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?

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?

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

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

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

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

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

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

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?

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.

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

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?

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?

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

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

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.

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?

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

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

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?

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

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.

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

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.

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

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

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

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?

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

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?

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?

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

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?

How many centimetres of rope will I need to make another mat just like the one I have here?

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?

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?

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.

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

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

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

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

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

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.

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.

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

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?