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

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

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 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.

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?

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

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?

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

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

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.

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

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

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.

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

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.

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?

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?

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?

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

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.

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?

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

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?

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

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.

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

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

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.

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?

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

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

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?

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?

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

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

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?

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?

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

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?

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

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

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

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?