If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five 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.
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 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.
Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?
How many different symmetrical shapes can you make by shading triangles or squares?
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?
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?
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?
Four friends must cross a bridge. How can they all cross it in just 17 minutes?
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?
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?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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?
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?
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.
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?
A Sudoku with clues as ratios.
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?
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?
The clues for this Sudoku are the product of the numbers in adjacent squares.
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?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
A few extra challenges set by some young NRICH members.
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?
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.
Can you coach your rowing eight to win?
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.
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.
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?
Each clue in this Sudoku is the product of the two numbers in adjacent cells.
Given the products of adjacent cells, can you complete this Sudoku?
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".
You need to find the values of the stars before you can apply normal Sudoku rules.
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?
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?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.
A challenging activity focusing on finding all possible ways of stacking rods.
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 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.
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.
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.