Packing small cubes into a cube of side twenty

Question

What is the largest number of cubes of side length $3$ that can be packed inside a cube of side length $20$ ?

Short answer

$216$ . Along any edge only $\lfloor 20/3\rfloor=6$ small cubes fit, since seven would need length $21$ . A $6\times6\times6$ grid achieves $6^3=216$ , and for axis-aligned packing this is the maximum.

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Lessons behind this question

Counting and Combinatorics
Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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