The Game to 20 is how I started teaching my Algebraic Thinking Unit when I taught pre-algebra to 7th grade students. This is one of my favorite games because the students discover that a simple pattern is the key to winning the game. The game to 20 is a version of the game NIM. There are multiple versions of this game that can be found online and in books with games and puzzles. This version can be described quickly and has some interesting patterns that can be used to gain a deeper understanding of the game.
Materials:
- Post-its (20 per pair) optional, but recommended
Rules:
- Two players will take turns counting from one to twenty.
- On a player’s turn, they can say one or two consecutive numbers following the previous player’s count.
For example, a game may start as shown below:
| Player 1 | Player 2 | Player 1 | Player 2 | Player 1 | … |
|---|---|---|---|---|---|
| “”One” | “Two, three” | “Four, five” | “Six” | “Seven” | … |
- The player who says “twenty” wins the game.
Opening:
Begin by asking for a volunteer. If you can, get an administrator to come in to help you model the game for students.
Work Session:
After seeing the game, students pair up to play the game with the idea that they are playing to notice what they can about strategies to win. I give pairs of students post-its to write the numbers 1-20 on so they can see the numbers as they say them (this has really helped students).
Students play the game for about 5 minutes.
Next, ask students to share what they noticed as they played the game.
Possible things students notice:
- You can win if you say the opposite of what your partner says. Like if they say one number, you say two numbers.
- If you get to 17, you’ll win.
- You can win if you you leave the post-its that say 18, 19, and 20.
- If you say 17 you always win.
When someone says something about saying 17, help them make this statement precise by having them explain how they know. “If we’re playing, and you say 17, what could I say?” What will you say, in each case?
At this point, I ask students to help me keep track of what we know with a table.
| Game to … | I know I’ll win if I say… |
|---|---|
| 20 | 17 |
So, in the game to 20, I know I can win if I say 17. So, how can I make sure I say 17? Students will suggest playing the game to 17 to find out.
Let students play the game to 17, have a discussion about what they notice and help them be precise with an explanation.
Fill in the table with the new information. At this point, students will likely ask to play the game to 14, even though some may think they see the pattern. As students discover the pattern of three, encourage them to keep playing as needed to finish the table, then discuss what the numbers in the table mean.
| Game to… | I know I’ll win if I say… |
|---|---|
| 20 | 17 |
| 17 | 14 |
| Game to… | I know I’ll win if I say… |
|---|---|
| 20 | 17 |
| 17 | 14 |
| 14 | 11 |
| 11 | 8 |
| 8 | 5 |
| 5 | 2 |
Closing:
Ask students to turn and talk about what the numbers in the table actually mean and how they could be used when playing the game.
Share. The goal is for students to discover that the pattern they discovered is the key to winning the game. Ask them to play the game as often as they wish, but don’t give away the pattern. They worked too hard for it.
Students will likely share that you beat the principal (or whoever modeled the game with you at the beginning of the lesson) because you knew the pattern and they didn’t. That’s partly true. But, did you memorize all of those numbers? Or is there an easy way to remember all of them?
Extensions:
- What would happen if you played so that whoever said 20 loses? would the pattern still be the same?
- Find some different Nim games. What are the patterns to win those games?
- What if you could say 1, 2, or 3 consecutive numbers? What if you could only say 2 or 3 consecutive numbers?
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