Why some passing moves have exactly one solution

By Marco Pereira · Jun 10, 2026

Turn a pitch into a constrained problem and the number of working orders collapses fast. Here is how constraints stack up to produce exactly one solution.

An open pitch offers hundreds of passing options. Add a requirement that every player must touch the ball exactly once and the count drops sharply. Add a wall of defenders, pass-line blocking, and a clean finish, and most boards have a single order that works.

That is not magic. It is the compound effect of overlapping constraints, each of which rules out a large fraction of the remaining possibilities.

The finish is the tightest constraint

Only one player takes the final shot, and the lane to goal has to be completely clear of defenders and earlier passes. That rules out most of the team immediately. Of the candidates who could realistically shoot, only one or two can reach the goal without crossing a line you have already drawn.

Working backwards from this tight constraint is faster than working forwards from the start, because it tells you who has to hold the ball last, and that determines what the second-to-last pass must be. Every board becomes easier once you have identified the finish.

Corner players run out of options quickly

A player standing near the touchline has fewer safe passing lanes than one in the middle of the pitch, because half their potential targets are blocked by the sideline or offside. Fitting those awkward players into the sequence early is usually the only way to get them in at all.

If you leave a corner player to the end, you often find that every pass to them crosses a line you drew earlier, or their only available onward pass has already been used. The move dies, and you have to unwind several steps. The pattern holds: constrained positions get served first.

Pass lines become obstacles

Once you draw a pass, that line blocks every future pass that would cross it. On the harder boards a move can look clear until the fourth or fifth touch, when you realize the pass you need is cut off by a line from two moves ago.

This is the constraint that stacks most sharply as the puzzle grows. In a four-player board you might never collide with your own lines. In a seven-player one you definitely will, and in a ten-player one the board is a lattice of used lanes. Solving it means planning the sequence so the lines you still need remain open.

Defensive pressure narrows the safe passes

A defender standing between two players blocks the direct pass, forcing it to route around them or skip them entirely. That rerouting costs options elsewhere, because now a different player has to receive first, and their onward pass might block something you needed later.

The tightest defences cluster in front of goal, which is also where you are trying to finish. Threading the ball through that zone without hitting a defender or doubling back is the hardest part of most boards, and it is why the naive approach, picking the shortest pass each time, almost never works.

The single-solution threshold

Small boards, three or four players, often have multiple solutions because the constraints are loose. Add two more players and the number of solutions drops from several to one or two. Add another two and boards with any solution at all become rare unless they are designed for it.

Machine-checking a board means trying every possible ordering and seeing which ones obey all the rules: every player touched, no defender or earlier line crossed, and a clean shot at the end. For a seven-player board that is five thousand orderings. For a ten-player one it is three million. Most of them fail on the first pass.

Key takeaways

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