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

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

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

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?

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

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 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?

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?

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. . . .

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.

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.

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

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

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

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

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

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.

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.

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

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?

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

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

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

Can all unit fractions be written as the sum of two unit fractions?

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

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

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

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?

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

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

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.

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

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

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?

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?

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?

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. . . .

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

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?

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

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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

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

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.

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

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

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