Imaginary Simplicity

Multiplication with the imaginary number √-1 accomplishes a counterclockwise 90° rotation, and that’s what today’s puzzle is based on. As casual readers will know, I enjoy simple things that rapidly get complicated. This puzzle is played on a rectangular checker board, with checker pieces in various colors placed on the squares. A move consists of rotating a checker about another checker by 90° either way. Below is the puzzle graph for a 3×3 board using two adjacent checkers, showing all possible positions and the possible moves indicated by edges that are colored by the checker you rotate about:

2example 01

It’s clear that in this case the two checkers have to stay together, and this poses no real challenge: Dealing with just two checkers is simple. Let’s add another one. Below you can see all possible six moves from the central position. You can either rotate about green or blue, but not (yet) about red.

Ex3colors3x3 01

The puzzle graph in this case is connected and has diameter 7. You will notice that the checkers cannot change their parity. As simple count will tell you that there are therefore 80=5x4x4 possible positions or vertices. Below is a simple example how a puzzle could look like, with an optimal solution: Use as few moves as possible to get from the left position to the right one.


Naturally, things get trickier with more checkers and larger boards. Below is an optimal solution for a 4 checker problem that realizes the diameter 7 of the game graph (that has 240 vertices).


Finally, here is a problem on a 3×4 board with 4 checkers. The shortest solution takes 9 moves, and takes place on a game graph with 900 vertices. Having choices is hard, isn’t it?


There are many variations possible. One can, for instance, designate left and right handed checkers that can only rotate one way, making the puzzle graph directed. One can also turn this into a two person game by letting two players take turns. More about this at a later point.


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