Challenge Level

Details are given of how check codes are constructed (using modulus arithmetic for passports, bank accounts, credit cards, ISBN book numbers, and so on. A list of codes is given and you have to check. . . .

Challenge Level

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

Challenge Level

Suppose an operator types a US Bank check code into a machine and transposes two adjacent digits will the machine pick up every error of this type? Does the same apply to ISBN numbers; will a machine. . . .

Challenge Level

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.

Challenge Level

On a "move" a stone is removed from two of the circles and placed in the third circle. Here are five of the ways that 27 stones could be distributed.

Challenge Level

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Challenge Level

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

Challenge Level

Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?

Challenge Level

You have worked out a secret code with a friend. Every letter in the alphabet can be represented by a binary value.

Challenge Level

You are given the method used for assigning certain check codes and you have to find out if an error in a single digit can be identified.

Challenge Level

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Challenge Level

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Challenge Level

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.

Challenge Level

What day of the week were you born on? Do you know? Here's a way to find out.

Challenge Level

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

An introduction to the notation and uses of modular arithmetic

Challenge Level

How many different ways can you arrange the officers in a square?

A Latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and exactly once in each column.

Challenge Level

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?

Challenge Level

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?

Challenge Level

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

Challenge Level

Ask a friend to choose a number between 1 and 63. By identifying which of the six cards contains the number they are thinking of it is easy to tell them what the number is.

Challenge Level

Find the values of n for which 1^n + 8^n - 3^n - 6^n is divisible by 6.

Challenge Level

Where will the point stop after it has turned through 30 000 degrees? I took out my calculator and typed 30 000 ÷ 360. How did this help?

Challenge Level

What remainders do you get when square numbers are divided by 4?

Challenge Level

Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?

In this article we shall consider how to solve problems such as "Find all integers that leave a remainder of 1 when divided by 2, 3, and 5."