Triangle peg game strategy
Most people who play the triangle peg game leave three or four pegs. Leaving one is a matter of a few habits, a good choice of starting hole and a little planning near the end.
1. Choose your starting hole
The triangle looks the same from all three sides, so the 15 holes fall into four groups that play identically: the three corners, the six holes next to a corner, the middle of each side, and the three inner holes. Every one of them can be solved, but they're far from equally easy.
- Middle of a side (holes 4, 6 and 13) is the friendliest. You get 4 different first jumps, none of them a mistake, and 7.4% of all possible games from there end with one peg, more than from anywhere else.
- A corner (holes 1, 11 and 15) is the traditional start and a fair one: about 5% of games end with one peg.
- Next to a corner (holes 2, 3, 7, 10, 12 and 14) is a little trickier, because one of its two first jumps already makes a one-peg finish impossible (see below).
- An inner hole (5, 8 and 9) is the real test. Only 1,550 of 137,846 possible games end with one peg, about 1 in 89, and the last peg can only ever finish in one particular hole.
2. Get the first jumps right
From a corner or the middle of a side, every first jump keeps a perfect game possible. From a hole next to a corner, it's different. Start with hole 2 empty and you have two choices: 7-2, up the left edge, or 9-2, across the middle. Only the first keeps one peg within reach.
The pattern is worth remembering: early on, prefer jumps that clear pegs from the edges over jumps that empty the middle. The middle holes are your highways. Every edge peg will need them later.
3. Deal with the corners and edges early
A peg in a corner can never be jumped over, because there's no hole beyond it to land in. The only way to remove it is for the corner peg itself to jump out, along one of the two edges. That needs a peg beside it and an empty hole two along. Pegs on an edge have a similar problem: they can only be jumped by a peg travelling along that same edge.
So corners and edges are where games get stuck. Each time you jump, glance at the three corners and ask whether each corner peg still has a way out. If you leave a corner peg alone until late, the neighbours it needs are often already gone.
4. Keep the pegs together
A peg with no neighbours can't jump and can't be jumped. Unless another peg later lands right next to it, it will still be there at the end. The surest way to finish with several pegs is to scatter them across the board.
Prefer jumps that land next to other pegs, and be wary of jumps that leave a peg on its own, especially far from the rest. A compact group of pegs can usually be whittled down; scattered pegs can't.
5. Plan the last few pegs
By the time four or five pegs remain, there are few enough possibilities to work them out in your head before you move. Look for pegs that will need a partner and make sure the partner will still be there. The very last jump needs two pegs side by side with an empty hole beyond.
Use the tools while you learn. With helpers on, the game tells you the moment a one-peg finish becomes impossible. Undo that jump and look for another. After a few games you'll start to see the mistakes before you make them.
By the numbers
The triangle is small enough for a computer to play out every possible game. Counting each complete sequence of jumps as one game, there are 7,335,390 games from the 15 starting holes, and 438,984 of them end with a single peg. Here is how the four groups compare:
| Starting hole | Possible games | End with one peg | Random jumps solve it | Worst finish |
|---|---|---|---|---|
| Corner (1, 11, 15) | 568,630 | 29,760 (5.2%) | 0.69% of the time | 8 pegs |
| Next to a corner (2, 3, 7, 10, 12, 14) | 294,543 | 14,880 (5.1%) | 0.34% of the time | 8 pegs |
| Middle of a side (4, 6, 13) | 1,149,568 | 85,258 (7.4%) | 0.71% of the time | 7 pegs |
| Inner hole (5, 8, 9) | 137,846 | 1,550 (1.1%) | 0.17% of the time | 10 pegs |
"Random jumps" is the chance of leaving one peg if, at every turn, you pick one of the available jumps at random. Whichever hole you start from, it's well under 1%, and the most likely result is 4 pegs. That is why the game has its reputation, and why a little strategy goes such a long way. The worst finish is the most pegs a game can get stuck with: starting from an inner hole, it's possible to run out of jumps with 10 pegs still on the board.
Want to see a perfect game? The solutions page steps through one from every starting hole. Or put these ideas into practice: play a game.