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Dots and Boxes xyz

This is an algebraic version of the popular two-player game dots and boxes created by Édouard Lucas in 1889. The rules we will play with here are a good starting point. All students should use these rules for their first games.

This game has now been improved upon. Go to Pillage and Profit.

First, select a board and give an initial value to all variables. Here we start with x=3, y=2 and z=1.

On each turn a player must connect two neighbouring points.

Connections van be made vertically or horizontally.

Once and only once in the game each player may choose not to connect and may instead increase or decrease one of the variables by 10. Players typically hold these special moves till later on in the game.

Moving forward in the game, we see that neither player has yet used their special turn to increase or decrease a variable by 10.

We are reaching a critical point in the game. Whenever a player completes one of the small square - they steal the contents of that square (whether they want to or not.)

In this case, the player on the left definitely wants the lower left box, so will complete that square in the next slide.

It is worth 5 points.

When a square is completed, the player must take another turn. Sometimes they will not want to do this.

In this case the left player will add a connector in to the bottom of the middle square on the next slide.

That is the end of the left players. turn. Let's skip ahead again...

It is Right's turn. On the next slide he will grab the 2x square.

The 2x square is completed...

Its contents are grabbed. Right must take another turn because he has completed a square. He chooses to complete the square around the x.

Now he grabs the contents of that square.

Right again must take another move because he's completed yet another square. This time he prepares to take his special move. He can only do this once throughout the whole game. He's going to choose to increase or decrease one of the variables by 10.

z is now worth minus 9 points!

That is the end of Right's turn. Let's skip ahead a couple of moves...

One last rule - after all the squares are taken you can no longer make your special move. In the present game both Left and Right have already made their special move (you can see that Left has increased the value of y to 12).

Now we'll skip to the end.

All that need to be done now is substitute the values for the variables to see who has won. Here we see that y is the most valuable and z actually gives negative points. Who wins?

Right wins!

After your students have played a few games they should feel free to agree on different rules before the game commences. Instead of the special move being increasing or decreasing by 10 it may be halving and doubling. Instead of the winner having the most points, it may be that the winner has got the fewest points. Let them get creative and agree on the rules before the game starts.

Here are written instructions and a variety of game-boards.

Dots and Boxes xyz

(Based on the game by Édouard Lucas, 1889)

Standards for Mathematical Practice

MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

MP1 Toughen up!

This is problem solving where our students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP3 Work together!

This is collaborative problem solving in which students discuss their strategies to solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

MP6 Be precise!

This is where our students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.

MP7 Be observant!

One of the things that the human brain does very well is identify pattern. We sometimes do this too well and identify patterns that don't really exist.


Please use MathPickle in your classrooms. If you have improvements to make, please contact us. We'll give you credit 😉

Gordon Hamilton

(MMath, PhD)


Lora Saarnio