# What's the maximum number of pegs you can leave on a 15-hole triangular peg solitaire?

Peg Solitaire is mostly known to me as the triangular board with 15 holes and 14 pegs (mostly because of the restaurant Cracker Barrel). The rules are simple: pegs jump each other in straight lines, removing the jumped peg and leaving an additional space each move.

The usual end goal is to remove all but one peg, but that's boring. Instead, I'd like to know how many pegs you can end with, while no longer having a move to make. You may start with any space being the open spot.

Text example (X is peg, O is space)

    O
X X
X X X
X X X X
X X X X X


It can be done in

4

jumps.

Here is the solution:

      x            x            x            x           x
x x          x x          x .          x .         x x
x . x        x x x        x . x        x x x       x . x
x x x x      x . x x      x x x x      x x . x     x . . x
x x x x x    x . x x x    x . x x x    x . x . x   x . x . x


If you start with the hole elsewhere, you need more moves to reach an end position. Here are all the optimal end positions for each case, but without the move sequence leading up to them.

  7 jumps
.           .
x x         . .
x x x       x x x
x x x x     . . . .
x x x x x   x x x x x

7 jumps
x           x
x .         x .
x x x       x . x
x x x x     x . x .
x x x x x   x . x . .

8 jumps
x           x
x x         . x
x x .       x . x
x x x x     . . . x
x x x x x   . x . . x

x           x
x x         x .
x x .       x . x
x x x x     x . . .
x x x x x   x . . . x

and their mirror images.

• Your second 7 jumps answer looks flawed to me, unless it isnt the final form i am looking at, there are still 2 possible moves – Jason V Aug 3 '17 at 15:51
• I must confess I used a computer. I have a page about Triangle Solitaire on my website, and still had the computer program I used to calculate the number of positions at each depth. I merely had to add a print statement to output any board state that has no more moves available. This adds no insight, however I maybe could have found the optimal solution using the SAX resource count which is explained on George Bell's site. – Jaap Scherphuis Aug 3 '17 at 15:53
• @Jason: Thanks, I copied that down wrong. It's fixed now. – Jaap Scherphuis Aug 3 '17 at 15:54