This week's content is provided by Erick Lee, an educator who is sharing a simple game that you can use with your child(ren). It's perfect for the holidays, a ride in the car, or a chilly weekend chat.
The subtraction game is a common game that has many variants and is a great way to practice counting and develop number sense. It is played with just a few simple rules and does not need dice, cards or a playing board. It is ideal for playing around the dinner table. You only need a few object to use as counters. I’ve used toothpicks, pennies, sugar packets at a restaurant, or a pile of pebbles at the playground.
How to Play
Start with a pile of 13 of toothpicks (or any other collection of similar objects). Two players take turns removing 1, 2 or 3 toothpicks from the pile. The winner of the game is the person who takes the last toothpick. There is an elegant strategy that can be discovered to guarantee a win for the first player in this game*. Once you’ve discovered the strategy for this game, you can alter it. You can start with a different number of objects or allow a different number of objects to be removed each turn (a version of this game even made an appearance on the television show Survivor: Thailand with two teams removing 1, 2 or 3 flags from a starting group of 21 flags.
You could also alter the rules of this game so that the player who takes the last object loses instead of wins (this is called a misère game).
*In the subtraction game described above, the first player can always win by taking the number that leaves the other player with a pile whose size is a multiple of 4. So on the first turn, the first player should take one object leaving 12 (3 x 4). Whatever the second player removes from this pile of 12, the first player should remove enough objects to bring the pile to a size of 8 (2 x 4). Whatever your opponent takes from the final pile of 4, you take the rest and win.
The Counting Game
The counting game is a variation of the subtraction game that is played orally instead of with physical objects. In this version, two or more players take turns counting up from one. The first player says “1” and then each player on their turn can say the next one, two or three numbers counting on from where the preceding player left off. The person who says “13” is out. The game starts again at “1” with the remaining players. Keep playing additional rounds until there is a final winner. An example game with two players is below:
Player 1: “one”
Player 2: “two, three”
Player 1: “four, five, six”
Player 2: “seven”
Player 1: “eight, nine”
Player 2: “ten”
Player 1: “eleven, twelve”
Player 2: “thirteen”... player 2 is out!
Since no equipment is needed, this game can be played anywhere... on car trips, while waiting in a long line, or sitting around a campfire. It is also a great way to practicing counting in another language. Like the subtraction game, this game can be changed to finish at any number or to allow each player to continue with a different amount of numbers on their turn. For example, "Add at most 4; lose on 21".
The subtraction game and the counting game are part of a broader class of games known as "Nim Games" in which each player removes some objects form a pile or piles. The most common game of this type is called Nim and is played by two players. This game dates back to medieval times. In this game there are 3 piles of objects. No two piles should have the same number of objects at the start of the game. For example, a game might start with 3 objects in the first pile, 5 objects in the second pile and 7 objects in the third pile. On each player’s turn, they choose one pile and remove as many objects as they'd like (but at least one) from that pile. The player that takes the last object wins. The winning strategy for this game is a bit more complicated than the subtraction game and was not published until 1902 by Harvard University professor Charles L. Bouton.
If you like these games, try checking out some similar games:
Additional Nim Resources:
Nim Like Games
If you have any questions, please feel free to contact me on Twitter (@TableTalkMath) or leave a comment below.
Thank you for taking the time to improve math fluency for children, one table talk conversation at a time. For previous newsletters, please check out www.tabletalkmath.com/previous-newsletters