Peg Solitaire — the classic 33-hole board, free in your browser

Thirty-three holes in the shape of a cross, thirty-two pegs, and one gap in the middle. A move is a jump: pick up a peg, hop it over a neighbour — up, down, left or right, never diagonally — and land in the empty hole directly beyond. The peg you jumped over comes off the board. Do it again. And again, for as long as a legal jump exists.

This is the English board — the 33-hole cross of the classic wooden set, not the 15-hole triangle — and it is far harder than it looks. Your score is the number of pegs still standing when nothing can move, so lower is better and one is perfect. Most first attempts strand five to ten pegs in different corners of the cross. Free, no download, unlimited undo, and it plays with two taps.

How to play

• Tap a peg to pick it up. Every hole it can legally jump to lights up green.
• Tap a green hole to jump. The peg you jumped over is removed from the board.
• Tap the selected peg again to put it down, or tap a different peg to switch.
• Undo is unlimited — step back as far as you like while the run is still alive.
• The run ends the moment no legal jump exists, and your score is the pegs left. Fewer is better; one is perfect.
• On a keyboard: arrow keys move the cursor, space or enter picks up and drops, U undoes.

Why the last peg has to land in the middle

Every jump removes exactly one peg, so the arithmetic is fixed before you start: 32 pegs down to 1 is exactly 31 jumps, and the counter in the corner is really telling you how far you have left to fall. What is much less obvious is that the finish is not free — you cannot leave that final peg wherever you like.

Number the rows and columns 0 to 6 and colour every hole by (row + column) mod 3. That splits the 33 holes into three interleaved diagonal stripes of exactly 11 each. Now look at what a single jump does: any jump — horizontal or vertical — spans three consecutive holes, and those three always carry all three different colours. So a jump takes one peg off two different stripes and puts one peg onto the third. All three counts change by one, which means all three parities flip together, on every single move. Track that from the opening position through 31 jumps and it forces the last peg onto a hole whose (row + column) is a multiple of 3. Repeat the whole argument with (row − column) mod 3 and you get the same constraint on the other diagonal. Only five holes on the board satisfy both: the centre, and the four tips at the far end of each arm.

Those four tips are then ruled out by a second classical tool, the pagoda function — a weighting of the holes chosen so that no jump can ever increase the total weight, which makes any finish that would require a heavier board than you started with impossible. A brute-force sweep of the board's 23,475,686 genuinely distinct positions agrees: no arm-tip finish exists. What survives is a single hole. If you get down to one peg, it is sitting in the centre — the hole you started by emptying. That is the perfect game, it genuinely exists, and this board will tell you when you have found it.

Three things actually help you get there. First, think in packages, not in single jumps. Compact clumps of pegs clean themselves up; scattered ones do not. The classic advice is to work in blocks of about six and clear each block completely before you open the next, rather than taking whatever jump happens to be available. Second, sweep the arms inwards. Each arm of the cross is a dead end, so pegs out there can only be resolved by pulling them toward the middle — nibbling at the edges of the centre first leaves the arms full and unreachable. Third, and this is the rule that decides most games: a peg with no orthogonal peg beside it can never move again. It has nothing to jump over, and nothing can jump over it. Every one of those is a point on your final score, locked in the moment you create it, so before you take a jump, check what it leaves stranded behind you.

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More puzzles with a provable best answer: the Tower of Hanoi and Lights Out. Or browse all games.