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.

There are 4 combinations of heads and tails for the quarters, not including inversions and rotations.

1  2  3  4
HH TH TT TH
HH HH HH HT

Likewise, there are also 4 combinations of flips that you can do each turn.

Each of the four combinations of flips can transform the four combinations of heads and tails to another combination (or the same combination). The chart below shows the possible resulting combination numbers, if you start with combination number shown on the top row and flip the combination number shown on the left column.

  | 1   2  3  4
1 | 1   2  3  4
2 | 2 134  2  2
3 | 3   2 14  3
4 | 4   2  3  1

Now, before minimizing the maximum, I'm going to try to just reach any maximum.

Flip#| combin.# | possible results
1    | 1        | 1,   2, 3,4
2    | 4        | -,   2, 3,1
3    | 3        | -,   2,14,-
4    | 4        | -,   2,-1,-
5    | 2        | -,1 34,--,-
6    | 4        | -,- 31,--,-
7    | 3        | -,-14-,--,-
8    | 4        | -,--1-,--,-

So I got it down to 8 turns.  Is it minimized?  I don't know.

Edited on July 17, 2004, 1:36 pm

Posted by Tristan on 2004-07-17 13:31:26