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The Two Envelopes Fallacy

During his Hamilton lecture in Dublin recently, Fields medalist Martin Hairer made a passing mention of the “Two Envelopes Paradox”. This is a well-known problem in probability theory that has led to much misunderstanding. It was originally developed in 1912 by the leading German number theorist Edmund Landau (see Gorroochurn, 2012). It is frequently discussed on the web, with much misunderstanding and confusion. I will try to avoid adding to that.


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Gravitational Waves & Ringing Teacups

Newton’s law of gravitation describes how two celestial bodies orbit one another, each tracing out an elliptical path. But this is imprecise: the theory of general relativity shows that two such bodies radiate energy away in the form of gravitational waves (GWs), and spiral inwards until they eventually collide.


Warning sign, described by Thomas Moore as a “geeky insider GR joke” [image from Moore, 2013].

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The “Napoleon of Crime” and The Laws of Thought

NewLightOnGBooleA fascinating parallel between a brilliant mathematician and an arch-villain of crime fiction is drawn in a forthcoming book – New Light on George Boole – by Des MacHale and Yvonne Cohen. Professor James Moriarty, master criminal and nemesis of Sherlock Holmes, was described by the detective as “the Napoleon of crime”. The book presents convincing evidence that Moriarty was inspired by Professor George Boole [TM151, or search for “thatsmaths” at].

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Listing the Rational Numbers III: The Calkin-Wilf Tree

Calkin-Wilf-TreeThe rational numbers are countable: they can be put into one-to-one correspondence with the natural numbers. In previous articles we showed how the rationals can be presented as a list that includes each rational precisely once. One approach leads to the Farey Sequences. A second, related, approach gives us the Stern-Brocot Tree. Here, we introduce another tree structure, The Calkin-Wilf Tree.

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Johannes Kepler and the Song of the Earth

Johannes Kepler, German mathematician and astronomer, sought to explain the solar system in terms of divine harmony. His goal was to find a system of the world that was mathematically correct and harmonically pleasing. His methodology was scientific in that his hypotheses were inspired by and confirmed by observations. However, his theological training and astrological interests influenced his thinking [TM150, or search for “thatsmaths” at].


The six planets known to Kepler [Image NASA].

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Saving Daylight with Hip-hop Time: a Modest Proposal

At 2:00 AM on Sunday 28 October the clocks throughout Europe will be set back one hour, reverting to Standard Time. In many countries, the clocks are put forward one hour in Spring and set back to Standard Time in the Autumn. Daylight saving time gives brighter evenings in Summer.


In Summer, the mornings are already bright before most of us wake up but, in Winter, the mornings would be too dark unless we reverted to Standard Time.

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Who Uses Maths? Almost Everyone!

In the midst of Maths Week Ireland, many students may be asking “What use is mathematics and what purpose is served by studying it?” Mathematicians often stress the inherent beauty and intellectual charm of the subject, but that is unlikely to persuade many people, who demand to know how mathematics can be of use and value to them. [TM149, or search for “thatsmaths” at].


In reality, mathematics is essential in numerous contexts: the diversity is remarkable, and you may be surprised how maths plays a vital role in the everyday work of so many people.

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