Some excellent ideas and approaches to this problem. The first part of the problem (when the triangle is larger than the square) is fairly accessible; as is the part where the two equal sides of the triangle shrink to less than half the length of the side of the square but the bit in between is tricky!!

The first solution offered below is based on the work of Thomas, Hannah, Elliot, Lizzie and Hannah of Madras College The group also spent time looking at what happens as the triangle shrinks (not so easy). Well done to all of you.

The second solution is from Andrei of School 205, Bucharest. Thank you for this Andrei

Other excellent solutions were received from Alison, Sheila and Shona, also of Madras College, who have partly answered one of the January problems (if you tackle this problem you can refer to what you did for tilting traingles). I think the January version is a little easier.

Solution one

Introduction

We believed that the triangle took up a quarter of the square, and that a total of four triangles could fit around the square. We created a moving example:

Explanation

We started by rotating a square inside the four triangles as this has the same effect as rotatiing the triangle (editors note:I "liked this bit of lateral thinking").

From our "moving" representation (Fig. 1) we could see that it is always possible to fit four right angled triangles around the centre of the square. This is because the centre of the square allows a 360