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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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