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?

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

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

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

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.

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

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

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

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.

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

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

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

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

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?

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?

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

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?

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. . . .

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?

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

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

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?

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

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

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?

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

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?

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. . . .

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

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

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

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

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

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

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

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.

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

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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?

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?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?