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Many of you may be familiar with a puzzle game that consists of a 15-peg triangle and is dubbed as "The Original IQ Test". The idea behind the game is that you fill 14 of the 15 holes with pegs, and the way to play is to "jump" another peg into an empty space adjacent to the peg that is being jumped over. After the jump is completed, the peg that is jumped over is removed from the triangle. The goal of the game is to jump the pegs in such a way that in the end, there is only one peg left.

My questions are:

  1. Is there a solution for every starting configuration i.e. no matter what hole is left unfilled in the beginning? If so, are there solutions for each hole for every $n$-peg triangle?

  2. Is there a definitive algorithm for completing the puzzle no matter what peg is left unfilled in the beginning?

  3. Is there only 1 unique solution for every starting configuration?

I have played around with the puzzle for a little bit, but I haven't been able to complete it with one peg left but rather two. Any help with this problem would be appreciated, thanks!

Here is a link to the picture of the board:

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Can you provide a picture, or a link to one of these as its not obvious from your description exactly how these holes are arranged. – Warren Hill Sep 4 '13 at 14:57
Just added, sorry for the confusion! @WarrenHill – joejacobz Sep 4 '13 at 15:00
If the initial missing peg can be placed at a that still allows a reflective symmetry such as the top peg then once you have found a solution imagine a line through this symmetry and try all the mirror image moves. This will solve the puzzle so the answer to 3 is No: there can be different solutions to some (if not all) starting positions. I have no idea about 1 and 2 but would be interested to find out. – Warren Hill Sep 4 '13 at 15:18
Ok, thank you! @WarrenHill – joejacobz Sep 4 '13 at 15:25
For a reference look at JD Beasley's book "The Ins and Outs of Peg Solitaire" or there is a chapter in "Winning Ways" by Berlekamp, Conway and Guy (which is less extensive). – Mark Bennet Sep 4 '13 at 16:04
up vote 5 down vote accepted

This is one of the classic peg solitaire layouts. Many resources are available on the web. The page I reference, under The Triangle (5) board, points to a solution and discusses what starting and stopping hole combinations are possible.

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