The golden coin game

In this take-away game you start with a pile of tokens and each turn remove between $1$ and $k$ of them; whoever takes the last token wins. You have already met the case $k = 2$, where the losing pile sizes are the multiples of $3$: whatever you take, the opponent completes the round to $3$ and hands the next multiple of $3$ straight back to you.

Nothing in that argument leaned on the number $2$. Whatever the largest legal take is, the opponent can always top the round up to one more than it, so the modulus travels with $k$. Work out which multiples are the losing ones for this game before you start playing, and messy play becomes a clean rule.