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.

What is the remainder when 2^{164}is divided by 7?

Can you find ways to put numbers in the overlaps so the rings have equal totals?

An introduction to the notation and uses of modular arithmetic

Data is sent in chunks of two different sizes - a yellow chunk has 5 characters and a blue chunk has 9 characters. A data slot of size 31 cannot be exactly filled with a combination of yellow and. . . .

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...

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

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

All strange numbers are prime. Every one digit prime number is strange and a number of two or more digits is strange if and only if so are the two numbers obtained from it by omitting either. . . .

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

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 guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

Can you explain why a sequence of operations always gives you perfect squares?

Which numbers can we write as a sum of square numbers?

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

Just because a problem is impossible doesn't mean it's difficult...

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

This article explains how credit card numbers are defined and the check digit serves to verify their accuracy.

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.