Visualising and representing

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 × 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural numbers.

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

In how many different ways can I colour the five edges of a pentagon so that no two adjacent edges are the same colour?

Place a red counter in the top left corner of a 4×4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves it will take to move the red counter to HOME?

Find out how many pieces of hardboard of differing sizes can fit through a rectangular window.

A 3×8 rectangle is cut into two pieces... then rearranged to form a right-angled triangle. What is the perimeter of the triangle formed?

A cube is made from smaller cubes, 5 by 5 by 5, then some of those cubes are removed. Can you make the specified shapes, and what is the most and least number of cubes required ?

Can you match each graph to one of the statements?