Puzzles & Problems 2

1. Colouring Regions (2)

The diagram shows a regular pentagon with its five diagonals drawn.

The diagonals split the pentagon into eleven regions.

Some of the regions can be shaded to make different patterns.

I can shade any number of regions (from 0 to 11 regions) to make the patterns.

Three different patterns are shown below.

How many different patterns can be made in total?

2. Tangent Sums

Can you find three angles A, B & C such that:

tan(A) + tan(B) + tan(C) = tan(A) tan(B) tan(C)

How many solutions exist and what can you say about them?

3. Powerful Numbers

The number 407 is an unusual number because:

Can you find any other numbers that are equal to the sum of the cubes of their digits?