to say parallelogram. I was thinking that one could make two triangles as shown , but use glue on one of them so it could be flipped upside down and placed next to the other one. I knew that I would have a base of one greater than the angled side. Since you had two you would divide the number that you got when you multiplied the N + 1 base times the N side. by two.

 

[Subduction Zone]

Tagging everyone who has given this a shot now that I'm posting the answer!

@Jedster @Subduction Zone @fantome profane @LuisDantas @ChristineM @exchemist @Shadow Wolf @bobhikes @SalixIncendium

1) To count the cups in the two-dimensional pyramid, we can treat each level as a term in an arithmetic sequence with a common difference of one, a first term = 1, and a last term = 12. The sum of an arithmetic series up to the nth term is given by this formula:

This makes the sum until the 12th term = 12[(1 + 12)/2] = 12(6.5) = 78 cups.

2) For the tetrahedral pyramid, we have to account for the four shared columns of cups, which make up the lateral edges:

The problem with multiplying 78 by 4 to get the answer is that it means we'll be counting the lateral edges eight times instead of four. This gives an extra 12 x 4 = 48 cups.

Instead, we will construct the outer structure made up of the four columns, which have 11 x 4 = 44 cups, and then calculate the number of cups in each of the four faces, excluding the columns:

You'll notice that the top cup is shared, which means the four columns, highlighted in red, each have 11 cups, so the outer structure has 4 x 11 = 44 cups.

For the inner triangle making up the four identical faces, we have an arithmetic series with a common difference of one, a first term = 1, and a last term = 10. The sum until the 10th (and last) term is 10[(1 + 10)/2] = 10(5.5) = 55. So the four faces have 55 x 4 = 220 cups.

Adding the cups from the four columns and the shared cup on top, the tetrahedral pyramid has 1 + 44 + 220 = 265 cups.

Okay I did not do the pyramid that way, but I was thinking of a solid one. So it would have 12^2 cups at the base 11^2 cups in the next level, all the way to the top.

 

[anna.]

@anna. You get a mention as well for your kind gesture of bumping the thread!

Aw... Thanks for your kindness. I took one look at that pyramid and said "nope..." The farthest I've gone in math is behavioral statistics, which I hope to never revisit in my lifetime... but I do like seeing people being able to answer questions like this!

 

[bobhikes]

The reason that half of 12 x 12 is not the number of cups in this case is that constructing a square around the triangle would give a square with double the area of the triangle, but the area doesn't account for the irregularities in the shape of the cups.



In a triangle formation, the cups can't be arranged to perfectly fill exactly half the area of the square. It's like pouring a given amount of batter into a square pan, where it perfectly takes the shape of the pan, versus pouring it into a circular one with the same area and then trying to fit the cake into the square pan. Although the area is the same, the different shape means that the cake will have to be cut at the corners and not fit perfectly.

I'll admit there are some assumptions. I used a cup as measurement and half a cup as a measurement but otherwise my answer was correct. If I split the triangle in half with a perpendicular line and flip it so the two halves form a rectangle I get 12 high by 6.5 wide again with some assumptions this equals an area of 78 cups. The only reason I would have to do this is for taking a test and the tests I take are multiple choice so as long as I don't have to show my work I would get the right answer.

 

[jhonsteve111]

I'm not really into math, and I’ve never tried something like this before. Can someone help me figure out how to count these cups without having to count each one individually?

 

[9-10ths_Penguin]

How can you count the number of cups without counting the cups individually?

Spoiler

Weigh one cup, then weigh the pyramid.

 

[Revoltingest]

Suppose you have this two-dimensional pyramid of cups:

How can you count the number of cups without counting the cups individually?

Bonus question: Assume this pyramid is turned into a pyramid with four faces, with each face looking like the pyramid in the picture. How many cups are there in that tetrahedral pyramid?

Please use spoiler tags for your answers!

Spoiler: How to do it.

Mentally add a 2nd pyramid to the 1st.
Now you have a parallelogram.
Multiply the 2 sides together.
Divide by 2.

Process is similar for a 4 sided pyramid.
It takes 3 pyramids to make a cube.
(You can imagine how to do this by sliding the top to reside over a corner.)
Multiple the 3 sides.
Divide by 3.

The above can be done equivalently using series added together.
In the 1st case, it results in (n + 1) added together n times.
This gets....
n ( n + 1) / 2

 

[It Aint Necessarily So]

How can you count the number of cups without counting the cups individually?

For 1+2=+3+4+....+n, where n+ 12 in this problem

Spoiler

The stack's edges have 12 cups, so n=12 and n(n+1)/2 = 12x13/2 =78

Assume this pyramid is turned into a pyramid with four faces, with each face looking like the pyramid in the picture. How many cups are there in that tetrahedral pyramid?

Now it's 1+3+6+10 + ...+n (where n=12 again)

Spoiler

Assuming that the pyramid is solid:

so, Sn= 12(13)(14)/6=364