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?
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.
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
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?
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
What happens when you round these numbers to the nearest whole number?
What happens when you round these three-digit numbers to the nearest 100?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
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?
This challenge extends the Plants investigation so now four or more children are involved.
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.
Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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.
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.
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?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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?
Each clue in this Sudoku is the product of the two numbers in adjacent cells.
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?
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?
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 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.
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?
The Zargoes use almost the same alphabet as English. What does this birthday message say?
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?
When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?
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?
These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.
Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
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?
Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.
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.
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 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?
A Sudoku that uses transformations as supporting clues.
You need to find the values of the stars before you can apply normal Sudoku rules.
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?
Given the products of adjacent cells, can you complete this Sudoku?
Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".