Brain Teasers
Pyramids By the Block
In a catalog, you read about a set of blocks. There are 1029 blocks; all are identical in volume. They can be assembled into several tiers which are 1 foot thick and stack to form a pyramid. (The pyramid has a square base; its four sides are equivalent isosceles triangles.) How tall is this pyramid?
Hint
You might want to start at the top.Answer
It is 7 feet tall.The top tier is also the peak. Suppose it is a single block and see how the numbers behave.
Its four triangular surfaces share edges with its square base; the length of each such edge we will call b. (Assuming this length is 1 foot will also work.) The peak is a smaller pyramid so its volume, v1 = ( b * b * 1 ) / 3. Because a second tier will not change any of the angles involved, new triangles will be similar to old and all changes in measurement of length will be proportional. A pyramid 2 tiers tall would have v2 = ( 2*b * 2*b * 2 ) / 3 = 8*v1. For 3 tiers, v3 = ( 3*b * 3*b * 3 ) / 3 = 27*v1. It becomes clear that a cube number must be a factor of the number of blocks.
1029 = 3 * 7^3, so there are 7 tiers. (The peak being formed by 3 separate blocks, every tier has 3 times as many blocks as in the supposition. Because 7 is a prime number, there is no smaller number of tiers which would satisfy the question.)
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Comments
A good one, but it is on the easy side of mathematics solving. Keep up the good work!
Good one for those who can do this type of math. How I could ever tell how many bushels any grain bin could hold, I never will know, but this one went right over my head. Nice job.
Good teaser I'd just woken up and was using ^2 instead of ^3 so I got the answer wrong
I got lost with the English (I speak Spanish...), but I would like to understand the problem... Is the pyramid hollow?
A solid pyramid, with 1 block at the top, 4 in the second tier, 9 in the third tier, 16..., needs 14 tiers to use 1015 blocks. How is it possible that a hollow pyramid uses more blocks in less tiers?
A solid pyramid, with 1 block at the top, 4 in the second tier, 9 in the third tier, 16..., needs 14 tiers to use 1015 blocks. How is it possible that a hollow pyramid uses more blocks in less tiers?
The problem could have been written more clearly. I also went with the assumption of an Egyptian-style pyramid, not a mathematical pyramid shape, so I also came up with 14 feet.
Nice teaser if I'd understood the question.
Nice teaser if I'd understood the question.
3 blocks for the top? Could someone explain how this works? And how it is a better solution than 14 levels with the top being a single block? I'm guessing this is a mathmatical pseudo pyramid or somthing but I can visualize it.
Some people have made the mistake of thinking the blocks must be cubes, but in order to form a true pyramid, all exterior blocks must have some element of prism or pyramid in their composition.
The top layer, by itself, must be a true pyramid. If you cut an "X" through it following the visible edges, you create four equal tetrahedrons (pyramids with triangular bases). With some additional cutting you could get twelve tetrahedrons of equal volume. Each of three blocks in the top layer could be composites four adjacent twelfths.
The top layer, by itself, must be a true pyramid. If you cut an "X" through it following the visible edges, you create four equal tetrahedrons (pyramids with triangular bases). With some additional cutting you could get twelve tetrahedrons of equal volume. Each of three blocks in the top layer could be composites four adjacent twelfths.
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