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

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

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. . . .

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?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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?

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

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

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

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.

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.

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.

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.

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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

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

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

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

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

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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?

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

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.

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?

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?

Can you find sets of sloping lines that enclose a square?

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

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

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

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.

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

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

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

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

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

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

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

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

Charlie has moved between countries and the average income of both has increased. How can this be so?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

It starts quite simple but great opportunities for number discoveries and patterns!

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

How many moves does it take to swap over some red and blue frogs? Do you have a method?

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?