I am exactly n times my daughter's age. In m years I shall be ... How old am I?
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.
Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2
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
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.
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.
An equilateral triangle is sitting on top of a square.
What is the radius of the circle that circumscribes this shape?
Can you work through these direct proofs, using our interactive
Investigate the sequences obtained by starting with any positive 2
digit number (10a+b) and repeatedly using the rule 10a+b maps to
10b-a to get the next number in the sequence.
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
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?
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
Can you discover whether this is a fair game?
Prove Pythagoras' Theorem using enlargements and scale factors.
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
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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. . . .
Can you explain why a sequence of operations always gives you perfect squares?
Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...
Can you make sense of these three proofs of Pythagoras' Theorem?
Can you make sense of the three methods to work out the area of the kite in the square?
The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
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/...)).
A introduction to how patterns can be deceiving, and what is and is not a proof.
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.
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. . . .
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
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.
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.
Some diagrammatic 'proofs' of algebraic identities and
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.
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
Tom writes about expressing numbers as the sums of three squares.
By proving these particular identities, prove the existence of general cases.
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.
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
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 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 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.
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.