# Boxing the Blocks

by Kevin J. Lin

There's nothing wrong with your nephew. There is no way to pack the twenty-seven
blocks into the box.

This fact was proven by the mathematician de Bruijn who demonstrated that collection
of bricks with the dimensions *a x ab x abc* can only be efficiently packed into
a volume measuring *ax x aby x abcz*. If you can visualize the relationship between
these volumes, you can see that in that case, the bricks can be packed uniformly-
all with the same orientation. So if a brick with dimensions *a x ab x abc* can
be packed into a volume, it can be packed simply.

Now in the case of your Nephew's blocks, can this be simply demonstrated without
knowing of de Bruijn's proof?

Consider a block packed with red and blue 2 x 2 x 2 bricks, as shown below:

If you were to place your nephew's blocks in the box, each block would occupy four
cubic inches of "blue brick space" and four cubic inches of "red brick
space." But there are more red bricks than blue bricks. So no matter how the
4 x 2 x 1 blocks are placed in the box, one will always be left that can't fit in
the remaining space.