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Math Magic Archive

Problem of the Month (July 2014)

This month we extend the problem of the December 1998 Math Magic to 3 dimensions. Consider the ways in which an n×n×n cube can be cut into smaller integer-sided cubes. Define:

f(n) = the largest possible size of the smallest cube

g(n) = the smallest number of cubes

h(n) = the smallest possible value of the largest multiplicity of cubes needed

When n is composite with smallest prime factor p, it appears that f(n) = n/p f(p) and g(n) = g(p). Can you find counter-examples? Thus we mostly concern ourselves with finding the values of f(n) and g(n) for n prime.

Can you beat the best known tilings below?

In particular, what is the smallest n for which f(n) = k? How fast does g(n) grow? What is the smallest n for which different packings are needed to illustrate f(n) and g(n)? What is limn→∞ h(n) ? It is known that no n has h(n)=1. Are there n for which h(n)=2?

And a meta-question: what is the best way to visualize complicated cube packings?


You can see all the best-known results here.

Submit your answers here.

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