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

This challenge extends the Plants investigation so now four or more children are involved.

A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

A man has 5 coins in his pocket. Given the clues, can you work out what the coins are?

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

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?

Can you find all the different ways of lining up these Cuisenaire rods?

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

How many different symmetrical shapes can you make by shading triangles or squares?

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

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

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

Find the values of the nine letters in the sum: FOOT + BALL = GAME

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

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

Try out the lottery that is played in a far-away land. What is the chance of winning?

How many solutions can you find to this sum? Each of the different letters stands for a different number.

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

How many different triangles can you make on a circular pegboard that has nine pegs?

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

A package contains a set of resources designed to develop students’ mathematical thinking. This package places a particular emphasis on “being systematic” and is designed to meet. . . .

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Can you find all the different triangles on these peg boards, and find their angles?

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

A mathematician goes into a supermarket and buys four items. Using a calculator she multiplies the cost instead of adding them. How can her answer be the same as the total at the till?

A few extra challenges set by some young NRICH members.

An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.

Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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.

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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.