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Many math puzzles and game rely on the mathematical study of shapes and spaces called topology.

Topology involves looking at the shapes that result through stretching, transforming, deforming, folding and twisting. Tearing, on the other hand, is not allowed!

Some of the shapes involved include polygons and circles which can be manipulated in space to create different shapes. These are a great way for students to explore 2-dimensional shapes in 3-dimensional space.

Trihexaflexagons are ingenious paper contraptions that change their faces as they are folded and flipped. Watch this series for a humorous history of hexaflexagons.

In this activity, students make a paper foldable that changes its faces as it is manipulated.


  • Develop mathematical reasoning.

  • Make connections and solve problems.

  • Explain the importance of observation when doing math.


Key Questions

  • How many different faces does the trihexaflexagon have? How can you tell?

What To Do

  1. Cut out the two large strips of paper along the solid line.
  2. Glue the white sides together, ensuring to match up the shapes.
  3. Crease the paper both ways along each dashed line.This will allow your trihexaflexagon to flex easily.
  4. Follow the steps below to fold your own trihexaflexagon and make your trihexaflexagon flex!

Lay your strip out as shown.

Fold the left hand-side up to begin forming the maple leaf image.

Fold the right side under to complete the maple leaf image.

Tuck the top flap over. Tape the blank triangle to the triangle with words on it.


  • Design your own trihexaflexagon. HINT: Determining which pieces of the puzzle go where in the shape may be easiest if you start by drawing a whole picture and cut the elements to glue in the spaces from the template.
  • Can you make a trihexaflexagon with two different images?