Finding p positions with 2 subtraction sets in the take-away game

Question

Find the set of P-positions for the takeaway game with the subtraction sets: $S = {1,3,5,7}$ $S = {1,2,4,8,16,32}$ Who wins each game when there are 100 tokens on the table to start, the first or the second player?

Would the P positions just be every position other than 6 and multiples of 6?

And would the first player always win with 100 tokens since he can just take 64 off the start leaving 36, which is a multiple of 6?

Answer 1

If a player can finish his turn with the remaining number being a multiple of 6, he can guarantee to win. Since there are no available subtraction options which are multiples of 6, and 1,2,3,4,and 5 are all options, whatever the other player does you can counteract with the appropriate number 1-5 to get it back to a multiple of 6. Therefore player 1 would be wise to take 4, 16, or 64 on his first move.

Answer 2

Since the wording of the problem says "who wins each game?", I assume the question should have said "takeaway games" not "takeaway game". If that's right, then the comment "the subtraction sets means that both players have access to both.. ie for a game with 100 tokens on their turn they can take 1,3,5,7 or any power of two amount of tokens off the board." is not correct, and there are two separate questions in one:

1. What's the set of $P$-positions for the game with subtraction set $\{1,3,5,7\}$? (And so who wins with $100$ tokens?)
2. What's the set of $P$-positions for the game with subtraction set $\{1,2,4,8,16,32\}$? (And so who wins with $100$ tokens?)

The answer to neither of these questions is "the multiples of $6$". For example, in the first game, $4$ tokens is a $P$-position since the next player to move removes $3$ or $1$, and then other player removes the other number to win. In the second game, $3$ tokens is a $P$-position for the analogous reason with $1$ and $2$.

I would recommend writing out who wins for small numbers of tokens (maybe from $1$ to $12$ since you were already looking at multiples of $6$) for both games and seeing what pattern you find. Then see if you can prove the pattern continues (if you need to for this book/class/exercise).

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If my first interpretation was wrong, and this is a single partizan subtraction game where one player has one subtraction set and the other player has the other subtraction set, then please edit your question to clarify.