### All in the Mind

Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube?

### Instant Insanity

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

### Is There a Theorem?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

# Counting Triangles

##### Age 11 to 14Challenge Level

This solution was submitted by Andrei Lazanu from School 205, Bucharest, Romania. Congratulations Andrei.

To calculate the total number of triangles, I use the combinations without repetition, because in a triangle it doesn't matter about the order of the vertices and without repetition I can't put one vertex of the cube in a triangle more than once. Now, I apply the formula for the combinations without repetition: $${C_n^k}= {{n!} \over {k!(n-k)!}}$$I now apply the formula for $k=3$ and $n=8$: $${C_8^3} = {{8!} \over {3!(8-3)!}} = {{8\times 7\times 6\times 5\times 4\times 3\times 2\times 1} \over {3\times 2\times 1\times 5\times 4\times 3\times 2\times 1}} = {56}$$

So, the total number of triangles is 56.

Now, I search for all the different types of triangles, knowing that the sides of the triangle could be sides of the cube (a), diagonals of the sides of the cube (b) and diagonals of the cube (c).

I write all the possibilities:

 Combination Observation aaa Impossible aab OK aac Impossible abb Impossible abc OK acc Impossible bbb OK ccc Impossible bbc Impossible bcc Impossible

So, there are only three different types of triangle.