Peg Solitaire Boards

When analyzing chessboards, we are often aided by the fact that the squares alternate in color, so a similar coloring scheme could help here, too. However, since every jump involves three horizontally or vertically adjacent holes, we use a third color, gray, and color the boards so that any one jump involves one hole of each color. The image below shows a possible such coloring for each given board.

In every jump, two filled holes become empty and one empty hole becomes filled. Because each jump involves one hole of each color, this means that the number of filled holes for two of the colors will decrease (the ones containing the "jumper" peg and the "jumped" peg) and the number of filled holes for the third color will increase (the one where the "jumper" peg lands.) But this also means that the parity (whether a number is odd or even) of the number of filled holes for every color will change with every jump.

Every board starts with every hole filled with a peg, except for the center hole. We count the starting number of filled holes of each color for each of the four boards.

• Board A: ${\color{#FFFFFF} \text{filler}}$ 13 Black, 10 White, 13 Gray
• Board B: ${\color{#FFFFFF} \text{filler}}$ 18 Black, 20 White, 18 Gray
• Board C: ${\color{#FFFFFF} \text{filler}}$ 22 Black, 20 White, 22 Gray
• Board D: ${\color{#FFFFFF} \text{filler}}$ 24 Black, 20 White, 24 Gray

We are looking for a board that could end with a single peg, i.e. with one color having an odd number of pegs (1) and the other two colors having an even number of pegs (0). Note that with boards B, C, and D, all three colors start with the same parity. But then after every jump, all three colors will still have the same parity, so it is impossible to end with just one single peg on any of these three boards. Thus board A, which starts with two colors having an odd number of pegs and one color having an even number, is the only board that could end up with a single peg.

[Of course, strictly speaking, we'd still have to show that at least one such game is possible on board A.]

Only Board A is null-class, and so is the only one for which a complementation solution might be possible. This board is known as Huber's 37-hole board or, more prosaically, as Square(3) + (3,3,1) , and a solution for this board can be found here .

I counted which had an odd number of blank spaces

I thought of geometrical shapes and the concept of war. That shape is similar to Sparta formation.

Try the game bcuz it actually works