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2 DICE & 9 CARDS Game, Puzzle: Odds, Probabilities, Strategy

The best strategy is to eliminate card 9 as soon as the two dice sum up to 9.


Here is a problem for you to solve, axiomatic one.

I have 9 playing cards, Ace to nine... and 2 dice. I throw the dice and can choose to either; take to number showing from either dice, or the total of both dice added together. i.e. lowest possible number is 1, highest is 12 (although there are only 9 cards, remember). I can then eliminate the respective card from the Ace to nine. Example:

  • I throw 2 dice and get a four and a three. I can choose to either eliminate the cards 3, 4 or 7... and repeat the process.

    The object of the game is to eliminate all 9 cards.

  • What are the probabilities of eliminating certain cards?
  • Do some cards have a higher probability of being eliminated than others?
  • What is probability of eliminating all 9 cards?
  • What formulae do I use to work out my odds?


    Parpaluck's Probability Response

    Kind of craps... that card-and-dice game.

    Probability requires number of favorable cases and total possible cases. We can calculate easily the number of all possible cases: 2 dice, 6 faces each, amount to 36 possibilities. I assume the Ace counts as 1 only. We can generate them possibilities easily even manually:

    We can notice easily that the faces 1, 2, 3 have the most eliminating power. We can also count the frequency of each card.

    A = 11 times; 2 = 11 times; 3 = 12 times; 4 = 14 times; 5 = 15 times; 6 = 16 times; 7 = 6 times; 8 = 5 times; 9 = 4 times.

    Not all cards are equal! Number 6 is the favorite: It appears 16 times among the 36 possibilities (44%). Number 9 fares the worst: 4 occurrences (11%).

    A quick strategy:

  • Eliminate card #9 as soon as the dice roll amounts to 9. If the roll is 3&6 or 6&3 eliminate card 9 right away; a 6 will come more often. In the next 3&6 or 6&3, eliminate the 3-card, as the 6-occurrence has a better frequency than 3.

    Good luck (to your sister, too)!

    Ion “Parpaluck” Saliu,
    Founder of Mathematics of Card Games

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    You can play a nice game with two dice and nine cards from Ace to 9.

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