# Math Game: Thirty-One

Players: best for two.
Equipment: one deck of math cards.

### How to Play

Lay out the ace to six of each suit in a row, face-up and not overlapping, one suit above another. You will have one column of four aces, a column of four twos, and so on‌—‌six columns in all.

The first player flips a card upside down and says its number value. Then the second player turns down a card, adds it to the first player’s number, and says the sum.

Players alternate, each time turning down one card, mentally adding its value to the running total, and saying the new sum out loud. The player who exactly reaches thirty-one, or who forces the next player to go over that sum, wins the game.

### Variations

For a shorter game, use only the ace to four of each suit. Play to a target sum of twenty-two.

Or play to thirty-one but don’t add to each other’s numbers. Let players keep their own separate running totals, as Rachel suggests in the comments below. How does that change the strategy?

### History (and a Puzzle)

Thirty-One comes from British mathematician Henry Dudeney’s classic book, The Canterbury Puzzles. According to Dudeney, “This is a game that used to be (and may be to this day, for aught I know) a favourite means of swindling employed by card-sharpers at racecourses and in railway carriages.”

Dudeney challenges his readers to find a rule by which a player can always win: “Now, the question is, in order to win, should you turn down the first card, or courteously request your opponent to do so? And how should you conduct your play?”

Dudeney, H. E. The Canterbury Puzzles, Thomas Nelson and Sons, 1919 (originally published 1907); available at Project Gutenberg or the Internet Archive.
http://www.gutenberg.org/ebooks/27635
https://archive.org/details/canterburypuzzle00dudeuoft

This post is an excerpt from my book Addition & Subtraction: Math Games for Elementary Students, available now from your favorite online book dealer.

## 6 thoughts on “Math Game: Thirty-One”

1. Almazea Fatima (AE) says:

Awesome! I love your post having card games! Thank you!

2. Rachel Davis says:

This was great! I played as written first, then tried the variation with my younger daughter and something seems off — neither of us reached target 22 before the cards ran out. We’re considering what the “correct” target should be to involve the most strategic thinking. Any updates?

1. In this game, both players are increasing the same score. Don’t keep separate scores. I’ve edited the post to clarify the rules.

Your alternative game, where each person keeps their own score, is a fantastic idea for modifying the original. I’ve added it to the post as a new variation. Thanks!

As Pam Sorooshian pointed out (quote here https://denisegaskins.com/2020/08/07/mathematicians-play/), changing game rules IS mathematical thinking!

When you change a rule, you want to pay attention to how the game itself changes. In your case, with the new rule, you ran out of cards. Can you figure out why you ran out in the shorter game, but not in the longer one? (“Why?” is the primary question of mathematics.) How will you adapt your new rules to make a more satisfying game?

3. C Ball says:

My clever boots Grade 3/4’s played as originally intended, then scrambled the cards (so they weren’t in order and you had to really look to see what numbers were left.) Then, they played it backwards, counting down from 31 to 0, using subtraction. Great fun was had by all. Thank you!

1. I’m so glad to hear your students enjoyed the game. Sounds like they did plenty of great mathematical thinking!

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