Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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.

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.

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 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.

I added together some of my neighbours house numbers. Can you explain the patterns I noticed?

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?”

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.

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

To avoid losing think of another very well known game where the patterns of play are similar.

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

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

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

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

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.

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

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.

An account of some magic squares and their properties and and how to construct them for yourself.

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

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

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

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

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

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

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

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

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

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?

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

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.

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?

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 loser is the player who takes the last counter.

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

What's the largest volume of box you can make from a square of paper?