Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). . . .

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

Rotate a copy of the trapezium about the centre of the longest side of the blue triangle to make a square. Find the area of the square and then derive a formula for the area of the trapezium.

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

Can you find the hidden factors which multiply together to produce each quadratic expression?

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to. . . .

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.