Brain Teasers
Cubes, Cubes, Everywhere!
In the far future, you find yourself a cube-maker. To make cubes, you would take a 12 by 12 by 12 Rubik's cube-like object. Normally, you would touch each tiny one by one by one cube and they would fall off. However, with new quantum technology, it has become more interesting. You may say the total number of possible cubes (not necessarily with side length one) in the twelve by twelve by twelve Rubik's cube, and hocus-pocus, cubes of all sizes would appear. Assuming that each side in the tiny cubes has length one and that you may only make cubes of integer side length, what number should you say for the new technology to work?
Hint
For 1^3+2^3+... n^3, use (1+2+...n)^2.Answer
6084.First, try a easier problem. Try a two by two by two Rubik's cube. There are 8 tiny cubes and one 2 by 2 by 2 cube. It should come to your notice that you take 2^3+1^3. But just to check, you look at a standard Rubik's cube with side length 3. There are, predictably, 27 one by one cubes, 8 two by two cubes, and one 3 by 3 cube. That was 3^3 +2^3 + 1^3. Now you can come to the conclusion that for any Rubik's cube with side length n, there are n^3+(n-1)^3 + ... 2^3+1^3 total cubes with integer side length. Therefore, there are 1^3+2^3+...12^3 cubes in the Rubik's cube with side length 12. The hint gives the formula for adding a sequence of cubes. Thus, you would get (1+2+3+4+...+11+12)^2 = 6084 cubes.
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