Cryptography and Geometry, Beyond the Triangular Grille

Duncan Chiu (

1. Divide an equilateral triangle into three congruent parts (fig. 1). There should be 27 small triangles in each.

2. Number 1 through 27 in each part. You may arrange the 27 numbers randomly, but they must be identical in the three parts (see fig. 1.)

3. Circle 1 through 27 only once (see fig. 1.) Cut out these 27 circles. This is your template for coding your secrete message.

4. Here is how you code your message:
a) Plut the template over a blank triangle of the same size (fig. 2).
b) Fill in your message in the circles from top to bottom and left to right.
c) After fiuing in the 27 circles, rotate the template 120 degree clockwise.
d) Fill in the circles again with the next 27 letters of your message.
e) Repeat steps 4c) and d).

5. To decode the message, you need to provide your friend with the template (the private key.)
a) Place the template over the message triangle.
b) Record the message from the circles.
c) Rotate the template and repeat step 5b) until done.

6. If the message is shorter than 81 characters, you may pack the remaining circles with other letters to confuse people.

7. Extension: To challenge your students, you can ask them to divide the triangle (fig. 1) into different congruent shapes. Each shape must contain 27 triangles and they must be contiguous. There are at least 10 different shapes! See if they can find the maximum number of such. (Note: Only rotation works in triangle.)