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

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

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?

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?

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

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

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

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

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.

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?

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

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

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

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.

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.

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

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?

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

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.

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?

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

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

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

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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.

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.

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

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.

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

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?

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?

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

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

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

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

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

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

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.

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.

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

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?

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.

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?