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

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?

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?

Find the sum of all three-digit numbers each of whose digits is odd.

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?

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

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?

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

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

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

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?

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

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?

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

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?

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

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?

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

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

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

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

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?

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

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.

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

What happens when you round these three-digit numbers to the nearest 100?

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

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.

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

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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

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?

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

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.

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?

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

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

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?

What happens when you round these numbers to the nearest whole number?

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

This challenge asks you to imagine a snake coiling on itself.

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

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.

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

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?

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. . . .

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?

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?