Three equilateral triangles ABC, AYX and XZB are drawn with the
point X a moveable point on AB. The points P, Q and R are the
centres of the three triangles. What can you say about triangle
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.
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. . . .
Can you discover whether this is a fair game?
A circle has centre O and angle POR = angle QOR. Construct tangents
at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q
lie inside, or on, or outside this circle?
Prove Pythagoras Theorem using enlargements and scale factors.
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 work through these direct proofs, using our interactive
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
The circumcentres of four triangles are joined to form a
quadrilateral. What do you notice about this quadrilateral as the
dynamic image changes? Can you prove your conjecture?
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
This is an interactivity in which you have to sort the steps in the
completion of the square into the correct order to prove the
formula for the solutions of quadratic equations.
This is an interactivity in which you have to sort into the correct
order the steps in the proof of the formula for the sum of a
Use this interactivity to sort out the steps of the proof of the formula for the sum of an arithmetic series. The 'thermometer' will tell you how you are doing
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
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
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
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.
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?
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.
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.
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.
A composite number is one that is neither prime nor 1. Show that
10201 is composite in any base.
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.
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Follow the hints and prove Pick's Theorem.
The diagram shows a regular pentagon with sides of unit length.
Find all the angles in the diagram. Prove that the quadrilateral
shown in red is a rhombus.
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
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.
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.
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 diagrammatic 'proofs' of algebraic identities and
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.
Tom writes about expressing numbers as the sums of three squares.
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. . . .
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.
An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum orf two or more cubes.
Take a number, add its digits then multiply the digits together,
then multiply these two results. If you get the same number it is
an SP number.
Peter Zimmerman, a Year 13 student at Mill Hill County High School
in Barnet, London wrote this account of modulus arithmetic.
Peter Zimmerman from Mill Hill County High School in Barnet, London
gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is
divisible by 33 for every non negative integer n.
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...
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
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.
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.
An article which gives an account of some properties of magic squares.