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?

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?

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?

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

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

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?

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 explain the strategy for winning this game with any target?

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

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?

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

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

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.

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?

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.

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

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

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?

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

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. . . .

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

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

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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?

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?

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

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

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

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?

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

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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

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

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

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?

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

Can you work out how to win this game of Nim? Does it matter if you go first or second?

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

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?

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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?

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?

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

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