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.
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?
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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.
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
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?
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.
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?
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?
The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?
Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.
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.
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?
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?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?
Find the values of the nine letters in the sum: FOOT + BALL = GAME
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.
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?
Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?
Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.
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?
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.
Pentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?
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?
An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.
This challenge extends the Plants investigation so now four or more children are involved.
The clues for this Sudoku are the product of the numbers in adjacent squares.
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
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?
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...
Use the differences to find the solution to this Sudoku.
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
A challenging activity focusing on finding all possible ways of stacking rods.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Find out about Magic Squares in this article written for students. Why are they magic?!
Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?
Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?
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.
You need to find the values of the stars before you can apply normal Sudoku rules.
Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".
This Sudoku, based on differences. Using the one clue number can you find the solution?
Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?
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.
Given the products of adjacent cells, can you complete this Sudoku?
A few extra challenges set by some young NRICH members.