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?

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?

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

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

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

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

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

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?

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

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

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?

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

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.

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

What happens when you round these three-digit numbers to the nearest 100?

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

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

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.

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

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

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

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?

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

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

What happens when you round these numbers to the nearest whole number?

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

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?

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.

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

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?

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

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

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

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

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

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

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

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.

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

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?

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

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

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. . . .

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.