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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.

"Ion,

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?

ANY HELP APPRECIATED SO I CAN BEAT MY SISTER!!!!"

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:

1&1 = eliminates cards A or 2
1&2 = eliminates cards A or 2 or 3
1&3 = eliminates cards A or 3 or 4
1&4 = eliminates cards A or 4 or 5
1&5 = eliminates cards A or 5 or 6
1&6 = eliminates cards A or 6 or 7
2&1 = eliminates cards A or 2 or 3
2&2 = eliminates cards 2 or 4
2&3 = eliminates cards 2 or 3 or 5
2&4 = eliminates cards 2 or 4 or 6
2&5 = eliminates cards 2 or 5 or 7
2&6 = eliminates cards 2 or 6 or 8
3&1 = eliminates cards A or 3 or 4
3&2 = eliminates cards 2 or 3 or 5
3&3 = eliminates cards 3 or 6
3&4= eliminates cards 3 or 4 or 7
3&5= eliminates cards 3 or 5 or 8
3&6= eliminates cards 3 or 6 or 9
4&1 = eliminates cards A or 4 or 5
4&2 = eliminates cards 2 or 4 or 6
4&3= eliminates cards 3 or 4 or 7
4&4 = eliminates cards 4 or 8
4&5 = eliminates cards 4 or 5 or 9
4&6 = eliminates cards 4 or 6
5&1 = eliminates cards A or 5 or 6
5&2 = eliminates cards 2 or 5 or 7
5&3= eliminates cards 3 or 5 or 8
5&4 = eliminates cards 4 or 5 or 9
5&5 = eliminates card 5
5&6 = eliminates cards 5 or 6
6&1 = eliminates cards A or 6 or 7
6&2 = eliminates cards 2 or 6 or 8
6&3= eliminates cards 3 or 6 or 9
6&4 = eliminates cards 4 or 6
6&5 = eliminates cards 5 or 6
6&6 = eliminates card 6

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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