Hula hoops were all the rage in 1958. Yo-yos, popular before World War II, were relaunched in the 1960s. Rubik's Cube, invented in 1974, quickly became a global craze. Sudoku, which had been around for years, was wildly popular when it started to appear in American and European newspapers in 2004. The latest fad is … Continue reading Why Waffle when One Wordle Do?
Tag: Recreational Maths
John Horton Conway: a Charismatic Genius
John Horton Conway was a charismatic character, something of a performer, always entertaining his fellow-mathematicians with clever magic tricks, memory feats and brilliant mathematics. A Liverpudlian, interested from early childhood in mathematics, he studied at Gonville & Caius College in Cambridge, earning a BA in 1959. He obtained his PhD five years later, after which … Continue reading John Horton Conway: a Charismatic Genius
Spiralling Primes
The prime numbers have presented mathematicians with some of their most challenging problems. They continue to play a central role in number theory, and many key questions remain unsolved. Order and Chaos The primes have many intriguing properties. In his article ``The first 50 million prime numbers'', Don Zagier noted two contradictory characteristics of the … Continue reading Spiralling Primes
Learning Maths without even Trying
Children have an almost limitless capacity to absorb knowledge if it is presented in an appealing and entertaining manner. Mathematics can be daunting, but it is possible to convey key ideas visually so that they are instantly accessible. Visiting Explorium recently, I saw such a visual display demonstrating the theorem of Pythagoras, which, according to … Continue reading Learning Maths without even Trying
Rambling and Reckoning
A walk on the beach, in the hills or along a river bank provides great opportunities for mathematical reflection. How high is the mountain? How many grains of sand are on the beach? How much water is flowing in the river? [TM156 or search for “thatsmaths” at irishtimes.com]. Daily average flow (cubic metres per second) at … Continue reading Rambling and Reckoning
Our Dearest Problems
A Colloquium on Recreational Mathematics took place in Lisbon this week. The meeting, RMC-VI (G4GEurope), a great success, was organised by the Ludus Association, with support from several other agencies: MUHNAC, ULisboa, CMAF-IO, CIUHCT, CEMAPRE, and FCT. It was the third meeting integrated in the Gathering for Gardner movement, which celebrates the great populariser of … Continue reading Our Dearest Problems
Discoveries by Amateurs and Distractions by Cranks
Do amateurs ever solve outstanding mathematical problems? Professional mathematicians are aware that almost every new idea they have about a mathematical problem has already occurred to others. Any really new idea must have some feature that explains why no one has thought of it before [TM155 or search for “thatsmaths” at irishtimes.com]. It is both difficult and … Continue reading Discoveries by Amateurs and Distractions by Cranks
Tom Lehrer: Comical Musical Mathematical Genius
Tom Lehrer, mathematician, singer, songwriter and satirist, was born in New York ninety years ago. He was active in public performance for about 25 years from 1945 to 1970. He is most renowned for his hilarious satirical songs, many of which he recorded and which are available today on YouTube [see TM147, or search for “thatsmaths” … Continue reading Tom Lehrer: Comical Musical Mathematical Genius
Kaprekar’s Number 6174
The Indian mathematician D. R. Kaprekar spent many happy hours during his youth solving mathematical puzzles. He graduated from Fergusson College in Pune in 1929 and became a mathematical teacher at a school in Devlali, north-east of Mumbai. Kaprekar is remembered today for a range of curious mathematical patterns that he discovered. The best known … Continue reading Kaprekar’s Number 6174
Learning Maths has never been Easier
Maths is hard: many people find it inscrutable and have negative attitudes towards maths. They may have bad memories of school maths or have been told they lack mathematical talents. This is unfortunate: we all have the capacity to apply reasoning and logic and we can all do maths. Given the vital role mathematics plays … Continue reading Learning Maths has never been Easier
Fractions of Fractions of Fractions
Numbers can be expressed in several different ways. We are familiar with whole numbers, fractions and decimals. But there is a wide range of other forms, and we examine one of them in this article. Every rational number $latex {x}&fg=000000$ can be expanded as a continued fraction: $latex \displaystyle x = a_0 + \cfrac{1}{ a_1 … Continue reading Fractions of Fractions of Fractions
Patterns in Poetry, Music and Morse Code
Suppose we have to ascent a flight of stairs and can take only one or two steps at a time. How many different patterns of ascent are there? We start with the simplest cases. With one step there is only one way; with two, there are two: take two single steps or one double step. … Continue reading Patterns in Poetry, Music and Morse Code
The Beer Mat Game
Alice and Bob, are enjoying a drink together. Sitting in a bar-room, they take turns placing beer mats on the table. The only rules of the game are that the mats must not overlap or overhang the edge of the table. The winner is the player who puts down the final mat. Is there a … Continue reading The Beer Mat Game
Torricelli’s Trumpet & the Painter’s Paradox
Evangelista Torricelli, a student of Galileo, is remembered as the inventor of the barometer. He was also a talented mathematician and he discovered the remarkable properties of a simple geometric surface, now often called Torricelli's Trumpet. It is the surface generated when the curve $latex {y=1/x}&fg=000000$ for $latex {x\ge1}&fg=000000$ is rotated in 3-space … Continue reading Torricelli’s Trumpet & the Painter’s Paradox
Numerical Coincidences
A numerical coincidence is an equality or near-equality between different mathematical quantities which has no known theoretical explanation. Sometimes such equalities remain mysterious and intriguing, and sometimes theory advances to the point where they can be explained and are no longer regarded as surprising. Simple Examples A simple example is the near-equality between 2 cubed … Continue reading Numerical Coincidences
Topology in the Oval Office
Imagine a room – the Oval Office for example – that has three electrical appliances: • An air-conditioner ( a ) with an American plug socket ( A ), • A boiler ( b ) with a British plug socket ( B ), • A coffee-maker ( c ) with a Continental plug socket ( … Continue reading Topology in the Oval Office
Metallic Means
Everyone knows about the golden mean. It must be one of the most written-about numbers, certainly in recreational mathematics. It is usually denoted by $latex {\phi}&fg=000000$ and is the positive root of the quadratic equation $latex \displaystyle x^2 - x - 1 = 0 \ \ \ \ \ (1)&fg=000000$ with the value $latex {\phi … Continue reading Metallic Means
That’s Maths Book Published
A book of mathematical articles, That's Maths, has just been published. The collection of 100 articles includes pieces that have appeared in The Irish Times over the past few years, blog posts from this website and a number of articles that have not appeared before. The book has been published by Gill Books and copies … Continue reading That’s Maths Book Published
Recreational Mathematics is Fun
We all love music, beautiful paintings and great literature without being trained musicians, talented artists or accomplished writers. It is the same with mathematics: we can enjoy the elegance of brilliant logical arguments and appreciate the beauty of mathematical structures and symmetries without being skilled creators of new theorems. [See TM097, or search for “thatsmaths” … Continue reading Recreational Mathematics is Fun
Lateral Thinking in Mathematics
Many problems in mathematics that appear difficult to solve turn out to be remarkably simple when looked at from a new perspective. George Pólya, a Hungarian-born mathematician, wrote a popular book, How to Solve It, in which he discussed the benefits of attacking problems from a variety of angles [see TM094, or search for “thatsmaths” … Continue reading Lateral Thinking in Mathematics
Bloom’s attempt to Square the Circle
The quadrature of the circle is one of the great problems posed by the ancient Greeks. This “squaring of the circle” was also an issue of particular interest to Leopold Bloom, the central character in James Joyce’s novel Ulysses, whom we celebrate today, Bloomsday, 16 June 2016 [see TM093, or search for “thatsmaths” at irishtimes.com]. The challenge … Continue reading Bloom’s attempt to Square the Circle
Mathematics Everywhere (in Blackrock Station)
Mathematics is everywhere. We are often unaware of it but, when we observe our environment consciously, we can see mathematical structures all around us. Recently, while waiting for a train in Blackrock Station (Co Dublin), I photographed various objects in and around the station. There were circles and squares all about, parallel planes and lines, hexagons … Continue reading Mathematics Everywhere (in Blackrock Station)
The Imaginary Power Tower: Part II
This is a continuation of last week's post: LINK The complex power tower is defined by an `infinite tower' of exponents: $latex \displaystyle Z(z) = {z^{z^{z^{.^{.^{.}}}}}} \,. &fg=000000$ The sequence of successive approximations to this function is $latex z_0 = 1 \qquad z_{1} = z \qquad z_{2} = z^{z} \qquad \dots \qquad z_{n+1} = z^{z_n} … Continue reading The Imaginary Power Tower: Part II
The Imaginary Power Tower: Part I
The function defined by an `infinite tower' of exponents, $latex \displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}} &fg=000000$ is called the Power Tower function. We consider the sequence of successive approximations to this function: $latex \displaystyle y_0 = 1 \qquad y_1 = x \qquad \dots \qquad y_{n+1} = x^{y_n} \,. &fg=000000$ As $latex {n\rightarrow\infty}&fg=000000$, the sequence $latex {\{y_n\}}&fg=000000$ converges for … Continue reading The Imaginary Power Tower: Part I
How many Christmas Gifts?
We all know the festive carol The Twelve Days of Christmas. Each day, “my true love” receives an increasing number of gifts. On the first day there is one gift, a partridge in a pear tree. On the second, two turtle doves and another partridge, making three. There are six gifts on the third day, … Continue reading How many Christmas Gifts?
Eccentric Pizza Slices
Suppose six friends visit a pizzeria and have enough cash for just one big pizza. They need to divide it fairly into six equal pieces. That is simple: cut the pizza in the usual way into six equal sectors. But suppose there is meat in the centre of the pizza and some of the friends … Continue reading Eccentric Pizza Slices
Golden Moments
Suppose a circle is divided by two radii and the two arcs a and b are in the golden ratio: b / a = ( a + b ) / b = φ ≈ 1.618 Then the smaller angle formed by the radii is called the golden angle. It is equal to about 137.5° or … Continue reading Golden Moments
Triangular Numbers: EYPHKA
The maths teacher was at his wits' end. To get some respite, he set the class a task: Add up the first one hundred numbers. “That should keep them busy for a while”, he thought. Almost at once, a boy raised his hand and called out the answer. The boy was Carl Friedrich Gauss, later … Continue reading Triangular Numbers: EYPHKA
Cartoon Curves
The powerful and versatile computational software program called Mathematica is widely used in science, engineering and mathematics. There is a related system called Wolfram Alpha, a computational knowledge engine, that can do Mathematica calculations and that runs on an iPad. Mathematica can do numerical and symbolic calculations. Algebraic manipulations, differential equations and integrals are simple, … Continue reading Cartoon Curves
Biscuits, Books, Coins and Cards: Massive Hangovers
Have you ever tried to build a high stack of coins? In theory it's fine: as long as the centre of mass of the coins above each level remains over the next coin, the stack should stand. But as the height grows, it becomes increasingly trickier to avoid collapse. In theory it is possible to … Continue reading Biscuits, Books, Coins and Cards: Massive Hangovers
The High-Power Hypar
Maths frequently shows us surprising and illuminating connections between physical systems that are not obviously related: the analysis of one system often turns out to be ideally suited for describing another. To illustrate this, we will show how a surface in three dimensional space --- the hyperbolic paraboloid, or hypar --- pops up in unexpected … Continue reading The High-Power Hypar
The Watermelon Puzzle
An amusing puzzle appears in a recent book by John A. Adam (2013). The answer is very surprising. The book argues in terms of simultaneous equations. A simpler argument, using the diagram below, should make all clear. The Watermelon Puzzle. A farmer brings a load of watermelons to the market. Before he sets out, he … Continue reading The Watermelon Puzzle
New Estimate of the Speed of Light
A team of German scientists have recently discovered a new method of measuring the speed of light using Einstein's famous equation E = m c2 Scientists from SFZ, the Spätenheim Forschungszentrum in Bavaria, assembled a group of twenty volunteer climbers at a local mountain, Schmerzenberg. Using high-precision Mettler balance equipment, each climber was weighed at … Continue reading New Estimate of the Speed of Light
Amazing Normal Numbers
For any randomly chosen decimal number, we might expect that all the digits, 0, 1 , … , 9, occur with equal frequency. Likewise, digit pairs such as 21 or 59 or 83 should all be equally likely to crop up. Similarly for triplets of digits. Indeed, the probability of finding any finite string of … Continue reading Amazing Normal Numbers
Joyce’s Number
With Bloomsday looming, it is time to re-Joyce. We reflect on some properties of a large number occurring in Ulysses. The Largest Three-digit Number What is the largest number that can be written using only three decimal digits? An initial guess might be 999. But soon we realize that factorials permit much greater numbers, and … Continue reading Joyce’s Number
Chess Harmony
Long ago in the Gupta Empire, a great-but-greedy mathematician, Grababundel, presented to the Maharaja a new game that he had devised, called Chaturanga. Thirty-two of the Maharaja's subjects, sixteen dressed in white and sixteen in black, were assembled on a field divided into 64 squares. There were rajas and ranis, mahouts and magi, fortiers and … Continue reading Chess Harmony
The Power Tower
Look at the function defined by an `infinite tower' of exponents: $latex \displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}} &fg=000000$ It would seem that for x>1 this must blow up. But, amazingly, this is not so. In fact, the function has finite values for positive x up to $latex {x=\exp(1/e)\approx 1.445}&fg=000000$. We call this function the power tower … Continue reading The Power Tower
ThatsMaths 