Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
You have twelve weights, one of which is different from the rest.
Using just 3 weighings, can you identify which weight is the odd
one out, and whether it is heavier or lighter than the rest?
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.
Take three whole numbers. The differences between them give you
three new numbers. Find the differences between the new numbers and
keep repeating this. What happens?
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.
The challenge is to find the values of the variables if you are to
solve this Sudoku.
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?
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?
Find out about Magic Squares in this article written for students. Why are they magic?!
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?
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?
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?
Bellringers have a special way to write down the patterns they
ring. Learn about these patterns and draw some of your own.
Solve the equations to identify the clue numbers in this Sudoku problem.
You need to find the values of the stars before you can apply normal Sudoku rules.
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?
A Latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and exactly once in each column.
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?
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.
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?
A Sudoku with a twist.
This is a variation of sudoku which contains a set of special clue-numbers. Each set of 4 small digits stands for the numbers in the four cells of the grid adjacent to this set.
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
Four numbers on an intersection that need to be placed in the
surrounding cells. That is all you need to know to solve this
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?
How many different symmetrical shapes can you make by shading triangles or squares?
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Find all the ways of placing the numbers 1 to 9 on a W shape, with
3 numbers on each leg, so that each set of 3 numbers has the same
in how many ways can you place the numbers 1, 2, 3 … 9 in the
nine regions of the Olympic Emblem (5 overlapping circles) so that
the amount in each ring is the same?
Label this plum tree graph to make it totally 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?
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.
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. . . .
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
A Sudoku with clues as ratios.
This sudoku requires you to have "double vision" - two Sudoku's for
the price of one
Solve this Sudoku puzzle whose clues are in the form of sums of the
numbers which should appear in diagonal opposite cells.
This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?
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.
Just four procedures were used to produce a design. How was it
done? Can you be systematic and elegant so that someone can follow
60 pieces and a challenge. What can you make and how many of the
pieces can you use creating skeleton polyhedra?
Four small numbers give the clue to the contents of the four
Use the differences to find the solution to this Sudoku.
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?
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. . . .