Triangle peg game solutions

A perfect game, leaving one peg, from every one of the 15 starting holes. Each solution was found and checked by computer: step through it on the diagram, or play it out yourself on the real board.

Holes are numbered 1 to 15, row by row from the top, and each jump is written as the hole it starts from and the hole it lands in: 4-1 means the peg in hole 4 jumps over hole 2 into hole 1. Pick a starting hole in each group to see its solution. Click any jump in the list to go straight to it.

Trying to work it out yourself first? Read the strategy guide instead; there are no spoilers there.

Starting from a corner (holes 1, 11, 15)

The classic start. The two possible first jumps are mirror images of each other, so it doesn't matter which you pick. Of the 568,630 possible games, 29,760 end with one peg.

Starting next to a corner (holes 2, 3, 7, 10, 12, 14)

Careful with the first jump here: of the two available, only the one along the edge keeps a perfect game in reach. From hole 2, 9-2 is already a mistake.

Starting from the middle of a side (holes 4, 6, 13)

The easiest start. All four first jumps work, and more of the possible games end with one peg than from any other hole.

Starting from an inner hole (holes 5, 8, 9)

The hardest start. Only 1,550 of 137,846 possible games end with one peg, and the last peg can only finish in one hole: from hole 5, that's hole 13.

Challenge: finish where you started

For a harder goal, try to leave the last peg in the very hole that was empty at the start. It's possible from the corners, the holes next to a corner and the middle of each side: 12 of the 15 holes. From the three inner holes it can't be done. Here's how it goes from hole 1, ending back in hole 1:

Got it? Now play it without looking.