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.

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?

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

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

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.

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

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

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

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

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

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

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.

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

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?

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

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?

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

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?

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.

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

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.

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

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?

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.

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?

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

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?

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?

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

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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

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

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.

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

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?

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

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.

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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?

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

Find out what a "fault-free" rectangle is and try to make some of your own.

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

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

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