There is a little known lovely geometric counterpart to the well
known Arithmetic/Geometric Means Inequality. The inequality says that
for every sequence of positive numbers *a _{1}, a_{2},
..., a_{n}*, the geometric mean of these numbers is smaller
than or equal to the arithmetic mean of these numbers:

If we multiply both sides of this inequality by *n* and raise to
the *n*th power, we come to:

This leads to a very natural question:

**Question. ** Is it possible to pack *n ^{n}*
rectangular

This question is much harder than it seems! It implies the inequality of the means, of course, but it is not implied by it. Here's almost all that is known:

**The case of n=2: ** Here one needs to place

**The case of n=3: ** Now this is a much harder problem. It
is quite impossible to solve it without a physical model, and you'll
only appreciate how hard it is if you get a physical model and spend a
few hours with it. Here's one made of wood, along with a failed attempt
to assemble it (the last wall of boxes wouldn't fit and there's too much
useless leftover space at the top):

You can also make a paper/cardboard model. Just click on the two images below to download two PDF files. Print the first once and the second seven times, cut along the full lines, fold along the dashed lines, and glue as marked. Thicker paper is better!

the big cube |
the boxes |

Let me say that again - fitting these 27 boxes inside this one cube
is a hard exercise! If you despair, check out the solution sequence
below. It's actually harmless to see the solution even if you intend
to work on the puzzle later on - the solution is quite unmemorable, and
unless you make a conscious effort to memorize it, you're going to forget
it right away. You can also click on the __Mosaic__ or __Layers__
buttons below and get a printable key to the solution.

First Previous Next Last Mosaic Layers

**The case of n=4: ** Somewhat surprisingly, the case of

**Hint. ** Use the case of *n=2* three times to translate the
following algebraic argument to a specific way to assemble *256*
boxes of sides *a* by *b* by *c* by *d*
inside a single cube of side *a+b+c+d*:

**The case of n=5 ** is wide open. If you can do it and all
higher unknown cases, you'll be famous! After all, we are in the 21st
century and its not so easy to come up with new proofs of high-school level
inequalities.

**Did you try to program it? ** For *n=5*, one has to place
5^{5}=3,125 boxes, and each one has 5!=120 possible orientations.
Thus there are 120^{3,125} possibilities to start with, and we
don't know how many of them are good (i.e., have no internal clashes and do
not extrude out of the cube. So an exhaustive search is out of question.
(In fact, merely testing if a given proposed solution is indeed a solution
is a non-trivial proposition. There are 171,584 pairs of diagonally
neighboring boxes so 171,584 potential internal clashes, and another 3,125
ways a box may extrude out of the cube.)

I did try to program it nevertheless. We don't know how many solutions
there are for *n=5*. Possibly none, possibly few, possibly very very
many. In the last case, we have a chance of finding one. So I wrote a
program that starts from some arbitrary proposed solution, and then uses
simulated annealing to try and lower the number of "problems" it has.
Here's a short summary of the results:

- The program takes no time at all to solve the (human difficult)
*n=3*case. - Starting with the known solution at
*n=4*and performing just 10 random moves, the program is unable to find its way back to a solution. - At
*n=5*the best I could find was an arrangement with 9,334 internal clashes of boxes. A far cry from the required 0.

While a failure, my programing attempt does raise some interesting mathematical questions. Some future edition of this document may contain a summary of those.

**The cases of n>5: ** These are divided in two:
the simple and the hopeless. An argument similar to that of the case of

**Reference. ** Everything in this page (except the pictures and the
program) comes from Berlekamp, Conway and Guy's *Winning Ways for Your
Mathematical Plays,* Academic Press, New York 1983.

Displayed equations by HTMX.

Some artwork and painting by Amir, Assaf and Itai.

Thanks to Kumar Murty and Ehud
de Shalit for pointing out a typo in an earlier version of this
page.