# The Game of 31

#### Thirty-One:

Last weekend, 77 recommended me to watch a TV shows, High Intelligent Players, 高能玩家.

Among the games they played, there is a game of 31, I think it is very interesting.

#### Game Rules:

There are 4 sets of 1 to 6.

 1🖤 2🖤 3🖤 4🖤 5🖤 6🖤 1❤ 2❤ 3❤ 4❤ 5❤ 6❤ 1🍁 2🍁 3🍁 4🍁 5🍁 6🍁 1🔷 2🔷 3🔷 4🔷 5🔷 6🔷

• Player A and Player B take turns to pick a card and annouce
the running total
• Continue playing until the player makes the total 31 wins
• If the player goes over 31 loses.

#### Can we search for a Winning Strategy for 31?

a player‘s strategy is any of the options which he or she chooses in a setting where the outcome depends not only on their own actions but on the actions of others. A player’s strategy will determine the action which the player will take at any stage of the game.

geralt / Pixabay

#### Solution:

Thought Progress:

After exploring the game, we notice if the play A arrives 24, then no matter which number(1-6) player B put down, player A can definitely call out 31, and wins the game. Continuing the same fashion, 17,10 are also the critical points for this game.

Question: MUST we call our 10,17,24 in order to win 31?

Answer: Not necessarily because there are only 4 sets of numbers.

Example 1:

 Player A 5 2 2 2 Player B 5 5 5 x

In this Game Player B has 10, 17, 24, theoretically, Player B can call out 31, however, the game ran out of 5 (only 4 set of numbers), so Player B has to either call 6, the sum will be 32, the game exploded.  Player B can call anything smaller than 5, then play A will call (7-x), Player A wins.

Example 2:

 Player A 2 4 4 4 x Player B 4 3 3 3

In this round, The player A in order to call 10, 17,24  used all the 4s. therefore, Player A either exploded in the 5th term or pick another number smaller than 4, either way, Player B will win this game.

Similarly, in this case:

 Player A 3 3 3 3 x Player B 4 4 4 4

or :

 Player A 1 1 1 Player B 1 6 6

It seems like in the above 4 cases, Player B has a better chance of winning, however, the game of 31 does Favor Player A (First Mover), if player A chooses the correct first number, and be aware of consuming too many of the same number.