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
Can you make sense of these three proofs of Pythagoras' Theorem?
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
There are 12 identical looking coins, one of which is a fake. The
counterfeit coin is of a different weight to the rest. What is the
minimum number of weighings needed to locate the fake coin?
This is the second article on right-angled triangles whose edge lengths are whole numbers.
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...
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?
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?
The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
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!
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.
Relate these algebraic expressions to geometrical diagrams.
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.
A, B & C own a half, a third and a sixth of a coin collection.
Each grab some coins, return some, then share equally what they had
put back, finishing with their own share. How rich are they?
The sum of any two of the numbers 2, 34 and 47 is a perfect square.
Choose three square numbers and find sets of three integers with
this property. Generalise to four integers.
Given that u>0 and v>0 find the smallest possible value of
1/u + 1/v given that u + v = 5 by different methods.
Can you find the value of this function involving algebraic
fractions for x=2000?
Janine noticed, while studying some cube numbers, that if you take
three consecutive whole numbers and multiply them together and then
add the middle number of the three, you get the middle number. . . .
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.
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.
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
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.
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.
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.
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.
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
Follow the hints and prove Pick's Theorem.
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?
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.
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
Find all positive integers a and b for which the two equations:
x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.
Prove that you cannot form a Magic W with a total of 12 or less or
with a with a total of 18 or more.
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 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.
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.
Some diagrammatic 'proofs' of algebraic identities and
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. . . .
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.
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?
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 article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
An article which gives an account of some properties of magic squares.
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.