Powers and roots

How many ways are there to count 1 - 2 - 3 in the array of triangular numbers? What happens with larger arrays? Can you predict for any size array?

Can you work out the value of x in this 'power-full' equation?

A wooden cube with edges of length 12cm is cut into cubes with edges of length 1cm. What is the total length of the all the edges of these centimetre cubes?

Find the smallest numbers a, b, and c such that: a^2 = 2b^3 = 3c^5 What can you say about other solutions to this problem?

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

Find the exact values of x, y and a satisfying the following system of equations: 1/(a+1) = a - 1 x + y = 2a x = ay

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

What is the last digit of the number 1 / 5^903 ?

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?