Just got a new group-theory puzzle, called Tsukuda's Square. It's
sort of like the 15/16 puzzle, but harder. It consists of a 4x4 matrix
in the center with the numbers from 1 to 16. On the left side are
4 plungers, one for each row. At the top is one plunger, which pushes
down columns 2, 3, and 4, all at once. Both the top and the side plungers
push the rows or columns over 1 square; releasing them causes the same row(s)
or columns to slide back. When the top plunger is pushed down, the number
1 row plunger cannot be pushed because of interference; that is the only
A typical move, for instance, would be to push in the #2 row plunger,
push the top, release the #2, release the top. This would change
1 2 3 4
5 6 7 8 (only looking at the top 2 rows)
1 3 4 8
2 5 6 7.
You can push 2 or 3 (or even4) side plungers at once. Pushing an
adjacent pair is even useful in solving this thing.
At any rate, it has the normal even parity problem - that is, you
cannot exchange a single pair, but you can exchange 2 pairs, or permute
three squares. Because of the limited moves, it is not nearly so simple
as it looks. I can permute 3 in one specific position, and it takes me
12 moves to do so (where a move is a push of 1 or more side plungers,
then push the top, then release them, or the reverse [push the top first]).
Can anyone develop some more efficient algorithms?