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

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

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?

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

Are these statements always true, sometimes true or never true?

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?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

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.

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

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?

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

This challenge asks you to imagine a snake coiling on itself.

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

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

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?

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

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

Benâ€™s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

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.

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.

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

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

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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 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.

Make some loops out of regular hexagons. What rules can you discover?

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

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

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

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?

Got It game for an adult and child. How can you play so that you know you will always win?

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

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

Are these statements always true, sometimes true or never true?

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?

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

A game for 2 players with similarities to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.

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?

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?

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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?

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

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

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

Try out this number trick. What happens with different starting numbers? What do you notice?