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

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

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?

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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?

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

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.

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

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.

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.

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

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?

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

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?

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

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

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

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?

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

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

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

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?

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?

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?

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. . . .

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

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

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

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

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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

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.

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

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

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

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.

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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

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

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

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.

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

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

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

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

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