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?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
An extra constraint means this Sudoku requires you to think in
diagonals as well as horizontal and vertical lines and boxes of
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?
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.
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.
This Sudoku, based on differences. Using the one clue number can you find the solution?
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?
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.
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?
Use the differences to find the solution to this Sudoku.
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?
Four small numbers give the clue to the contents of the four
How many solutions can you find to this sum? Each of the different letters stands for a different number.
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.
How many different symmetrical shapes can you make by shading triangles or squares?
A pair of Sudoku puzzles that together lead to a complete solution.
Bellringers have a special way to write down the patterns they
ring. Learn about these patterns and draw some of your own.
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?
A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.
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.
The challenge is to find the values of the variables if you are to
solve this Sudoku.
Label this plum tree graph to make it totally magic!
Can you use your powers of logic and deduction to work out the missing information in these sporty situations?
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.
Each of the main diagonals of this sudoku must contain the numbers
1 to 9 and each rectangle width the numbers 1 to 4.
This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?
This sudoku requires you to have "double vision" - two Sudoku's for the price of one
Use the clues about the shaded areas to help solve this sudoku
60 pieces and a challenge. What can you make and how many of the
pieces can you use creating skeleton polyhedra?
Two sudokus in one. Challenge yourself to make the necessary
Rather than using the numbers 1-9, this sudoku uses the nine
different letters used to make the words "Advent Calendar".
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
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.
A Sudoku with clues given as sums of entries.
A man has 5 coins in his pocket. Given the clues, can you work out
what the coins are?
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 Sudoku with a twist.
You need to find the values of the stars before you can apply normal Sudoku rules.
Each clue number in this sudoku is the product of the two numbers in adjacent cells.
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?
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?
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.
Four numbers on an intersection that need to be placed in the
surrounding cells. That is all you need to know to solve this