Prove that the shaded area of the semicircle is equal to the area of the inner circle.
This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.
What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?
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. . . .
The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!
The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .
It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
This is the second article on right-angled triangles whose edge lengths are whole numbers.
An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?
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.
Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?
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. . . .
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
Can you discover whether this is a fair game?
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. . . .
Some diagrammatic 'proofs' of algebraic identities and inequalities.
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
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. . . .
Can you use the diagram to prove the AM-GM inequality?
Mark a point P inside a closed curve. Is it always possible to find two points that lie on the curve, such that P is the mid point of the line joining these two points?
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. . . .
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 vertices?
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
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.
Can you make sense of the three methods to work out the area of the kite in the square?
Can you make sense of these three proofs of Pythagoras' Theorem?
Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.
Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry
The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?
Prove Pythagoras' Theorem using enlargements and scale factors.
Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.
This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.
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 number...
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. . . .
ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.
Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.
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.
The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.
Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
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.
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.
Solve this famous unsolved problem and win a prize. Take a positive integer N. If even, divide by 2; if odd, multiply by 3 and add 1. Iterate. Prove that the sequence always goes to 4,2,1,4,2,1...