There are 24 hours in a day. Why?

Why aren't there 25? ... 10?


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Take a square pyramid. Say it is of cannonballs or oranges. At the top there is one round shape. Below it there are four. Below that, nine.

Start unpacking. Fill a square with round shapes. First, one. After that, five. Going down a level further there will be fourteen.

Is there a level so that when all shapes are taken off and placed in a square, there is a completely filled four-sided shape?

There is. The only square that can be completely filled by deconstructing a pyramid has 576 elements. The deconstructed pyramid will have 24 levels.

The number 576 is the square of 24.
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Find any three positive integers a, b, c such that

a³ + b⁴ = c⁵

Note that
2ⁿ + 2ⁿ = 2 (2ⁿ) = 2^{(n + 1)}

Use this if n is a multiple of 3 and 4 and n + 1 is a multiple of 5.

Smallest such n = 24.

2²⁴ + 2²⁴ = 2²⁵

(2⁸)³ + (2⁶)⁴ = (2⁵)⁵

a = 2⁸ = 256,

b = 2⁶ = 64,

c = 2⁵ = 32