# Inequality of the means

**Question**Is is possible to pack rectangular -dimensional boxes each of which has side lengths inside an -dimensional cube with side length ?

Taking the arithmetic/geometric mean inequality multiplying both sides by and then raising both sides to the power yields: So, in the above question, the volume of the cube is at least the sum of the volumes of the rectangular boxes. Furthermore, a positive solution to this question would yield a strengthening of the arithmetic/geometric mean inequality.

For the problem is trivial, for it is immediate, and for it is tricky, but possible. It is also known that a solution for dimensions and can be combined to yield a solution for dimension . Thus, the question has a positive answer whenever has the form . It is open for all other values.

See Bar-Natan's page for more.

## Bibliography

[BCG] E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways for Your Mathematical Plays, Academic Press, New York 1983.

* indicates original appearance(s) of problem.