# Challenge Your Middle Schoolers with Mathematical Strategy Games!

## Help learners develop the ability to think logically and mathematically, in a really fun and engaging way.

By Barry Nitikman

Posted

There are a lot of math games out there that will help students develop basic skills, such as addition, subtraction, reducing fractions, and so on. But true mathematical strategy games, where the player is challenged to discover the strategy that will guarantee a victory every time, are really rare. Here are two of the best.

I suggest you try these two games in the order they are presented here, as there will be a logical progression in how learners approach finding a strategy. Trust me!

### The 100 Game

There are two contestants, and the goal is to reach 100 before your opponent. Both players start out with zero. Player A chooses any number, 1-10, and his/her opponent follows suit by adding another number, 1-10, and announces the total. The game will progress until they reach 100, and the last player to add a number is the winner.

Here's how a typical game might proceed:

Step 1: Player A chooses the number 5.

Step 2: Player B adds 7, announces a sum total of 12.

Step 3: Player A adds 10, announces a new sum total of 22.

Step 4: The game continues until the sum total reaches 100, and the player who added the last number wins.

Pupils will have to play a few times before they start working towards the correct strategy, which guarantees victory every time. Here is the sequence of numbers that must be followed to ensure victory:

1, 12, 23, 34, 45, 56, 67, 78, 89 (game over)

All you must do, is make sure you get to the numbers in the pattern on each of your turns. This will result in your opponent being forced to arrive at a number from which they cannot prevent you from reaching 100. Regardless of what number your opponent adds, you can’t lose.

### How to Present the Games

To me, the great value of these games is that they force children to think mathematically in order to develop a winning strategy. Winning is obviously a big motivator for most kids, but regardless, my experience has shown me that *most* kids will respond positively to this kind of challenge; possibly only for the mere fact that it’s fun! Also, I have found that children really become attuned to thinking closely and carefully about numbers and strategy.

Here’s a very effective way to introduce both of these games:

- Start out by playing a game with you against the class. Let them begin, and as it gets close to 89, use whatever number you need to hop onto the pattern so that you win the game. However, do this in such a way that is not obvious so they won’t catch on right away.

- Without fail, there will be a few kids who think they’ve beat you right away. Tell them they’re on the right track, but they’re not there yet. Prove it by playing them again and demonstrate that their strategy is incomplete; just hop onto the pattern at a safe point and make it impossible for them to get to their goal of 89. The reaction to this is what I call a delighted frustration. They are frustrated at finding that their strategy has failed them, but they are also delighted and challenged, once again, to figure out how to make it foolproof.

The beauty of this approach is that as the rest of the group starts to gradually get it. I have found that even if there are one or two kids who never quite get the concept by themselves, they still are delighted when they find out how it’s done; it's almost like learning a magic trick to them. If they didn’t get it during class, they can still feel clever by playing it with other people.

**Important note:** I believe you should limit the amount of time the game is played to about 10 minutes each time; maybe 15. The reason is that while many pupils will be completely engaged in the challenge the entire time, others will inevitably lose focus after a few rounds. It is far better to have them intrigued for a short time, than to risk their losing interest. You may even decide to have them fill out a little scoreboard. This will keep them focused on the task, rather than on the freedom to roam and talk to friends. Here’s an example you might use:

Game # | Who won? | Strategy? |

### Variations of the 100 Game

The first and most obvious variation is to switch the objective to forcing the *other* person to reach 100 first. In this case, you want the other person to go first. The pattern changes, and looks like this:

11, 22, 33, 44, 55, 66, 77, 88, 99 (In other words, multiples of 11)

Of course, once you’re at 88, the other player will recognize that their goose is cooked. (No matter what number between 1-10 they choose, you can always leave them with the last number, which they must take, and they lose!)

I always present this variation to advanced learners once they’ve solved the original game so that they can continue to be presented with challenges that will intrigue them. I introduce the variation to the whole class as soon as a fair amount of them have achieved mastery of the original game. This one won’t take them nearly as long to figure out, primarily because they are now aware of what is required when they aren’t looking to get to 89; they are looking specifically for a pattern that they know will do the trick. What adds so much to their motivation, obviously, is the fact that discovering this pattern allows them to win; this is a strong motivator.

### Using Different Numbers

A further variation is to have them try other numbers, such as 1-8 or 1-9. If the players can select numbers 1-9, and the winner is the one who gets to 100, the strategy is simple: let them go first, and then make sure you always wind up with multiples of 10 (10, 20, 30........you wind up at 90, and no matter what they pick, you can always take the last number!)

For example:

- You are at 89. If the other player takes 1 to get to 90, you take 9 to leave him the last number. If he takes 2 to get to 91, then you will take 8 to leave him once again at 99, tasting the bitter ashes of defeat…
- The pattern is: Let the other player go first, and then get to 9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 89.

In all of these 100 Game variations, you can use the same approach to teaching these to the whole class: demonstrate, challenge, they disperse to attempt to figure out strategy, and report back to you as they work out their methods.

This approach works beautifully with all strategy games. Try it on the following series of games based on the classic, ancient game of NIM.

### NIM

There are countless variations of this game, dating back thousands of years. The difficulty of the variations extends from fairly basic to staggeringly complex. The version(s) I will be including here are of the more basic variety. In fact, the first version is a well-known bar bet that has been used for untold years by hustlers to fleece unsuspecting bar patrons. In any case, this game provides much enjoyment and challenge, and above all, excellent practice in mathematical thinking, problem-solving, and logic.

**Materials:** 20 toothpicks, matches, beans, paper clips, or pretty much anything similar. I use toothpicks in my classroom.

**Procedure:** There is a pile of 20 toothpicks on the table. Players take turns, taking anywhere from 1-3 matches at a time. (Each Player *must* take 1-3 on his or her turn; you may not pass.) The winner is the Player who takes the final toothpick.

Example:

- Player A takes 2
- Player B takes 1
- Player A takes 3
- Player B takes 3
- Player A takes 2
- Player B takes 3

At this point, only six toothpicks remain. Player A then takes 3 more, leaving 3 left in the pile. Player B takes the final 3 toothpicks and wins the game. Of course, Player A’s last move was a bonehead play, since it obviously leaves the game wide open for Player B to take the last three. In real life, your gifted learners will see this coming by the second or third game they play.

### Winning Strategy

I have found a tried and true strategy: Player B allows A to go first and simply matches their opponents take with a quantity that will complete a multiple of four.

For example:

- Player A takes 2
- Player B takes 2 ( 2 + 2 =4)
- Player A takes 3
- Player B takes 1 (3 + 1 = 4)
- Player A takes 1
- Player B takes 3
- Player A takes 3
- Player B takes 1

(At this point, only four toothpicks remain, and Player A is stymied; no matter what quantity they take, Player B will be able to take the final toothpick.

What you will find after playing The 100 Game, is that subsequent games will be approached by learners in a much more organized and methodical way. They will begin to strategize, use logic, and not just aimlessly throw out numbers and guesses. This is a very significant and valuable realization, and one that will hopefully begin to lead them into an overall appreciation and love for the beauty and excitement of mathematics. Instilling positive feelings about math is important to me.

As a child, I was a poor math student. I have no memories of it ever being fun or exciting. This is the antithesis of what I want to achieve for my classes– math *can* be fun, it can be exciting, and it can be fascinating. By using games and other activities, I feel that even for those learners who struggle with math, not only will their skill development be enhanced, but they will learn that math *can* be something wonderful. That, in itself, is a worthy and valuable goal.