Four Triangles - The Art of Problem Solving

I came across a new problem from NRICH last week. It is called Four Triangles Puzzle. I realised it would be a perfect hands on activity for the Roadshow and also is a lovely problem in its own right. If you haven't seen it before have a go at it now before you read on.

Four Triangles Puzzle

If you cut a square diagonally from corner to corner you get four right-angled isosceles triangles.
How many different shapes can you make by fitting the four triangles back together?

You may only fit long sides to long sides and short sides to short sides.

The whole length of the side must be joined.

As a task as it is open to a range of approaches and also can be presented in a variety of ways. I cut the pieces out of acrylic as I wanted a hands on aspect for the Roadshow but you can just as easily cut the pieces out of card, use a visualising approach, doodle on squared paper or use the online interactive.

Visualising: Initially it is nice to spend some time visualising the shapes in your head. There was often an initial focus on shapes that had names; square, parallelogram, trapezium etc. This led to the interesting realisation that there is more than one way to build a parallelogram.

Exploring and Noticing Structure: Next there is a decision to make as to whether we are counting arrangements of pieces or the outlines of shapes. I decided that the shape itself was the important thing and hence these two diagrams count as the same shape. Also rotations and reflections would count as the same. 

Recording: When recording the process I chose dotty paper as I felt I needed the structure. Plain paper would have perhaps been preferable as it gives more freedom and doesn't limit orientation. Here is an example of some initial thoughts. 

Thinking Strategically: When looking for all arrangements that are possible there are many different routes but one thing that it seemed important to notice was that there are three ways of arranging two triangles:

Working systematically: It seemed important here to develop a system that would ensure that no shapes were missed out. I have seen several strategies applied at this point. Here are a few examples:

A common approach was to look for the arrangements of three triangles. There are four possible arrangements. Then one more triangle is added in all orientations to each initial shape. Then duplicates can be removed.

This next approach is to consider each combination of two triangles with each other combination of two triangles. For example square with square, square with triangle, square with parallelogram etc.

Another approach was to label the joins and then find all ways of joining 4 triangles with 3 joins, care was taken to notice where there was more than one possibility and then again to eliminate repeats.

The process of eliminating duplicates seemed important and being able to recognise reflections and rotations led to interesting descriptions of each shape.

I enjoyed sharing this puzzle as there are more solutions than I imagined though few enough that they can be found in a reasonable amount of time, an evening in the pub for example or a lesson.