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

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?

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

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

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.

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

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

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

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

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

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

To avoid losing think of another very well known game where the patterns of play are similar.

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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

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

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

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.

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

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.

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.

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 for. . . .

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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.

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

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

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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.

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

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

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

Can you describe this route to infinity? Where will the arrows take you next?

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

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.

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

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.

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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?

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

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. . . .

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .