Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?

How many different sets of numbers with at least four members can you find in the numbers in this box?

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

"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?

There are ten children in Becky's group. Can you find a set of numbers for each of them? Are there any other sets?

Ben’s class were making cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?

Can you find a way to identify times tables after they have been shifted up?

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

When Charlie asked his grandmother how old she is, he didn't get a straightforward reply! Can you work out how old she is?

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

The number 8888...88M9999...99 is divisible by 7 and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

Katie and Will have some balloons. Will's balloon burst at exactly the same size as Katie's at the beginning of a puff. How many puffs had Will done before his balloon burst?

How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .

Find the words hidden inside each of the circles by counting around a certain number of spaces to find each letter in turn.

The sum of the first 'n' natural numbers is a 3 digit number in which all the digits are the same. How many numbers have been summed?

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

Using the digits 1, 2, 3, 4, 5, 6, 7 and 8, mulitply a two two digit numbers are multiplied to give a four digit number, so that the expression is correct. How many different solutions can you find?

Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?

What is the lowest number which always leaves a remainder of 1 when divided by each of the numbers from 2 to 10?

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

I throw three dice and get 5, 3 and 2. Add the scores on the three dice. What do you get? Now multiply the scores. What do you notice?

The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?

Can you find any perfect numbers? Read this article to find out more...

Nearly all of us have made table patterns on hundred squares, that is 10 by 10 grids. This problem looks at the patterns on differently sized square grids.

Twice a week I go swimming and swim the same number of lengths of the pool each time. As I swim, I count the lengths I've done so far, and make it into a fraction of the whole number of lengths I. . . .

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

A game that tests your understanding of remainders.

I am thinking of three sets of numbers less than 101. They are the red set, the green set and the blue set. Can you find all the numbers in the sets from these clues?

I am thinking of three sets of numbers less than 101. Can you find all the numbers in each set from these clues?

Explore the relationship between simple linear functions and their graphs.

Becky created a number plumber which multiplies by 5 and subtracts 4. What do you notice about the numbers that it produces? Can you explain your findings?

Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?

Can you order the digits from 1-6 to make a number which is divisible by 6 so when the last digit is removed it becomes a 5-figure number divisible by 5, and so on?

An investigation that gives you the opportunity to make and justify predictions.

This article for teachers describes how number arrays can be a useful reprentation for many number concepts.

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

How can you use just one weighing to find out which box contains the lighter ten coins out of the ten boxes?

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

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

Factor track is not a race but a game of skill. The idea is to go round the track in as few moves as possible, keeping to the rules.

These red, yellow and blue spinners were each spun 45 times in total. Can you work out which numbers are on each spinner?

What is the smallest number of answers you need to reveal in order to work out the missing headers?

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?