# Packing 9 Pieces in a Box

Copyright © Karl Dahlke, 2023

This section of the book
might not be science proper - perhaps math, or analytics, or logic.
However, learning to think clearly in an abstract setting
is important, in science, and in every endeavor.
Besides, some of these abstract puzzles are, to me, quite lovely.
This one comes from the mind of John Conway.

You have a 3 by 3 by 3 box. Maybe 3 inches on a side; any unit you like.
You have 9 pieces to fill the box.
3 of them are single cubes, an inch on each side. I'll call them singlets. If you had 27 of these they would fill the box, but you have 3.
Then you have 6 squares, each square is 2 by 2 by 1. Like 4 cubes put together in a square.
4×6 + 3 = 27 so the volume is correct.

How do these 9 pieces fill the box - and prove there is only one solution, up to reflections and rotations.
You don't need a computer or even pencil ad paper.
You can do it in your head.

## Solution

Start with the top layer of the box.
If a square is flat in the layer, it consumes 4 cubes; if it is vertical it consumes 2 cubes.
Either way it consumes an even number of cubes.
There are 9 cubes in the top layer, so we need 1 singlet, or 3.
If all 3, we have a problem, for there are none left for the middle layer or the bottom layer.
Each layer has one singlet.
Similarly, the back plane, middle plane, and front plane each contain 1 singlet,
and the left plane, middle plane, and right plane each contain 1 singlet.

Put the singlet in the center of the top layer.
It is surrounded by 4 vertical squares.
Below that singlet is another singlet, but we said we needed 1 singlet in each of the back middle and front planes.
Therefore, our singlet in the top layer can't be in the center.

Put the singlet in the back middle position of the top layer.
It has a vertical square on either side.
Now there must be a singlet adjacent to this one, in the same plane, and that is a contradiction.

Therefore, spinning the box as we wish, the singlet in the top layer is at the back left.
Using the same reasoning, the singlet in the bottom layer is in one of the 4 corners.
Yet it can't be in the same plane as the singlet on top, so it is in the bottom front right.

Each of the 3 middle perpendicular planes must contain a singlet, so the last singlet is in the center.
Fill in with the 6 squares, and you are done.

If you enjoy packing shapes into rectangles or boxes,
you might like my section on
polyominoes.