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

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 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.

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

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?

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?

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

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?

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

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 ...?

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

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

Ben’s class were making cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

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.

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

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

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

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?

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.

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.

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

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

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

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

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?

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

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.

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 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?

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

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

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?

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

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?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

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

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

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

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?

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?

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

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.

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?