Several pupils from The Mount School in York attempted this problem. Two pupils began to try to explain how they knew they had found all the solutions. They said:

"If you've got a base of 1 unit and a height of 1 unit then there are 3 triangles possible,

if you've a base of 1 and a height of 2 then there are another 3 possible triangles and

a base of 1 with a height of 3 gives another 3.

So you've got 9 triangles with a base of 1"

Here are two diagrams to illustrate this:

This is a good and convincing start - they made 27 triangles but do not appear to have considered triangles whose bases are not horizontal.

Can anyone develop these excellent beginnings? Perhaps the students at The Mount School could put their ideas together to come up with a more "complete" solution.

Solution to Triangular Tantaliser This solution is offered by
Year 8 pupils at Hethersett High School (Andrew, Chris H and Chris
W.

Thank you for your efforts and I am pleased that you were being
systemmatic; thinking of each of a range of different types of
triangle.

It is a good start but I still need some convincing! Some editors
are hard to please!! For instance, I am not sure you have all the
right angled triangles. How can you pursuade me? This certainly
gets us on our way!

Solution

In total we found 26 different triangles without rotating,
reflecting or translating. We did this in groups of different
triangle.

We started off finding all the right-angled traingles: there were
7.

Then we found all the isosceles triangles, then all the irregular
triangles, giving us a total of 26.