Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?
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.
Make a set of numbers that use all the digits from 1 to 9, once and
once only. Add them up. The result is divisible by 9. Add each of
the digits in the new number. What is their sum? Now try some. . . .
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?
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?
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
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.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
A huge wheel is rolling past your window. What do you see?
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.
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
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.
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?
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. . . .
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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. . . .
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. . . .
If you know the sizes of the angles marked with coloured dots in
this diagram which angles can you find by calculation?
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
Blue Flibbins are so jealous of their red partners that they will
not leave them on their own with any other bue Flibbin. What is the
quickest way of getting the five pairs of Flibbins safely to. . . .
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
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. . . .
The nth term of a sequence is given by the formula n^3 + 11n . Find
the first four terms of the sequence given by this formula and the
first term of the sequence which is bigger than one million. . . .
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Use the numbers in the box below to make the base of a top-heavy
pyramid whose top number is 200.
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
Can you find all the 4-ball shuffles?
The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Here are some examples of 'cons', and see if you can figure out where the trick is.
Eight children enter the autumn cross-country race at school. How
many possible ways could they come in at first, second and third
Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . .
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.
Do you know how to find the area of a triangle? You can count the
squares. What happens if we turn the triangle on end? Press the
button and see. Try counting the number of units in the triangle
now. . . .
Find the smallest positive integer N such that N/2 is a perfect
cube, N/3 is a perfect fifth power and N/5 is a perfect seventh
Can you fit Ls together to make larger versions of themselves?
ABC is an equilateral triangle and P is a point in the interior of
the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP
must be less than 10 cm.
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .
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. . . .
Can you discover whether this is a fair game?
Find the area of the annulus in terms of the length of the chord
which is tangent to the inner circle.
Can you convince me of each of the following: If a square number is
multiplied by a square number the product is ALWAYS a square
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. . . .
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?