Algebra Middle School

The Puzzle Solutions You’ve Been Waiting For….

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.


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.  

With 23 dots to start the game, I will offer the student a choice; who plays first.
If they let me play first (which they often do because they want to be polite), then I erase 3 dots (leaving 20) and wait for their eventual demise.  If they play first, then I need them to slip up and make it so that I can control the multiples of 4.  If they erase 2 on their first move, then I remove 1, leaving 20.  
It often takes a few days of playings at the end of class for kids to develop the strategy.  They will first realize that being left with 4 is sure doom.  Then 8….then eventually they get it.
Have fun
And here are some ways to tweak the game, and see if students can develop strategies:
  • 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…)
I posted about the Hot Seat game recently.  Time to reveal the secrets to this interesting pattern game –

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.



By Bob Lochel

HS Math Teacher. Hatboro-Horsham School District, Horsham, PA.

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