Can you discover whether this is a fair game?
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 PQR?
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?
Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry
Investigate circuits and record your findings in this simple introduction to truth tables and logic.
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.
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
Keep constructing triangles in the incircle of the previous triangle. What happens?
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
Can you work through these direct proofs, using our interactive proof sorters?
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. . . .
Here the diagram says it all. Can you find the diagram?
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. . . .
Can you correctly order the steps in the proof of the formula for the sum of a geometric series?
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.
Prove Pythagoras' Theorem using enlargements and scale factors.
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.
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.
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.
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...
Kyle and his teacher disagree about his test score - who is right?
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.
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.
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.
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. . . .
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. . . .
Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.
Follow the hints and prove Pick's Theorem.
This is the second article on right-angled triangles whose edge lengths are whole numbers.
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
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.
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 article which gives an account of some properties of magic squares.
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 of two or more cubes.
Some diagrammatic 'proofs' of algebraic identities and inequalities.
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.
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.
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
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.
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.
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.
Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.
With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?