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

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

Watch the video of Fran re-ordering these number cards. What do you notice? Try it for yourself. What happens?

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

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?

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

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?

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

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

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.

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?

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?

Ben and his mum are planting garlic. Can you find out how many cloves of garlic they might have had?

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

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

Florence, Ethan and Alma have each added together two 'next-door' numbers. What is the same about their answers?

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?

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.

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

Stop the Clock game for an adult and child. How can you make sure you always win this game?

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

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

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

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?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

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

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

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

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

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

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.

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?

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

What happens when you round these three-digit numbers to the nearest 100?

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

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

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.

Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

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

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

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.

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?

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

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

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