On each corner of a square is a quarter. Your task is to have all four quarter heads-up or tails-up at the end of a turn.

You are blindfolded at the start, and you do not know which are heads-up and which are tails-up. Each turn, you may flip however many of them you want and then ask if you are done (and no, you cannot tell, by touch, whether it is heads- or tails-up). The square is then rotated a random, undisclosed number of quarter spins (multiple of 90 degrees), and you may take another turn.

Minimize the maximum number of turns required to be assured you will complete the task.

(In reply to Good start by Tristan)

That indeed must be the minimum.  At the beginning, even keeping track of the identities of the four corners (without the random turning), there are 2^4 = 16 combinations of flips, of which 2 will satisfy the task.  So 8 combinations must be tried (since it's blindfolded), with the 8 combinations chosen so as not to include complements (i.e., if the top left is to be flipped alone as one combination, then top-right/lower-right/lower-left as a combination is not to be tried).  No fewer combinations will work.

It's just amazing that a set can be made, not only blindfolded, but with random turns, and in the minimum, 8, yet.

Posted by Charlie on 2004-07-17 15:18:30