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?

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

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

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?

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

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?

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

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

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!

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

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

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

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

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

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

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?

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

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

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

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

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

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

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

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?

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.

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

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

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

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?

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.

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

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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

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

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?

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

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.

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

Are these statements 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?

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

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

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.

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

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?