[1,2]-3-[4,5]-6-[7,8]...21 Helping Children Devise and Understand Winning Strategies

Do you remember playing those fun counting games in elementary school? No, well, play along as if you do! The teacher or a friend would go first and always seem to win or you would go first and always lose. You knew there was a trick and if you figured it out it was exhilarating - like understanding the key to a magic trick.

Like most parlor games, there's genuine mathematics underlying these counting games. In this post we will describe a few of these and an investigation to help students not only devise a winning strategy (or algorithm) but to come to an understanding how division and remainders play a significant role.

Variation #1: The Game of '21'

Age Group: Certainly appropriate for children even as early as 1st grade (however, devising winning strategies and explaining why they work might be a bit ambitious!)

# of players: 2 is best

Object: To win, make your opponent say some target number like 21

Rules: First player starts counting from 1 and says either '1' or '1-2'; Other player then says the next number or the next two numbers; play continues in this way until someone is forced to say the number '21'. Verbal or written directions here are far more confusing than just demonstrating actual play.

Sample Play: See title of post for a partial play

Winning Strategy (partial): If you go first, say '1,2'. If you don't, your opponent can beat you if she/he knows the strategy.

Further Discussion: For the younger children, let them play against each other in pairs for a few minutes to allow them to feel comfortable with the game. Then you can ask if anyone wants to 'challenge the master' - you, that is! Tell them because you are older, you deserve the courtesy of going first (that will last for about 30 seconds or less!). After playing against students for a while, they will figure out that part of the winning strategy is to go first and say '1,2' but most will not pick up on the rest of the method. To mystify them even more, you can let them go first. You most likely will still win because you know the strategy and they will most likely not catch on for some time! There's always one sharp youngster even in the primary grades whose eyes will start glowing and will say, "Let me go first. I can beat you." At that point, you may want to say, "Game over!"

Winning Strategy: Those of you who are familiar with these kinds of counting games, know that they are all variations on the same basic theme and are simpler versions of the classic game, NIM. In this version of '21', some children will quickly see that, whoever gets to 2o has to win. It will take them a little longer to work backwards from there to see that to get to 20 you have to get to reach 17, which is 3 less than 20. To get to 17, you have to reach 14, which is 3 less than 17. Thus, working backwards, the winning positions, or 'magic numbers' if you will, are 20-17-14-11-8-5-2. Reversing this provides you with a guaranteed win but of course you need to go first and say '1,2'! But learning and using this strategy does not imply that the child understands WHY it works!

Using questions to help children begin to grasp the underlying idea: Children will immediately see why '20' is a winning position but ask them to explain why 17 also is (Possible student response: "Because if you say '17, then the other person can only get to 19 and you will be able to get to 20"). Continue to subtract 3 to obtain other 'magic numbers.' Ask the children why subtracting 3 is critical. Why 3? Children, even older ones, will soon see what is going on. Some may ask if one has to memorize all of these numbers. Don't answer that! Just smile and let them figure it out for themselves. Allow the children to practice the winning strategy on each other until they feel comfortable. They will surely want to try this out on other friends, teachers or family members!

Underlying Concept: At what grade level are children expected to grasp the essential idea that repeated subtraction is equivalent to division? Thus, in our problem, working backwards, starting from the winning position of 20 and continually subtracting 3, is equivalent to dividing 20 by 3:
20 ÷ 3 = 6 with a remainder of 2.
This can be interpreted to mean that after performing six subtractions by 3, the number 2 will remain! Of course, the repeated subtractions reveal all of the winning positions so children may not be appreciate the benefit of division. Help them to see that the remainder does reveal that one needs to go first and say '1,2' to guarantee a win.

A Million Variations
Well, maybe not that many in this post, but I'm sure you can see the possibilities are endless. You may want to ask children to devise their own version and a winning strategy as an outside project or assignment. They may invent something really cool no one has thought of! You might want to first ask the group how they could modify the game: "If you were going to invent your own game, what might you change about the game of 21?"
Some suggested variations:
(1) Whoever says '21' wins
(2) '21' loses but this time students can say the next number or the next two numbers or the next three numbers. Thus, if you go first and say '1,2,3', I would say '4'; then you might say '5,6' and I would say '7,8'. Am I guaranteed to win if I play correctly?
(3) Start from some number like 50 and allow children to subtract any number from the set {1,2,3,4,5,6}. Then you subtract one of these 6 numbers from the result and repeat play until one player reaches the number '1'. That player wins. This 'Game of 50' is also famous and will mystify adolescents as much as younger children! I'll let our readers explain the strategy and why it works! By the way, don't underestimate how much reinforcement of basic subtraction skill this game provides!