The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
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?
How many solutions can you find to this sum? Each of the different letters stands for a different 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?
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?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Find out about Magic Squares in this article written for students. Why are they magic?!
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.
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.
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 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?
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?
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.
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?
Special clue numbers related to the difference between numbers in
two adjacent cells and values of the stars in the "constellation"
make this a doubly interesting problem.
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.
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.
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?
This Sudoku, based on differences. Using the one clue number can you find the solution?
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?
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?
Label this plum tree graph to make it totally magic!
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?
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?
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 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?
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
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.
How many different symmetrical shapes can you make by shading triangles or squares?
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?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
The puzzle can be solved with the help of small clue-numbers which
are either placed on the border lines between selected pairs of
neighbouring squares of the grid or placed after slash marks on. . . .
A challenging activity focusing on finding all possible ways of stacking rods.
Can you coach your rowing eight to win?
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.
A package contains a set of resources designed to develop
students’ mathematical thinking. This package places a
particular emphasis on “being systematic” and is
designed to meet. . . .
This challenge extends the Plants investigation so now four or more children are involved.
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. . . .
A pair of Sudokus with lots in common. In fact they are the same problem but rearranged. Can you find how they relate to solve them both?
Can you use your powers of logic and deduction to work out the missing information in these sporty situations?
Use the differences to find the solution to this Sudoku.
60 pieces and a challenge. What can you make and how many of the
pieces can you use creating skeleton polyhedra?
Just four procedures were used to produce a design. How was it
done? Can you be systematic and elegant so that someone can follow
This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?
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?
Four small numbers give the clue to the contents of the four
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?