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?