Problem #WSP-000467

Problem

Naomi and Rory get tired of playing Nim, so decide to change the rules to mix it up. They call their new variant ‘Wonim’. There are two piles of four matchsticks each. They take it in turns to take matchsticks. Each player has to take at least one matchstick, and they can take as many as they like from one pile only.

Except, their new rule is that a player cannot take the same number of matchsticks that their opponent just did. For example, consider Wonim($5$,$10$). If Naomi’s first move is to take $4$ matchsticks from the pile of size $5$, turning the game to Wonim($1$,$10$), then Rory cannot take $4$ matchsticks - he has to take more or less. A player loses if they cannot go - this can happen if there are no matchsticks left, or if there are matchsticks left, but they can’t take any since their opponent took that number. e.g. Wonim($1$,$1$), Naomi takes $1$, Rory faces Wonim($1$) but can’t move since he’s not allowed to take $1$.

In the game Wonim($4$,$4$) with Naomi going first, who has the winning strategy?