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

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?

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?

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

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

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

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

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

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

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

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

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?

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?

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

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?

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?

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

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

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

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

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

Delight your friends with this cunning trick! Can you explain how it works?

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

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?

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?

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

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?

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.

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?

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.

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

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?

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

Can you find the values at the vertices when you know the values on the edges?

It would be nice to have a strategy for disentangling any tangled ropes...

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. . . .

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

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

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?

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

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

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.

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

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.

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