# How to Solve a Triangle Peg Board Puzzle

By Kira Jaines

The 15-hole triangle peg board game is a modern version of a game that has been played in Europe since the end of the 17th century. In the United States, Herbert M. Smith patented a triangular version of the game in 1891. It is also known as peg solitaire or the Cracker Barrel puzzle. The puzzle gained in popularity when the restaurant put one on every table to amuse patrons waiting for their food. The game consists of 14 pegs set in a triangle shape with 15 holes like bowling pins, but with one more row. One of the spaces in the triangle is left empty, and the object is to jump pegs, removing each peg jumped, until you are left with only one.

Place 14 pegs in the 15-hole triangular board, leaving one hole empty. In your mind, number the holes from 1 to 15, going left to right, so that 1 is the tip of the triangle and 15 is the right-most hole in the bottom row.

Remove the peg from hole 1, leaving the tip of the triangle empty. Jump pegs in the following order, removing each peg jumped: from 4 to 1, from 6 to 4, from 1 to 6 and from 7 to 2. You will be left with holes 1, 2, 3 and 5 empty in a diamond shape. Jump from 13 to 4, from 2 to 7, from 10 to 8, from 7 to 9, from 15 to 13, from 12 to 14, from 6 to 13, from 14 to 12 and from 11 to 13. This solution will also work with starting points at holes 11 and 15 by rotating the board to place those holes at the top.

Repeat Step 1. Remove the peg from hole 2. Jump from 7 to 2, from 13 to 4 and from 11 to 13. Rotate the board to the right so that hole 11 now becomes hole 1, and you will see the same diamond-shaped pattern of vacant holes. Continue playing as in Step 2, from the fourth move. By mirroring the first three moves of Step 3, this solution will also work from a beginning empty position at holes 3, 7, 10, 12 and 14.

Repeat Step 1. Remove the peg from hole 4. Jump from 13 to 4, from 10 to 8 and from 15 to 13. Rotate the board to the left, so that hole 15 is now hole 1. Note the diamond shape of empty holes and continue from the fourth move in Step 2. With mirroring of the first three moves, this solution will also work for vacant positions at holes 6 and 13.

Repeat Step 1. Start at the midpoint of an edge for another possible solution. Remove the third peg from any one side, so that there are two pegs on either side of a vacant hole. Align the board so this vacant hole is on the bottom of the triangle, and is, specifically, hole 13. Jump from 11 to 13, from 14 to 12, from 4 to 13, from 1 to 4, from 12 to 14, from 6 to 13, from 15 to 6, from 7 to 2, from 2 to 9, from 3 to 10, from 10 to 8, from 14 to 12 and from 12 to 5.

Repeat Step 1. Add variety to the Step 2 solution by again leaving the tip of the triangle empty. Jump the pegs from 4 to 1, from 6 to 4, from 15 to 6, from 3 to 10, from 13 to 6, from 11 to 13, from 14 to 12, from 12 to 5, from 10 to 3, from 7 to 2, from 1 to 4, from 4 to 6 and from 6 to 1. You will be left with one peg in hole 1, which was vacant at the beginning of the game.

Repeat Step 1. Challenge yourself by beginning from the interior. Remove the peg at hole 5. Jump from 12 to 5, from 14 to 12, from 10 to 8, from 3 to 10, from 2 to 9, from 7 to 2, from 15 to 6, from 6 to 13, from 12 to 14, from 1 to 4, from 4 to 13, from 14 to 12 and from 11 to 13. This solution works backward from the solution in Step 5.