Impossible picture?
Under what circumstances can you rearrange a big square to make three smaller squares?
Age
14 to 16
Challenge level
Problem
Take a look at the picture below:
Image
It looks like you can dissect a square into 6 pieces, and reassemble it into 3 new squares:
Image
In practice, this only works when the dissection has particular proportions.
Can you describe when it is possible?
Click below to reveal some hints.
Use this interactive to see how the pieces could change.
Can you describe when the orange parts would make a square?
Can you describe when the orange parts would make a square?
You might need to use Pythagoras' Theorem.
You might need to express your answer using surds.
You might need to express your answer using surds.
What is the ratio of the areas of the 3 squares?
With thanks to Don Steward, whose ideas formed the basis of this problem.
Getting Started
Try cutting some squares of paper up and moving the pieces as shown in the pictures.
Teachers' Resources
Why do this problem:
In this curious problem the diagram shows a seemingly impossible situation but by showing some perseverence and when provided with appropriate scaffolding, students can make sense of what's going on. This problem forces us to explore our assumptions when working with diagrams. Diagrams can be misleading, and in investigating this problem students will need to question all our assumptions
carefully and justify their strategies.
Possible approach :
Project the first diagram on the board. "Here is a square that has been cut up into 6 pieces. the 6 pieces rearrange to give 3 squares." Show the second picture.Give each student a few squares of paper to experiment with and challenge them to cut their square up as shown in the picture. It is unlikely that anyone will manage to make 3 exact squares from the pieces, and even if they do, the correct dimensions may not be obvious.
After students have had a a chance to explore bring the class together and discuss what they have found out. You may wish to show this interactive to see how the pieces change. Invite students to think about what constraints are needed to make the third shape a square.
Once students have discussed the necessary conditions for the third shape to be a square, invite them to try to find appropriate dimensions. Some students may use Pythagoras' Theorem; others might try to get as close to a square as they can using trial and improvement.
Finally you could ask your students to find the ratios of the areas of the 3 squares.
Key questions :
What assumptions have you made?Can you describe what properties the diagram has when it is possible to make 3 squares?
How would you convince someone else?
Possible extension :
Students could create their own version of the interactive in GeoGebra.