Penny Pinching

This week’s Riddler Classic is indeed a classic! Here it goes (paraphrased to make it a bit more general):

The game starts with $n$ pennies, which I then divide into two piles any way I like. Then we alternate taking turns, with you first, until someone wins the game. For each turn, a player may take any number of pennies he or she likes from either pile, or instead take the same number of pennies from both piles. Each player must also take at least one penny every turn. The winner of the game is the one who takes the last penny.

If we both play optimally, what starting numbers of pennies guarantee that you can win the game?

Here is my solution:
[Show Solution]

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