The Ulamog Gambit, by Craig Wescoe. A Magic puzzle that tests multilevel thinking.

His logic feels wrong. Attacking at any point prior to my opponent have more than 20 cards in his deck will potentially give me a sub 93.75% chance of winning. The right answer is to just wait for my opponent to take 6 draws, then attack.

It doesn't even matter if my opponent buried the RT. It's a hypergeometric stats question. The population of cards in the equation starts at 20, with 1 success in the population, a sample size of 1, and one success possible in the sample, and the population decreases by 1 each draw my OP takes.

If he buried RT, my % chance of winning is already 100%, assuming I wait. But since I can't safely make that assumption, I know I need to force him to draw, but that's okay because the odds are massively in my favor to win at this point.

  • My Turn 1: I pass
  • Op Turn 1: 20 meaningful cards remain in library, he draws. He had a 5% chance of drawing RT My chance of winning is now 95%
  • My Turn 2: I pass
  • Op Turn 2: 19 meaningful cards remain in library, he draws. He had a 5.3% chance of drawing RT. My chance of winning is now 94.7%
  • My Turn 3: I pass
  • Op Turn 3: 18 meaningful cards remain in library, he draws. He had a 5.5% chance of drawing RT. My chance of winning is now 95.1%
  • My Turn 4: I pass
  • Op Turn 4: 17 meaningful cards remain in library, he draws. He had a 5.8% chance of drawing RT. My chance of winning is now 94.2%
  • My Turn 5: I pass
  • Op Turn 5: 16 meaningful cards remain in library, he draws. He had a 6.25% chance of drawing RT. My chance of winning is now 93.75%
  • My Turn 6: I attack, bury his library, 0 cards remain in library. My change of winning is now 100%

If I never attack until he has 20 cards in his library, the average of my chances to win across those 5 draws he has to make was 94.375%.

The odds aren't cumulative, at least not the way the author presents. Each turn represents a new hypergeometric equation, where the population size decreases by one. That shifts the odds a very small amount.

In other words, each draw of my opponent is a new roll of a d100, and each time I'm hoping he doesn't roll above a 93. This whole multilevel strategy is a waste of mental effort in this scenario and can only decrease your chance of winning.

Then again I could be totally screwing up the stats, but I don't think I am.

/r/magicTCG Thread Link - magic.tcgplayer.com