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. . . .

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

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

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?

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.

Delight your friends with this cunning trick! Can you explain how it works?

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?

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

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

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 numbers to the nearest whole number?

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.

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?

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

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?

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

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

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

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

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

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

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

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

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

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?

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

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

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!

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 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?

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

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.

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?

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

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

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

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 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.

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

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

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.

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.

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

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?

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