9. Advanced Nim
class: closer | impressiveness: 3/5 | factors: game is more fair, just as crooked | requires: 15 counters | watch full episode
Memorize a few configurations, and you’ll hold the key to another unbeatable game and loads of free drinks:
I mentioned that Nim is simultaneously totally fair and completely unbeatable. This advanced version of the game is even simpler for the sucker to play, but just as stacked in your favor.
The Rules: Set up three piles of markers, with the starting numbers of 3, 5, and 7. On each player’s turn, they can take as many markers as they want from any one pile. Whoever takes the last marker loses.
To Win: There are five “pole positions,” winning configurations that you can always achieve in just one move (as long as you go second). Once you’ve hit one these pole positions, no matter what move they make, you’ll always be able to hit the next pole position on your turn. Keep following the pole positions, and you’ll be guaranteed to win the game.
You’ll have to memorize the winning configurations, but they’re easier to remember than it looks (oh, and don’t worry about the order of the piles, only the values):
Winning Configurations:
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1, 2, 3
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2, 4, 6
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1, 1, 1
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1, 4, 5
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…or two even piles.
To help remember all the configurations, I just think about guns: The initial setup is 357 (as in the .357 Magnum). The only tricky pole position is 145, so I think of “one colt .45.” The rest are super straightforward: 111, 123, 246, and two even piles. When you first start practicing you can use this book as your cheat sheet, but you won’t need it for very long.
If you go first, take only one marker. It won’t significantly alter the setup, and after their first turn you should be able to hit one of the winning configurations. The only way you’ll lose is if they’re already hip to this scam and beat you to the punch, hitting all the pole positions (if that happens, team up and take on the rest of the bar).
Remember that once you hit a winning configuration, you’ll always be able to continue hitting them. As you near the end of a game, make sure to pay attention: it’s easy to get caught up in trying to hit the next pole position and miss an opportunity to win the game.
There have been bajillions of math papers and computer simulations written around different versions of nim, but the first time I saw this take on it was thanks to Opie Houston at the TAOM convention a few years ago. Mega-thanks go to him for suggesting we cover it on Scam School.