Here’s an introduction to the one-cut challenge using triangles, from my Fun with Folding series, suitable for students of all ages (including teachers!). This is a rich, compelling problem that touches on a lot of sophisticated ideas in geometry, but is simple enough to start playing around with right away.
The one-cut challengeis as follows: given a shape made up of connected straight line segments (i.e. a polygonal chain), can you produce the shape as a cut-out using only straight folds and a single straight cut?
A good place to start is with an equilateral triangle. This is a fairly easy problem to solve, given the inherent symmetry in the figure. Fold across any line of symmetry to produce a new figure that looks like two line segments meeting at an angle. Fold those together along their vertex, and cut!
Now the kicker: try it out on a scalene triangle! No more lines of symmetry, and all of the sudden this is a pretty challenging problem!
Have more Fun With Folding!