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?

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

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.

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

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

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

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

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?

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

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 pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

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

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.

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

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?

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

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

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

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

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?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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

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?

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?

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

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

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. . . .

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

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

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

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

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?

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.

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.

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

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

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

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?

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

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