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?

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

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

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?

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.

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.

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

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

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

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?

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

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

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?

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

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.

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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?

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?

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

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

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

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.

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

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

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?

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

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

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

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

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

Can you find all the ways to get 15 at the top of this triangle of numbers?

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

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?

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

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

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

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.

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.

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

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?

This task follows on from Build it Up and takes the ideas into three dimensions!

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?

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.

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

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

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