**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.

- Activity sheet whiteboard: four square problem sheet in sheet protectors.
- Whiteboard markers.
- Erasers or rags.

**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

- Four Square worksheet (pdf).
- Julia Robinson Math Festival Difference Engine worksheet (pdf).
- Peter Trapa’s Diffy Boxes, iterations of the ducci four number game notes (pdf) .
- NY Times Numberplay article (pdf) including discussion of the risk of praise.