# Transmuting Pentominoes

Even the 12 pentominoes themselves can be quite fascinating, ignoring how they might be assembled into a shape. The fact that there are 12 different pentomino shapes no doubt has helped them gain popularity among puzzlers. A dozen pieces is enough to earn a complexity that tantalizes someone, but not so great that the number of combinations becomes daunting.

Each being composed of 5 squares thus affords a total of 60 cells when building shapes. 60 is quite a friendly number in the sense that it has many divisors and forms many possible rectangles: 6x10, 5x12, 4x15, and 3x20 -- all of which can, in fact, be formed with the pieces.

 I-pentomino L-pentomino Y-pentomino
I've also been interested in how the pieces, their shapes, relate to one another. I'm sure there's a game or a puzzle waiting to happen here, but I haven't yet figured it out. But it's still worth noting how easily one pentomino may be "transmuted" into the others.

I'll define "transmuting a pentomino" as taking said shape, removing a single square from it, and then replacing that square so as to form another pentomino (if placed in the original position, the original pentomino is reformed, of course).

For example, the long, straight I-pentomino can be transmuted by taking one of its squares off its end and reattaching the square elsewhere. There are three (non-trivially) distinct possibilities:

1. Place the square back on one of the ends (either the original place or the other end of the currently 4-long line of squares). This obviously reforms the I-pentomino.
2. Place the fifth square adjacent to one of the end squares, but on the "side" of the 4-long line of squares. This will form the L-pentomino in some orientation.
3. Place the fifth square along the edge but closer to the center of the line of 4 squares remaining. This will form a Y-pentomino in some orientation.
These examples are illustrated to the right (the red square is the replaced piece -- the symmetric cases should be clear).

Note that the L- and Y-pentominoes are the only ones that can be transmuted from the I-pentomino in one step (other than reverting back to the I- itself). I will say that those two pieces are "1 degree (of transmutation)" away from the I-pentomino (the I-piece is 0 degrees away from itself).

This terminology allows us to then show how closely all of the pieces are "related" to one another in terms of degrees of transmutation. Here is a table:

Degrees of Transmutation Between Pentominoes
F I L N P T U V W X Y Z sum
F 0 2 1 1 1 1 1 1 1 1 1 1 12
I 2 0 1 2 2 2 2 2 3 2 1 2 21
L 1 1 0 1 1 1 1 1 2 2 1 1 13
N 1 2 1 0 1 1 1 1 1 2 1 1 13
P 1 2 1 1 0 1 1 1 1 1 1 1 12
T 1 2 1 1 1 0 2 1 2 1 1 1 14
U 1 2 1 1 1 2 0 1 2 2 1 1 14
V 1 2 1 1 1 1 1 0 1 2 1 1 13
W 1 3 2 1 1 2 2 1 0 2 2 1 18
X 1 2 2 2 1 1 2 2 2 0 1 2 18
Y 1 1 1 1 1 1 1 1 2 1 0 1 12
Z 1 2 1 1 1 1 1 1 1 2 1 0 13

The sum column to the right gives an indication of the overall "friendliness" of a piece to its peers. The F-, P-, and Y- shapes are most easily transmuted to all of the other shapes, while the I- is certainly the most difficult. The I- and W- pentominoes are the most distantly related: the only combination requiring 3 steps of transmutation.

Again, I'm not sure where all of this information leads, but it does keep my mind occupied at times.

- Eric Harshbarger, 12 January 2005