Peg Boards

Peg boards are simple puzzles, easy enough to construct that many farming families may still have some that great-grandfather made to keep the children busy, consisting of a wooden triangle, a particular number of evenly spaced holes, and a number of pegs one less than that number. A typical arrangement is 15 holes with 14 pegs, the holes arranged in rows of 1,2, 3,4, and 5.

The goal is to jump pegs in order to reduce the board to as few pegs as possible. Pegs are jumped by selecting a peg that is two spaces away from an empty hole, and “jumping” over the peg between it and the hole; the jumped-over peg is then removed from the board. This move can often strand a peg in an area surrounded by holes. As with many puzzles, opening moves are easy, but the more moves you make, the more complicated things become.

The player can usually arrange the peg board however he likes prior to play, and some arrangements are harder than others. Because of the possibilities of movement, with a standard 15-hole (5 × 5) triangle, the position with the fewest winning combinations (the hardest game) is with the empty hole “in the middle,” that is, in the center of the third row. Unbeknownst to most players, the position with the most winning combinations is in the middle of the left side: third row, first hole.

It can be helpful to know that there are only four possible end positions for the last peg in winning games (games that reduce the game to one peg): fifth row middle, fourth row either end (first or fourth hole), and the single hole in the top row. Of those, the fifth row middle is the easiest to reach. The worst possible game leaves 10 pegs on the board—this is actually hard to accomplish.

Math classes will occasionally discuss peg boards, but because the math involved is more complicated than is usually used in grade school, they are more likely to be used to demonstrate the potential of math than to suggest actual problems and solutions.