This post will present solutions to two puzzles I have presented here on the blog, and some ideas for extending the learning with your students.
THE TAKE-AWAY GAME:
This is a game I proposed a while back on the blog, and it took my friend Anthony from Twitter’s gentle nudging to ask for a solution. Here’s a re-cap of the game:
- 23 marks are placed on a board
- On each turn, a player must remove 1, 2 or 3 marks
- The player who clears the board wins
I often challenge classes with this game, letting kids play me and try to figure out the secrets. The beauty here is that my students are often polite souls, and will let me go first…which leads them to doom. It will often take a few days before a student can conquer me.
Here’s a video example:
And here is the message I sent back to Anthony, who requested the secrets to the game.
Here’s the secret to the take-away game – if I can make it so that the number of dots remaining is a multiple of 4, then I will win the game – guaranteed. If they take away 3, I remove 1. When they remove 2, I remove 2. If they remove 1, I remove 3. The game is all about groups of 4.
- Change the number of starting marks
- Change the number of marks allowed to be removed
- Play with 3 players (actually, I’ve never tried this…)
This table shows the winning chair in the game, based on the number of chairs which start the game. Note that if the number of chairs is a power of 2, the winning chair will equal the number of chairs. From any multiple of 2, the winning chair goes up by 2, until the next power of 2 is reached.
How would your students express this pattern? What vocabulary would they need to use in order to communicate the pattern? After a class develops the data table, challenge students to write a concise rule which will identify the winning chair. Then, have students trade their explanations and critique them.
Could an algebra student develop a formula which outputs the winning chair, if the number of chairs is given?
Take a look at the winning chairs, plotted vs the number of chairs. Can we write a function which follows the pattern? Share your function ideas in the comments, and enjoy the challenge.