# Combinatorics Game help

I need help with this question. Im not too sure how to do it so an explanation would be helpful

Two people play the following game, starting with three piles of mathces. In a turn, a player moves any positive number of matches from one of the piles to a larger pile. The player who can’t make a move loses (the player who makes the last move wins). For example, in the position 17–12–12 you can move any number of matches from either of the piles of 12 to the pile of 17, but these are the only possible moves. In the position 9–6–3 you can move matches from the 3-pile to either of the other piles, and from the 6-pile to the 9-pile. Describe all final, losing and winning positions for this game (for any triple of numbers as pile sizes). Give a clear and concise argument for why your description is correct, that is, why these positions satisfy the conditions overleaf.

• So we can't move from the highest pile to lower piles ? What if all piles are same ? – A---B Mar 17 '17 at 15:56
• You have posted the question verbatim, which is helpful. Can you give more context? For examples, What theorems/facts do you know to help you analyze games like this? What did you try and then where did you get stuck? Have you (and/or your teacher/textbook) done any similar problems successfully? Etc. – Mark S. Mar 17 '17 at 15:57
• You cannot move highest piles to lower piles. If they are the same the player who makes all piles equal (moved last) has won. – Conbinatorics Mar 17 '17 at 15:58
• In previous problems we mapped out using examples of it being winning or losing positions. Im not too sure how to do the question or how to do previous ones – Conbinatorics Mar 17 '17 at 15:59
• @Conbinatorics: A move cannot make "all piles equal" since moving from a smaller pile to a larger pile only makes them less equal. Since there are three piles, it seems the game terminates when all "matches" have been transferred to one pile, unless the piles were all equal to begin with. – hardmath Mar 17 '17 at 16:28