A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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?

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?

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

Different combinations of the weights available allow you to make different totals. Which totals can you make?

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

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

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

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.

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

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

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

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?

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

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

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.

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?

Find out about Magic Squares in this article written for students. Why are they magic?!

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 Sudoku based on clues that give the differences between adjacent cells.

This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?

The clues for this Sudoku are the product of the numbers in adjacent squares.

A Sudoku that uses transformations as supporting clues.

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

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

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.

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.

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.

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?

This sudoku requires you to have "double vision" - two Sudoku's for the price of one

Two sudokus in one. Challenge yourself to make the necessary connections.

Two sudokus in one. Challenge yourself to make the necessary connections.

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

Countries from across the world competed in a sports tournament. Can you devise an efficient strategy to work out the order in which they finished?

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.

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

A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

A pair of Sudoku puzzles that together lead to a complete solution.

Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

A challenging activity focusing on finding all possible ways of stacking rods.

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?

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

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

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.