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?

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

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?

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

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?

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.

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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

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?

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

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

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.

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

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.

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

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

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

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

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

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?

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

Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain 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?

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?

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

Are these statements relating to odd and even numbers always true, sometimes true or never true?

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

Can you describe this route to infinity? Where will the arrows take you next?

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

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?

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

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.

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?

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

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

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.

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.

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

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

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.

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

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

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