The 31 Puzzle

 1  2  3  4  5  6
 1  2  3  4  5  6
 1  2  3  4  5  6
 1  2  3  4  5  6

We start with a regular card deck, but remove all cards with a face value higher than 6. This leaves us with 24 cards, numbered one (ace) to 6, with each value showing up four times.

The 24 cards are spread out on a table, face up, so that each card is visible. Two players now begin the game of "31".

A turn consists of taking one card from the table and adding it to your hand. If a player draws a card and makes the sum of 31 exactly, the player wins and the game is over.

The game ends in a draw if there is no winner, which happens when the deck has been exhausted with neither player having hit 31 exactly, or if both players have exceeded 31.

Which of the following statements is correct?

  1. There is a strategy by which the first player always wins;
  2. There is a strategy by which the second player always wins;
  3. There is no strategy by which the first or second player always wins.

This puzzle was provided by Yuen-Yick Kwan.

I give up, show me the solution.

Last revised on 06 March 2011.