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

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?

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

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 encourages you to explore dividing a three-digit number by a single-digit 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?

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?

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.

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

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

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?

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

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

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

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?

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?

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

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

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.

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

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

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

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

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?

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

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?

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.

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.

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.

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?

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.

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?

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. . . .

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.

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?

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.

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?

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.

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?

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?

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?