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?

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

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?

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

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

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

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

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?

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.

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

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?

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

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?

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

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

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?

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

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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.

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

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .

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

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.

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

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?

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

Can you see how to build a harmonic triangle? Can you work out the next two rows?

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

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

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

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 Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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

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.

Can all unit fractions be written as the sum of two unit fractions?

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

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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

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. . . .

How many centimetres of rope will I need to make another mat just like the one I have here?

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

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 out what a "fault-free" rectangle is and try to make some of your own.

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

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