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?

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

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

Can you work out how to win this game of Nim? Does it matter if you go first or second?

It would be nice to have a strategy for disentangling any tangled ropes...

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.

Can all unit fractions be written as the sum of two unit fractions?

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

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

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. . . .

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

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

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

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.

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 � 1 [1/3]. What other numbers have the sum equal to the product and can this be so. . . .

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.

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.

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

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

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.

How many moves does it take to swap over some red and blue frogs? Do you have a method?

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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?

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

Can you find the values at the vertices when you know the values on the edges?

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. . . .

Can you find sets of sloping lines that enclose a square?

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

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

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

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

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

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

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

These tasks give learners chance to generalise, which involves identifying an underlying structure.

This article for primary teachers discusses how we can help learners generalise and prove, using NRICH tasks as examples.

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

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.

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

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

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

Can you figure out how sequences of beach huts are generated?

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.

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

Can you find a way of counting the spheres in these arrangements?