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

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

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

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

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

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.

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?

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

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?

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?

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

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

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

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

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.

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

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.

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?

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?

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

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?

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.

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?

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

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?

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

Can you work out how to win this game of Nim? Does it matter if you go first or second?

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.

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

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

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

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

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?

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?

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. . . .

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

Are these statements always true, sometimes true or never true?

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

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

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

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

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

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

Watch this animation. What do you see? Can you explain why this happens?

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

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