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?

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

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?

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

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?

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

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

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

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

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

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.

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

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

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.

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.

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?

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

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

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

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

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?

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.

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.

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

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

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

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

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?

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

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?

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

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?

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

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?

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

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.

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?

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?

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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?

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

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. . . .

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

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