1-2-3 Nim on a 10-Frame

Topics: logic, patterns, addition to 10, counting to 10, subtraction
Materials: One 10-frame and at least ten counters per pair of students
Recommended Grades: K, 1
Common Core: K.CC.A.2, K.CC.B.4, K.CC.B.5, K.OA.A.1, K.OA.A.2, K.OA.A.4, K.OA.A.5,
1.OA.B.4, MP1, MP7

How to Play

Players take turns placing one, two, or three counters on the open squares of a 10-frame.
Whoever fills up the 10-frame wins. This game is for two players only.
Example Game
Player 1 plays three counters, leaving 7 spaces on the 10-frame.
Player 2 plays two counters, leaving 5 spaces.
Player 1 plays one counter, leaving 4 spaces.
Player 2 plays one counter, leaving 3 spaces.
Player 1 plays three counters, filling the 10-frame and winning the game.

Why We Love It

This Nim variation is a great way to give kids more experience with 10-frames, while practicing
the addition, subtraction and strategy of Nim in a dynamic way. Kids are rewarded by thinking
ahead, and the game corrects their errors in strategy (and arithmetic) as they’re beaten, and
have to play again.
There is a way to play 1-2-3 Nim perfectly, and your students may figure it out if they have
enough time. Completely unlocking the game is an exciting and powerful achievement for a
student.

Variations

  1. Poison: Whoever fills up the 10-Frame loses.
  2. 1-2 Nim: Players can take only one or two counters.
  3. Play on a 5-Frame for a simpler game.
  4. Add a 10-Frame to play up to 20 rather than 10 for a more complex game.

Questions

The Central Question: how can you win 1-2-3 Nim? What would a perfect strategy look like?
Good questions for the teacher to ask students:
  • What move should I (the teacher) make?
  • How did you/they/I win that game?
  • What do you think they will do if you/I take three counters?
  • Would you like to take back your move?
Possible student conjectures, true and false, that may arise:
  • Whoever goes first wins.
  • Whoever goes second wins.
  • Odd vs. Even determines your strategy.
  • It matters/doesn’t matter where the counters go in the 10-Frame.
  • Whoever can give their opponent four open spaces wins.

Tips For The Classroom

  1. Demonstrate the game with volunteers for at least three games (or many more!), until you are certain everyone understands it and is excited to play.
  2. When demonstrating 1-2-3 Nim, narrate the game out loud, using mathematical language, and leaving empty space for students to chime in: “My opponent just added two counters, leaving… [wait for students] eight open spaces. I think I will add three counters, leaving…[wait for students] five open spaces.”
  3. Remind students that they will lose many games as they play, and that every loss is an opportunity to learn. Can they steal the strategy of the person who just beat them? Point out how students are trying out new strategies as they play you in demonstration games.
  4. As kids play each other, circulate to see what strategies they are developing. Challenge them to play you, and see if they can beat you.
  5. Use counters in two colors, or reversible counters, so students can see who made what moves in a game.
  6. This game is great for station work once students already know how to play it.
  7. Homework: have kids teach 1-2-3 Nim to a friend or family member. Then beat them at it!

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