Baker, Cooper, Jones and Smith are four people whose occupations
are teacher, welder, mechanic and programmer, but not necessarily
in that order. What is each person’s occupation?
After some matches were played, most of the information in the
table containing the results of the games was accidentally deleted.
What was the score in each match played?
This jar used to hold perfumed oil. It contained enough oil to fill
granid silver bottles. Each bottle held enough to fill ozvik golden
goblets and each goblet held enough to fill vaswik crystal. . . .
Write down a three-digit number Change the order of the digits to
get a different number Find the difference between the two three
digit numbers Follow the rest of the instructions then try. . . .
Here are some examples of 'cons', and see if you can figure out where the trick is.
What can you say about the angles on opposite vertices of any
cyclic quadrilateral? Working on the building blocks will give you
insights that may help you to explain what is special about them.
Three teams have each played two matches. The table gives the total
number points and goals scored for and against each team. Fill in
the table and find the scores in the three matches.
In the following sum the letters A, B, C, D, E and F stand for six
distinct digits. Find all the ways of replacing the letters with
digits so that the arithmetic is correct.
We have exactly 100 coins. There are five different values of
coins. We have decided to buy a piece of computer software for
39.75. We have the correct money, not a penny more, not a penny
less! Can. . . .
Eight children enter the autumn cross-country race at school. How
many possible ways could they come in at first, second and third
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
Use the numbers in the box below to make the base of a top-heavy
pyramid whose top number is 200.
If you know the sizes of the angles marked with coloured dots in
this diagram which angles can you find by calculation?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
A huge wheel is rolling past your window. What do you see?
Find the area of the annulus in terms of the length of the chord
which is tangent to the inner circle.
A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.
Carry out cyclic permutations of nine digit numbers containing the
digits from 1 to 9 (until you get back to the first number). Prove
that whatever number you choose, they will add to the same total.
From a group of any 4 students in a class of 30, each has exchanged
Christmas cards with the other three. Show that some students have
exchanged cards with all the other students in the class. How. . . .
What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.
What are the missing numbers in the pyramids?
Nine cross country runners compete in a team competition in which
there are three matches. If you were a judge how would you decide
who would win?
Can you fit Ls together to make larger versions of themselves?
Six points are arranged in space so that no three are collinear.
How many line segments can be formed by joining the points in
I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?
I start with a red, a blue, a green and a yellow marble. I can
trade any of my marbles for three others, one of each colour. Can I
end up with exactly two marbles of each colour?
Choose any three by three square of dates on a calendar page...
This addition sum uses all ten digits 0, 1, 2...9 exactly once.
Find the sum and show that the one you give is the only
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number.
Cross out the numbers on the same row and column. Repeat this
process. Add up you four numbers. Why do they always add up to 34?
Here are three 'tricks' to amaze your friends. But the really
clever trick is explaining to them why these 'tricks' are maths not
magic. Like all good magicians, you should practice by trying. . . .
A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.
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 use the diagram to prove the AM-GM inequality?
These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .
A, B & C own a half, a third and a sixth of a coin collection.
Each grab some coins, return some, then share equally what they had
put back, finishing with their own share. How rich are they?
Replace each letter with a digit to make this addition correct.
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
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?
Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are
natural numbers and 0 < a < b < c. Prove that there is
only one set of values which satisfy this equation.
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.
Which of these roads will satisfy a Munchkin builder?
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?
Can you discover whether this is a fair game?
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.