## The chocolate bar game: losing positions in binary

Recall the chocolate bar game from my last post, whose winning and losing positions can be visualized like this:

Here’s a list of some losing positions on or above the main diagonal (dark blue squares in the above picture), ordered by $x$-coordinate, along with their binary representations. Since the game is symmetric, if $(x,y)$ is a losing position then so is $(y,x)$. What patterns do you notice? Can you connect them to the above visualization?

x y x (binary) y (binary)
1 1 1 1
1 3 1 11
1 7 1 111
1 15 1 1111
1 31 1 11111
2 2 10 10
2 5 10 101
2 11 10 1011
2 23 10 10111
2 47 10 101111
3 3 11 11
3 7 11 111
3 15 11 1111
3 31 11 11111
4 4 100 100
4 9 100 1001
4 19 100 10011
4 39 100 100111
5 5 101 101
5 11 101 1011
5 23 101 10111
6 6 110 110
6 13 110 1101
6 27 110 11011
7 7 111 111
7 15 111 1111
7 31 111 11111
8 8 1000 1000
8 17 1000 10001
8 35 1000 100011
9 9 1001 1001
9 19 1001 10011
9 39 1001 100111
10 10 1010 1010
10 21 1010 10101
11 11 1011 1011
11 23 1011 10111
12 12 1100 1100
12 25 1100 11001
13 13 1101 1101
13 27 1101 11011
14 14 1110 1110
14 29 1110 11101
15 15 1111 1111
15 31 1111 11111
16 16 10000 10000
16 33 10000 100001
17 17 10001 10001
18 18 10010 10010
19 19 10011 10011
20 20 10100 10100
21 21 10101 10101
22 22 10110 10110
23 23 10111 10111
24 24 11000 11000
25 25 11001 11001