Breakthrough — the Game That Cannot Draw
Play Breakthrough against a computer, and see why it has no repetition rule: pawns only move forward, so no position can ever recur.
Pawns move one square straight forward into an empty square, or diagonally forward to capture. Reach the far row — or take every enemy pawn — to win. You are blue, moving up.
Pawns only go forward, so no position can ever repeat
Add up how far every pawn has come from its own back row. On a quiet move that total rises by exactly one — checked over 5,778 of them without exception, because a pawn advances one row and nothing else moves.
A capture breaks it, and that is worth being precise about rather than waving through: taking a pawn deletes whatever advancement it had carried, so the total can fall. The quantity that never decreases is
advancement + (rows − 1) × captures so far
because a captured pawn can have been carrying (rows − 1) at the very most, so the constant paid per capture always covers what was lost. That rises strictly on every move of every kind — 7,057 moves checked, none flat, none falling. A number that only goes up can never return to where it was, so Breakthrough has no repetition rule because it cannot have a repetition. The game cannot be drawn.
And who wins depends on the ranks, not just the size
| Board | One rank each | Two ranks each |
|---|---|---|
| 4 × 3 | first player | second player |
| 5 × 3 | first player | second player |
| 6 × 3 | first player | first player |
| 4 × 4 | first player | second player |
| 5 × 4 | first player | not solved |
| 6 × 4 | first player | not solved |
With a single rank the first player wins every board that could be settled — six for six. Add a second rank and it flips: the second player wins 4×3, 5×3 and 4×4, and then it flips back on 6×3 once the board is tall enough for the extra pawns to be worth using.
So there is no rule of thumb, and stating one from the three smallest boards would have been wrong. 5×4 with two ranks runs past four million positions and is left open here rather than guessed at — the boards above that size are played by search, not solved.
How to use
- Move a pawn one square straight forward into an empty square.
- Or one square diagonally forward to capture.
- Reach the far row, or take every enemy pawn, to win.
- Try one rank against two — the winner changes.
Frequently asked questions
What are the rules of Breakthrough?
Every piece is a pawn. It moves one square straight forward into an empty square, or one square diagonally forward to capture. You win by getting any pawn to the far row, or by taking every enemy pawn. There is nothing else to learn.
Can Breakthrough end in a draw?
No, and not by convention — it is impossible. Pawns only ever move forward, so a quantity that measures total progress rises on every single move, and a number that only goes up can never return to a value it has already had. No position can repeat, so there is nothing to declare a draw about.
Does the total advancement really rise every move?
On a quiet move, yes, by exactly one. A capture is different: taking a pawn deletes whatever advancement it was carrying, so the raw total can fall. The quantity that never decreases is advancement plus one board-height per capture, since a captured pawn can have been carrying at most that much.
Who wins Breakthrough with perfect play?
On the small boards that can be settled exactly, it depends on how many ranks of pawns you start with. With one rank the first player wins all six boards solved here. With two ranks the second player wins four by three, five by three and four by four — and then the first player wins six by three.
Why does the winner flip with an extra rank?
Because the extra rank is not simply a defensive bonus, it is an advantage that needs room to be used. On short boards it slows the attacker down and the defender profits; make the board taller and the attacker has space to bring the extra pawns to bear, and it flips back.
Is the full-size game solved?
No. Even five by four with two ranks runs past four million positions, and the standard eight by eight is far beyond exact solution, so the opponent on larger boards searches rather than knows. The table on this page states only what was actually settled.
How long can a game last?
Not long. Because progress only ever increases and has a ceiling of one row short of the board per pawn, the game is bounded from the start, and in practice ends far sooner than that limit.
Does this send anything anywhere?
No. The game and the search both run entirely in your browser.
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