![](/files/happy5.png)
Inequality of the means
![$ n^n $](/files/tex/699fff17a123a14fe73275fc0a636f46688029d9.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ a_1,a_2,\ldots,a_n $](/files/tex/e7587d53923e6a1289b1a25bce813c459b94a973.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ a_1 + a_2 + \ldots a_n $](/files/tex/e1dd2dd4a0225f650d6b7185f7abde703d0fbdbd.png)
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.