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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?
Find out about Magic Squares in this article written for students. Why are they magic?!
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
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?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
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?
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?
Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all.
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?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?
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.
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?
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
Charlie and Lynne 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?
The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?
How many different symmetrical shapes can you make by shading triangles or squares?
Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.
Find the values of the nine letters in the sum: FOOT + BALL = GAME
A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?"
A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?
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?
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.
In the multiplication sum, some of the digits have been replaced by letters and others by asterisks. Can you reconstruct the original multiplication?
Can you replace the letters with numbers? Is there only one solution in each case?
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?
Find out what a "fault-free" rectangle is and try to make some of your own.
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.
Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.
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?
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 package contains a collection of problems from the NRICH website that could be suitable for students who have a good understanding of Factors and Multiples and who feel ready to take on some. . . .
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?
An investigation that gives you the opportunity to make and justify predictions.
Can you substitute numbers for the letters in these sums?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
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?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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?
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?
In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?
Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.
Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?
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.
Your challenge is to find the longest way through the network following this rule. You can start and finish anywhere, and with any shape, as long as you follow the correct order.
If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?
Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.
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...