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?

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.

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

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

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?

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

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

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

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.

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

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

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?

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 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?

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

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

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

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 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?

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?

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

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

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?

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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

Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?

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

Charlie has moved between countries and the average income of both has increased. How can this be so?

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

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

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.

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. . . .

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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. . . .

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.

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.

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?

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.

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

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?

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

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

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