Why are there only a few lattice points on a hyperbola and infinitely many on a parabola?

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.

Is this a fair game? How many ways are there of creating a fair game by adding odd and even numbers?

Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.

Which of these continued fractions is bigger and why?

Find the sum, f(n), of the first n terms of the sequence: 0, 1, 1, 2, 2, 3, 3........p, p, p +1, p + 1,..... Prove that f(a + b) - f(a - b) = ab.