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

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

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

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

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.

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

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.

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

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.

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.

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

An AP rectangle is one whose area is numerically equal to its perimeter. If you are given the length of a side can you always find an AP rectangle with one side the given length?

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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?

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

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

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

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?

Are these statements relating to odd and even numbers always true, sometimes true or never true?

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

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

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

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

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

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

How many centimetres of rope will I need to make another mat just like the one I have here?

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.

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

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

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

Explore the effect of combining enlargements.

Can all unit fractions be written as the sum of two unit fractions?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

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

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

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

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

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

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

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?

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 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?

Charlie has moved between countries and the average income of both has increased. How can this be so?