A puzzle consists of a number of tiles, T, colored red on one side and blue on the other.

Start with all red side up. The goal is to get them all blue side up in the fewest number of rounds.
A round consists of flipping exactly N of them.

Find a rule for when the puzzle is impossible for given values of (T,N) with N≤T.

Find a rule for the number of rounds it will take when the puzzle is possible.

It's certainly possible if N divides T, and the number of turns required is T/N.

It can be done if T and N are relatively prime; example: {N,T}={3,5} - but not always: {2,3}

It can be done if T and N have a common divisor; example: {N,T}={4,6} - but not always: {6,9}

It can be done if N is odd, and T exceeds N by 1: example: [N,T}={7,8}

Conjecture: It is not possible if T and N are relatively prime, and T is odd and  N is even. The converse is not implied.

Counter-examples welcome!

Edited on April 9, 2015, 9:57 am

Posted by broll on 2015-04-09 00:55:35