Suppose A always beats B and B always beats C, then would you
expect A to beat C? Not always! What seems obvious is not always
true. Results always need to be proved in mathematics.
Prove that you cannot form a Magic W with a total of 12 or less or
with a with a total of 18 or more.
How many different cubes can be painted with three blue faces and
three red faces? A boy (using blue) and a girl (using red) paint
the faces of a cube in turn so that the six faces are painted. . . .
The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . .
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. . . .
Let a(n) be the number of ways of expressing the integer n as an
ordered sum of 1's and 2's. Let b(n) be the number of ways of
expressing n as an ordered sum of integers greater than 1. (i)
Calculate. . . .
Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?
The sum of any two of the numbers 2, 34 and 47 is a perfect square.
Choose three square numbers and find sets of three integers with
this property. Generalise to four integers.
How many tours visit each vertex of a cube once and only once? How
many return to the starting point?
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.
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?
The tangles created by the twists and turns of the Conway rope
trick are surprisingly symmetrical. Here's why!
A 'doodle' is a closed intersecting curve drawn without taking
pencil from paper. Only two lines cross at each intersection or
vertex (never 3), that is the vertex points must be 'double points'
not. . . .
The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .
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. . . .
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. . . .
Explain why, when moving heavy objects on rollers, the object moves
twice as fast as the rollers. Try a similar experiment yourself.
Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2
Four jewellers possessing respectively eight rubies, ten saphires,
a hundred pearls and five diamonds, presented, each from his own
stock, one apiece to the rest in token of regard; and they. . . .
Can you find the value of this function involving algebraic
fractions for x=2000?
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Start with any triangle T1 and its inscribed circle. Draw the
triangle T2 which has its vertices at the points of contact between
the triangle T1 and its incircle. Now keep repeating this. . . .
Janine noticed, while studying some cube numbers, that if you take
three consecutive whole numbers and multiply them together and then
add the middle number of the three, you get the middle number. . . .
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
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. . . .
Find all positive integers a and b for which the two equations:
x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
Can you discover whether this is a fair game?
A blue coin rolls round two yellow coins which touch. The coins are
the same size. How many revolutions does the blue coin make when it
rolls all the way round the yellow coins? Investigate for a. . . .
A point moves around inside a rectangle. What are the least and the
greatest values of the sum of the squares of the distances from the
In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.
This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.
Some diagrammatic 'proofs' of algebraic identities and
Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .
Some puzzles requiring no knowledge of knot theory, just a careful
inspection of the patterns. A glimpse of the classification of
knots and a little about prime knots, crossing numbers and. . . .
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
An article about the strategy for playing The Triangle Game which
appears on the NRICH site. It contains a simple lemma about
labelling a grid of equilateral triangles within a triangular
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.
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
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
These proofs are wrong. Can you see why?
Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.
Can you make sense of these three proofs of Pythagoras' Theorem?
Relate these algebraic expressions to geometrical diagrams.
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
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?
A connected graph is a graph in which we can get from any vertex to
any other by travelling along the edges. A tree is a connected
graph with no closed circuits (or loops. Prove that every tree. . . .