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I read once about a variant of the game Sprouts called Brussels Sprouts. Instead of placing dots on a plane, one places $n$ $+$ signs instead. Each player, in turn, connects any two free legs, either on the same $+$ sign or on different signs, but moves that would cross a pre-existing line (or $+$ sign) are not permitted. A new hash mark is drawn normal to the newly drawn line, providing two new free legs. The game ends when a player cannot move -- the other player is then declared the winner.

The name is a deliberate joke. That's because, according to the article I read, no matter what moves are made, each game will always have exactly $5n-2$ moves. I have no idea how to prove this but it seems to be correct for the examples I checked.

Does anyone know how to prove this claim?

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    $\begingroup$ There is a proof here. $\endgroup$ – David May 29 at 1:35

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