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Real Nim Variations (Posted on 2003-05-18) Difficulty: 3 of 5
The puzzle “A Game of Nim” asked for the strategy in a game in which two persons take turns removing cards from a deck, limited to 1, 2 or 3 cards.

However, there is a more interesting version of Nim, in which players can take as many as they want, but from only one pile or set of objects in a turn, where there are several piles or sets present.

If done with cards, one way of setting the rule could be that in one turn you might take as many as you want from any one suit. Another choice for the rule might be to take as many as you want of a given denomination. The object is again to be the person to take the last card.

If you were to play first (take the first set of cards), which of the two rules of the preceding paragraph would you want to use and why? (i.e., would you rather it limit one player’s turn to one suit or to one denomination?)

Finally, if the rules were that each play the player can take as many as desired of any one suit, but play were to start with the king of clubs removed, as well as the king and queen of hearts, what would be the strategy to win then? This is the equivalent of having a different number of objects in two of the four piles: 13, 13, 12 and 11.

See The Solution Submitted by Charlie    
Rating: 4.3333 (3 votes)

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re: Solution? | Comment 5 of 7 |
(In reply to Solution? by Trevor Leitch)

That makes perfect sense; and for the given situations that seems to be ironclad.
I was trying to work backwards, more or less. If there is only one card, or only cards in one pile, left, then just take it/them and win. If you leave the other person with two cards in different piles, they must take one of them and leave you with just one card. So, if you have one card in one pile and one in another, take all but one card to leave that same situation. Similarly, if you leave the other person with two cards each in two different piles, you can win no matter what they do. And so on...
Once you have that the piles have to be the same (in the case of two piles left) or symmetrical, then doing whatever you need to make it that way (or keep it that way) comes naturally. Seems too simple, but it will always work.
I'm not sure how you would manage it if the piles weren't symmetrical to begin with (I'm still trying to figure out why/if Gamer's works).
  Posted by DJ on 2003-05-18 18:13:50

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