Follow this recipe for sieving numbers and see what interesting patterns emerge.

Can you make lines of Cuisenaire rods that differ by 1?

Complete the following expressions so that each one gives a four digit number as the product of two two digit numbers and uses the digits 1 to 8 once and only once.

Find the highest power of 11 that will divide into 1000! exactly.

In how many ways can the number 1 000 000 be expressed as the product of three positive integers?

Take any pair of numbers, say 9 and 14. Take the larger number, fourteen, and count up in 14s. Then divide each of those values by the 9, and look at the remainders.

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

The flow chart requires two numbers, M and N. Select several values for M and try to establish what the flow chart does.

Lyndon chose this as one of his favourite problems. It is accessible but needs some careful analysis of what is included and what is not. A systematic approach is really helpful.

A number N is divisible by 10, 90, 98 and 882 but it is NOT divisible by 50 or 270 or 686 or 1764. It is also known that N is a factor of 9261000. What is N?

6! = 6 x 5 x 4 x 3 x 2 x 1. The highest power of 2 that divides exactly into 6! is 4 since (6!) / (2^4 ) = 45. What is the highest power of two that divides exactly into 100!?

Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?

The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

Factors and Multiples game for an adult and child. How can you make sure you win this game?

A game in which players take it in turns to choose a number. Can you block your opponent?

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. . . .

A man has 5 coins in his pocket. Given the clues, can you work out what the coins are?

An introduction to proof by contradiction, a powerful method of mathematical proof.