Explore what happens when you draw graphs of quadratic equations
with coefficients based on a geometric sequence.
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.
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
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. . . .
An inequality involving integrals of squares of functions.
To find the integral of a polynomial, evaluate it at some special
points and add multiples of these values.
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.
The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
Given any two polynomials in a single variable it is always
possible to eliminate the variable and obtain a formula showing the
relationship between the two polynomials. Try this one.
What can you say about the common difference of an AP where every term is prime?
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
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.
A polite number can be written as the sum of two or more
consecutive positive integers. Find the consecutive sums giving the
polite numbers 544 and 424. What characterizes impolite numbers?
A composite number is one that is neither prime nor 1. Show that
10201 is composite in any base.
Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?
Find all the solutions to the this equation.
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.
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.
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.
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.
Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?
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. . . .
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
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...
The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?
Peter Zimmerman, a Year 13 student at Mill Hill County High School
in Barnet, London wrote this account of modulus arithmetic.
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.
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.
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.
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
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.
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
A introduction to how patterns can be deceiving, and what is and is not a proof.
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 I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?
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.
The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .
By proving these particular identities, prove the existence of general cases.
Follow the hints and prove Pick's Theorem.
This is the second article on right-angled triangles whose edge lengths are whole numbers.
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.
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
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?
These proofs are wrong. Can you see why?
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.