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?

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

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?

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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

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 explain the strategy for winning this game with any target?

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?

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

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?

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?

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

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'.

Are these statements relating to odd and even numbers 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? Many opportunities to work in different ways.

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

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?

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

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

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?

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

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

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

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

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.

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

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.

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.

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?

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

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?

Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

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

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?

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?

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

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

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?