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

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?

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

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

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?

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

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

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

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. 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.

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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

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

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?

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

A game for 2 players with similaritlies 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.

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

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.

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?

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

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

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.

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

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?

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

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

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

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.

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

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

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?

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.

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.

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.

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

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?

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

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

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

What happens when you round these numbers to the nearest whole number?

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

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

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

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