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

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?

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.

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

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?

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?

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

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

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

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

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.

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

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

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?

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.

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?

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?

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

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

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

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

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?

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

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

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?

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.

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?

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

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

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

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

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

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

Can you find all the ways to get 15 at the top of this triangle of 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.

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

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

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?

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.

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

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

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

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.