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?

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

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

Ben’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?

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

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

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

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

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?

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 ...?

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

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

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.

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

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

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

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.

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

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

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.

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

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 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.

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

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

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

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

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?

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?

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 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 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.

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.

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

Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?

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

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

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?

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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

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

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

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