Published February 11, 2016
Tags: Games, Geometry, History, Probability
Franc-carreau is a simple game of chance, like the roll-a-penny game often seen at fairs and fêtes. A coin is tossed or rolled down a wooden chute onto a large board ruled into square segments. If the player’s coin lands completely within a square, he or she wins a coin of equal value. If the coin crosses a dividing line, it is lost.
The playing board for Franc-Carreau is shown above, together with a winning coin (red) contained within a square and a loosing one (blue) crossing a line. As the precise translation of franc-carreau appears uncertain, the name “fair square” would seem appropriate.
The question is: What size should the coin be to ensure a 50% chance of winning?
Continue reading ‘Franc-carreau or Fair-square’
Published January 28, 2016
Tags: Algorithms, Number Theory
Once again the record for the largest prime number has been shattered. As with all recent records, the new number is a Mersenne prime, a number of the form
Mp = 2p – 1
where p itself is a prime. Participants in a distributed computing project called GIMPS (Great Internet Mersenne Prime Search) continue without rest to search for ever-larger primes of this form.
Most of the recent large primes have been found in the GIMPS project (for an earlier post on GIMPS, click Mersennery Quest. The project uses a search algorithm called the Lucas-Lehmer primality test, which is particularly suitable for finding Mersenne primes. The test, which was originally devised by Edouard Lucas in the nineenth century and extended by Derek Lehmer in 1930, is very efficient on binary computers.
Continue reading ‘Prime Number Record Smashed Again’
Published January 21, 2016
Tags: Algorithms, Music
An ingenious method of tuning pianos, based on the concept of entropy, has recently been devised by Haye Hinrichsen of Würzburg University. Entropy, which first appeared in the mid-nineteenth century in thermodynamics and later in statistical mechanics, is a measure of disorder. Around 1948 Claude Shannon developed a mathematical theory of communications and used entropy as an indicator of information content [TM084, or search for “thatsmaths” at irishtimes.com].
Tuning curve showing the stretch for high and low notes (Image: Wikimedia Commons).
Continue reading ‘Entropy Piano Tuning’
Published January 14, 2016
Tags: Analysis, Probability
Next week there will be a post on tuning pianos using a method based on entropy. In preparation for that, we consider here how the entropy of a probability distribution function with twin peaks changes with the separation between the peaks.
Continue reading ‘Twin Peaks Entropy’
Modern weather forecasts are made by calculating solutions of the mathematical equations that express the fundamental physical principles governing the atmosphere [TM083, or search for “thatsmaths” at irishtimes.com]
The solutions are generated by complex simulation models with millions of lines of code, implemented on powerful computer equipment. The meteorologist uses the computer predictions to produce localised forecasts and guidance for specialised applications.
Continue reading ‘Richardson’s Fantastic Forecast Factory’
Published December 31, 2015
The circular functions occur throughout mathematics. Fourier showed that, under very general assumptions, an arbitrary function can be decomposed into components each of which is a circular function. The functions get their name from their use in defining a circle in parametric form: if
then , the usual equation for a circle in Cartesian coordinates. In the figure, we plot the familiar sinusoid, which has a period of .
Continue reading ‘Squaring the Circular Functions’