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?

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

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.

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.

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

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.

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?

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

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

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?

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?

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

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. . . .

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

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

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'.

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

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

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

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?

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

A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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.

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

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

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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?

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

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?

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.

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

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

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?

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.

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

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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?

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?

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

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.

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?

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?