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?

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

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

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

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 find a way of counting the spheres in these arrangements?

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?

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?

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

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

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 ...?

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

Watch this animation. What do you see? Can you explain why this happens?

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

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

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

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?

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

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

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

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

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

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

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

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.

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.

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?

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 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.

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

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

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.

Explore the effect of reflecting in two intersecting mirror lines.

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

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

Explore the effect of combining enlargements.

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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

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

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?

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

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?

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.