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.

Well, nobody has tried yet to find a rule for the number of rounds it will take.

As a start, I considered N = 3.

When T = 3k, it will take k turns.

When T = 3k+1, the minimum possible is obviously k + 1 turns, where one tile is flipped 3 times and the remaining 3k tiles are flipped once. But this is not possible if T = 4, because this scheme requires 2 "once-flipped" tiles to be flipped each time the "thrice-flipped" tile is flipped, and there are only 3 "once-flipped" tiles available . (T,N) = (4,3) requires 4 flips.

When T = 3k+2, the minimum possible is obviously k + 2 turns, where two tiles are flipped 3 times and the remaining 3k tiles are flipped once. This does work for T = 5, because with this scheme only one "once-flipped" tile needs to be flipped on each turn.

So for N = 3, the formula is

When T < 3 IMPOSSIBLE

When T = 4 4

Otherwise FLOOR(T/3) + MOD(T,3)

*Edited on ***April 10, 2015, 12:55 pm**