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.

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.

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

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

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

Can you explain the strategy for winning this game with any target?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

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

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?

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

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.

Got It game for an adult and child. How can you play so that you know you will always win?

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

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?

Are these statements always true, sometimes true or never true?

Nim-7 game for an adult and child. Who will be the one to take the last counter?

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 activity involves rounding four-digit numbers to the nearest thousand.

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

Here are two kinds of spirals for you to explore. What do you notice?

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

This task follows on from Build it Up and takes the ideas into three dimensions!

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

Can you find all the ways to get 15 at the top of this triangle of numbers?

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?

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Are these statements relating to odd and even numbers always true, sometimes true or never true?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

What happens when you round these three-digit numbers to the nearest 100?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?