One-Cut Challenge: Triangles

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 challenge is 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!

The next step is trying this with an isosceles triangle, whose single line of symmetry still allows this approach to work.

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!

Happy folding!

Have more Fun With Folding!

World Stats Counter

worldometersThis website provides running tallies on several world-wide statistics:

http://www.worldometers.info/

Data on Population, Energy, Economics, and Health are all constantly “updating”, brought you you by the Real Time Statistics Project.

In addition to the obvious questions one could ask, like “At what point will the world’s population grow to over 10 billion?” or “When will the earth run out of oil?”, there are interesting meta-questions like “Where do these models come from?” and “What assumptions are being made to calculate the amount of money spent on weight-loss programs?”.

Another nice resource to play around with!

2011 AIME A #8: Triangular Tables

Foam Table 1I was inspired to have some more fun with folding by a question from this year’s American Invitational Mathematics Examination (AIME) that turned triangles into tables and asked “How high can the table go?”.  (You can find the question here).

Investigating the problem seemed like more fun than solving it, so I cut out a triangle from some foam board and scored lines near the vertices.

Foam Triangle 2

Then I folded the corners and made the following table with an irregular hexagonal top!

Foam Fold

I made a few, to see what kinds of heights I could get.

Foam Tables

There are so many fun questions to explore here!  What comes to mind?

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