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Going First
This article shows how abstract thinking and a little number theory throw light on the scoring in Go.
History of Morse
This short article, gives an outline of the origins of Morse code and its inventor and how the frequency of letters is reflected in the code they were given.
The Knapsack Problem and Public Key Cryptography
An example of a simple Public Key code, called the Knapsack Code is described in this article, alongside some information on its origins. A knowledge of modular arithmetic is useful.
Calendars
Calendars were one of the earliest calculating devices developed by civilisations. Find out about the Mayan calendar in this article.
Learn about Number Bases
We are used to writing numbers in base ten, using 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Eg. 75 means 7 tens and five units. This article explains how numbers can be written in any number base.
When the Angles of a Triangle Don't Add up to 180 Degrees
This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the triangle.
Latin Squares
A Latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and exactly once in each column.
The Harmonic Triangle and Pascal's Triangle
The harmonic triangle is built from fractions with unit numerators using a rule very similar to Pascal's triangle.
Muggles, Logo and Gradients
Logo helps us to understand gradients of lines and why Muggles Magic is not magic but mathematics. See the problem Muggles magic.
A Knight's Journey
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
Public Key Cryptography
An introduction to the ideas of public key cryptography using small numbers to explain the process. In practice the numbers used are too large to factorise in a reasonable time.
Ding Dong Bell
The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.
Vectors - What Are They?
The article provides a summary of the elementary ideas about vectors usually met in school mathematics, describes what vectors are and how to add, subtract and multiply them by scalars and indicates why they are useful.
Multiplication of Vectors
An account of multiplication of vectors, both scalar products and vector products.
Playing Squash
Playing squash involves lots of mathematics. This article explores the mathematics of a squash match and how a knowledge of probability could influence the choices you make.
Transitivity
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.
Logic, Truth Tables and Switching Circuits Challenge
Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record your findings.
Some Circuits in Graph or Network Theory
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
What Are Complex Numbers?
This article introduces complex numbers, brings together into one bigger 'picture' some closely related elementary ideas like vectors and the exponential and trigonometric functions and their derivatives and proves that e^(i pi)= -1.
About Pythagorean Golden Means
What is the relationship between the arithmetic, geometric and harmonic means of two numbers, the sides of a right angled triangle and the Golden Ratio?
Small Groups
Learn about the rules for a group and the different groups of 4 elements by doing some simple puzzles.
Golden Mathematics
A voyage of discovery through a sequence of challenges exploring properties of the Golden Ratio and Fibonacci numbers.
Euclid's Algorithm II
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.
A Rolling Disc - Periodic Motion
Imagine a rectangular tray lying flat on a table. Suppose that a plate lies on the tray and rolls around, in contact with the sides as it rolls. What can we say about the motion?
Curvature of Surfaces
How do we measure curvature? Find out about curvature on soccer and rugby balls and on surfaces of negative curvature like banana skins.
Continued Fractions I
An article introducing continued fractions with some simple puzzles for the reader.
Approximations, Euclid's Algorithm & Continued Fractions
This article sets some puzzles and describes how Euclid's algorithm and continued fractions are related.
Continued Fractions II
In this article we show that every whole number can be written as a continued fraction of the form k/[1+k/[1+k/etc.]].
Euclid's Algorithm I
How can we solve equations like 13x + 29y = 42 or 2x +4y = 13 with the solutions x and y being integers? Read this article to find out.
Divisibility Tests
Tim Rowland takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.
Fractional Calculus II
Here explore some ideas of how the definitions and methods of calculus change if you integrate or differentiate n times when n is not a whole number.
Pythagorean Triples I
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!
Fractional Calculus III
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
Pythagorean Triples II
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Sums and Products of Digits and SP Numbers
This article explores the search for SP numbers, finding the few that exist and the proof that there are no more.
Whole Number Dynamics I
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
Whole Number Dynamics II
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.
Euler's Formula and Topology
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 relationship between Euler's formula and angle deficiency of polyhedra.
How Many Geometries Are There?
An account of how axioms underpin geometry and how by changing one axiom we get an entirely different geometry.
Whole Number Dynamics III
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.
Sums of Squares and Sums of Cubes
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 orf two or more cubes.
Whole Number Dynamics IV
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?
Whole Number Dynamics V
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.
Impossible Polyhedra
Is it possible to make an irregular polyhedron using only polygons of, say, six, seven and eight sides? The answer (rather surprisingly) is 'no', but how do we prove a statement like this?
Fractional Calculus I
You can differentiate and integrate n times but what if n is not a whole number? This generalisation of calculus was introduced and discussed on askNRICH by some school students.
Geometry and Gravity 1
This article (the first of two) contains ideas for investigations. Space-time, the curvature of space and topology are introduced with some fascinating problems to explore.
Geometry and Gravity 2
This is the second of two articles and discusses problems relating to the curvature of space, shortest distances on surfaces, triangulations of surfaces and representation by graphs.
Telescoping Functions
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.
The Dodecahedron
What is the shortest distance through the middle of a dodecahedron between the centres of two opposite faces?
Divided Differences
When in 1821 Charles Babbage invented the `Difference Engine' it was intended to take over the work of making mathematical tables by the techniques described in this article.
Magic Squares
An account of some magic squares and their properties and and how to construct them for yourself.
Frieze Patterns in Cast Iron
A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.
Links and Knots
Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots, prime knots, crossing numbers and knot arithmetic.
Try to Win
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...
Picturing Pythagorean Triples
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.
Sum the Series
This article by Alex Goodwin, age 18 of Madras College, St Andrews describes how to find the sum of 1 + 22 + 333 + 4444 + ... to n terms.
Why Stop at Three by One
Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.
8 Methods for Three by One
Two 18 year old students from Madras College St Andrews in Scotland produced eight different proofs of one result using (separately) Tan Angle Sum Formula, Sin Angle Sum Formula, Cosine Rule, Vectors, Matrices, Complex Numbers, Pure Geometry and Coordinate Geometry. Brilliant stuff.
Modulus Arithmetic and a Solution to Differences
Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.
Modulus Arithmetic and a Solution to Dirisibly Yours
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.
Recent Developments on S.P. Numbers
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.
A Method of Defining Coefficients in the Equations of Chemical Reactions
A simple method of defining the coefficients in the equations of chemical reactions with the help of a system of linear algebraic equations.
The Eternity Puzzle
A big prize was offered for solving The Eternity Puzzle, a jigsaw with no picture and every piece is the same on both sides. The finished result forms a regular dodecagon (12 sided polygon).
Earth Shapes
What if the Earth's shape was a cube or a cone or a pyramid or a saddle ... See some curious worlds here.
The Use of Mathematics in Computer Games
An account of how mathematics is used in computer games including geometry, vectors, transformations, 3D graphics, graph theory and simulations.
Impuzzable
This is about a fiendishly difficult jigsaw and how to solve it using a computer program.
An Introduction to Complex Numbers
A short introduction to complex numbers written primarily for students aged 14 to 19.
An Introduction to Galois Theory
This article only skims the surface of Galois theory and should probably be accessible to a 17 or 18 year old school student with a strong interest in mathematics.
Sperner's Lemma
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 frame.
To Prove or Not to Prove
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
A Computer Program to Find Magic Squares
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.
An Alphanumeric
Freddie Manners, of Packwood Haugh School in Shropshire solved an alphanumeric without using the extra information supplied and this article explains his reasoning.
The Kth Sum of N Numbers
Yatir from Israel describes his method for summing a series of triangle numbers.
2^n -n Numbers
Yatir from Israel wrote this article on numbers that can be written as 2^n-n where n is a positive integer.
Dancing with Maths
An article for students and teachers on symmetry and square dancing. What do the symmetries of the square have to do with a dos-e-dos or a swing? Find out more?
Mathematics in the Financial Markets
Financial markets mean the business of trading risk. The article describes in simple terms what is involved in this trading, the work people do and the figures for starting salaries.
Infinite Continued Fractions
In this article we are going to look at infinite continued fractions - continued fractions that do not terminate.
Conic Sections
The interplay between the two and three dimensional Euclidean geometry of conic sections is explored in this article. Suitable for students from 16+, teachers and parents.
Sprouts
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 significant food for thought.
Infinity Is Not a Number - It's a Free Man
Infinity is not a number, and trying to treat it as one tends to be a pretty bad idea. At best you're likely to come away with a headache, at worse the firm belief that 1 = 0. This article discusses the different types of infinity.
An Introduction to Mathematical Structure
An introduction to the sort of algebra studied at university, focussing on groups.
Mathematical Induction
This article gives an introduction to mathematical induction, a powerful method of mathematical proof.
Euler's Formula
Some simple ideas about graph theory with a discussion of a proof of Euler's formula relating the numbers of vertces, edges and faces of a graph.
The Chinese Remainder Theorem
In this article we shall consider how to solve problems such as "Find all integers that leave a remainder of 1 when divided by 2, 3, and 5."
Proof by Contradiction
An introduction to proof by contradiction, a powerful method of mathematical proof.
Corresponding Sudokus
This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
Where Do We Get Our Feet Wet?
Jenny Piggott chose this article. Professor Körner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
Drawing Doodles and Naming Knots
This article for students introduces the idea of naming knots using numbers. You'll need some paper and something to write with handy!
How Many Elements Are There in the Cantor Set?
This article gives a proof of the uncountability of the Cantor set.
Randomness and Brownian Motion
In Classical times the Pythagorean philosophers believed that all things were made up from a specific number of tiny indivisible particles called ‘monads’. Each object contained a different number of particles, and so they believed that ‘everything was number’.
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