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?

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

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

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

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.

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

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?

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

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

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

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. . . .

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

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

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?

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?

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.

Investigate the different ways that fifteen schools could have given money in a charity fundraiser.

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.

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

Surprise your friends with this magic square trick.

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

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.

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

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?

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

A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.

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

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

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

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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

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

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

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?

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.

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

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

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

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?

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.

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

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.

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?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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

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

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?

Find the sum of all three-digit numbers each of whose digits is odd.