A subtraction game with moves 1, 3 and 4

Idea. A position is losing iff all its moves lead to winning positions. Compute the losing positions and find the pattern.

Claim. The losing positions are exactly the $n$ with $n\equiv0$ or $2\pmod7$.

Proof by strong induction on $n$.
- *From a position in $L$ every move leads outside $L$:* if $n\equiv0$, the moves lead to $n-1\equiv6$, $n-3\equiv4$, $n-4\equiv3$; if $n\equiv2$, they lead to $n-1\equiv1$, $n-3\equiv6$, $n-4\equiv5$. None of these is $0$ or $2$ mod $7$.
- *From a position outside $L$ there is a move into $L$:* $n\equiv1$: remove $1$; $n\equiv3$: remove $1$; $n\equiv4$: remove $4$; $n\equiv5$: remove $3$; $n\equiv6$: remove $4$.

Together with $n=0$ losing, positions in $L$ are exactly the losing ones.

Answer. The first player wins iff $n\not\equiv0,2\pmod7$. For $n=100\equiv2\pmod7$ the first player loses.

Lesson. Compute small cases, guess the periodic set of losing positions, then verify the two properties.