IBM Ponder This #315 · July 2024

This riddle was suggested by Evert van Dijken - thanks, Evert!

Consider square tiles made of $4\times 4$ squares that can be either black (0) or white (1), such that each tile has an equal number of black and white squares.

We call two tiles equivalent if one can be rotated and/or reflected in order to obtain the other. It is easy to see that a tile can have 1, 2, 4 or 8 equivalent tiles (including itself).

Your goal: Find a tiling of a 16\times 16 squares board (4 x 4 of such tiles, with 4 x 4 squares in each tile) such that:

1. In each row and column, there are no more than two consecutive squares of the same color.

2. The total number of pairs of adjacent same color squares is minimal. Remember to count both vertical and horizontal pairs.

3. All tiles included in the tiling must be non-equivalent

4. As stated earlier, each tile must have the same number of black and white squares. Give your solution by first writing the number of pairs from constraint 2, and then the $16\times 16$ grid of squares (no need to describe the individual $4\times 4$ tiles). e.g.

113
1101001010101001
1010110101010110
0101001010101001
0010110101010110
1101001010101101
0010110101010010
0101100110110100
1010011001001011
0101100100110100
1001001010010010
0010010101101101
1101101101101011
0010010010010010
0011001100110100
1100110101001011
1011011011011011

A bonus "*" will be given for solving the above problem, but for a $20\times 20$ squares board with the additional constraint that one can only use tiles which have exactly 4 equivalent ones as defined above.

Solution

Best opened after a real attempt

To be added.