Four Square Problem

This problem is perfect for students who need practice in basic arithmetic and are up for a challenge. The students repeat subtraction over and over again and the more they engage in the game the harder subtraction problems they will assign to themselves. In addition, as a Ducci Four Number Game this activity has a solid connection to advanced mathematical theory.

In addition to the description of the game this lesson plan includes activity notes of required supplies, discussion on the mathematics included in the activity, guiding questions, and a list of additional resources to support the lesson.


How to play the game

Start with a square. On each corner of the square write a number. Then on the mid-point of each edge write the positive difference between two corner numbers. Continue the game with your mid-point squares until you reach a square of all zeros.


Activity Notes

Supplies needed

This activity can be done with just blank paper and pencils. You can also use a whiteboard setup described below and then have the students record results on blank paper.

Outreach pro tip: 1 box of sheet protectors is a great low-cost way to make a classroom set of individual whiteboards that can be used with different inserts and new problems

Outreach pro tip: Use blue tape or similar device on your whiteboard markers so you don’t steal any markers from the school or room you are visiting.

Where’s the Math?

The level of student will determine the level of mathematics to be introduced. Possible topics include:

  • Basic arithmetic – For this problem students have to subtract two numbers over and over.
  • Extensions of the problem provide opportunities for students to manipulate fractions and think about the patterns for longest string of numbers and different board shapes.

A list of Guiding Questions

  • What starting numbers results in the longest gameplay?
  • Are there patterns on the game board?
  • What happens if you change the rules? Think about if instead of the positive difference you allow the use of negative numbers?
    What about if you use fractions?

Additional Resources