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?
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
I am exactly n times my daughter's age. In m years I shall be ... How old am I?
Can you discover whether this is a fair game?
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.
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.
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
Can you invert the logic to prove these statements?
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.
Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2
Can you work through these direct proofs, using our interactive
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...
Here the diagram says it all. Can you find the diagram?
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.
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. . . .
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
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
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 serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
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 has. . . .
Follow the hints and prove Pick's Theorem.
Can you work out where the blue-and-red brick roads end?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
These proofs are wrong. Can you see why?
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
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
Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
Have a go at being mathematically negative, by negating these
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.
Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
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.
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.
Can you explain why a sequence of operations always gives you perfect 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.
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
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.
Tom writes about expressing numbers as the sums of three squares.
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 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?
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!
Take any prime number greater than 3 , square it and subtract one.
Working on the building blocks will help you to explain what is
special about your results.
We only need 7 numbers for modulus (or clock) arithmetic mod 7
including working with fractions. Explore how to divide numbers and
write fractions in modulus arithemtic.
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
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.
Peter Zimmerman, a Year 13 student at Mill Hill County High School
in Barnet, London wrote this account of modulus arithmetic.
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.
An article which gives an account of some properties of magic squares.