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A little puzzle I came up with yesterday: You have a queen, a king, a knight, a rook and a bishop. What is the smallest rectangular chessboard (i.e., of minimal area) you can position them on, such that every piece attacks exactly one other piece, every piece is attacked by exactly one other piece, and such that the attacks form a cycle of length 5? (so basically Rock-Paper-Scissors style)

It has a cool name! Just wanted to point that out

I think I figured it out, though I should probably doublecheck. I may have dismissed some scenarios a bit too quickly.

Are you taking anwers on PM?

Forseti wrote:

Are you taking anwers on PM?

Sure.

Forseti's solution is correct.

There are in fact 2 distinct solutions of this size.

In the meanwhile I figured out Erich Friedman dealt with a similar (broader) question in his Math Magic of February 2007 (edit: the linked page is now updated and thus contains spoilers), but these optimal solutions are not in his results.

Also, as a bonus question: He lists a 5x5 solution for the cycle "knight->queen->rook->king->bishop->", but it is not optimal. Can you improve it?

I can do it slightly smaller, just one less square.

That's the best I can find too. 1 square smaller.

One square less is indeed the best one can do.

I might as well post the proof:

An optimality proof should demonstrate it can not be done in a rectangle of area smaller than 24. Let's first draw the queen and the knight attacking it and mark the 7 other spots attacked by the knight as blocked.

The rook should be attacked by the queen diagonally, otherwise it attacks it back. Mark as potential positions for the rook, all positions under attack by the queen diagonally that do not make the configuration span a rectangle of area more than 23 and that are not direct horizontal or vertical neighbors of the knight and that are not attacked by the knight. I count only 6 of them.

Also we can mark all other squares under attack by the queen and not making them span more than 11 in any direction, as blocked, because she should only attack one piece (the rook). In one of the potential positions of the rook it attacks the knight with no possibility for the king to interfere. In one of them its attack on the knight can be blocked by the king, but at such a close range the king attacks the rook as well. Both rook positions can be marked as blocked.

Interestingly, there are now only 4 positions left from where the bishop could attack the knight while not spanning a rectangle of area larger than 23 with the queen, the knight and any of the 4 potential rook positions.

One bishop position is under attack by two neighboring rook positions, while the other two rooks are so far away that the configuration would span an area of 25. Eliminate this bishop position. Now one rook can not be paired with any of the remaining bishop positions without spanning a too large rectangle, so this position can also be eliminated.

The king should be a neighbor of the bishop and in line with a rook, but it should not attack the knight or the rook. This is not possible any longer. This concludes the proof.

That's very much like what I did, except I went with a 24 square limit instead of 23 squares, to actually construct a solution of less than 25 squares. (That was the bonus question.

) As a by-product, I found that there wasn't a smaller one than 24.

Interesting link, and I'm impressed that a page dated februari 2007 is still being updated so promptly.