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?

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

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

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?

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

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

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can. . . .

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

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?

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

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

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. . . .

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

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.

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

There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?

The discs for this game are kept in a flat square box with a square hole for each disc. Use the information to find out how many discs of each colour there are in the box.

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

Given the products of adjacent cells, can you complete this Sudoku?

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

"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 ...?

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

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?

Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?

Have a go at balancing this equation. Can you find different ways of doing it?

Can you work out some different ways to balance this equation?

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

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

The clues for this Sudoku are the product of the numbers in adjacent squares.

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

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

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 do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

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

An environment which simulates working with Cuisenaire rods.

Complete the magic square using the numbers 1 to 25 once each. Each row, column and diagonal adds up to 65.

Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

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