Pyramids By the BlockMath brain teasers require computations to solve.
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?
HintYou might want to start at the top.
AnswerIt 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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