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.

## Posts Tagged 'Puzzles'

### The Two Envelopes Fallacy

Published November 29, 2018 Occasional Leave a CommentTags: Probability, Puzzles

### The Flight of the Bumble Bee

Published August 23, 2018 Occasional Leave a CommentTags: Algebra, Puzzles

Alice and Bob, initially a distance *l* apart, walk towards each other, each at a speed *w*. A bumble bee flies from the nose of one to the nose of the other and back again, repeating this zig-zag flight at speed *f *until Alice and Bob meet. *How far does the bumble bee fly?*

### Grandi’s Series: Divergent but Summable

Published July 12, 2018 Occasional Leave a CommentTags: Analysis, History, Puzzles

** Is the Light On or Off? **

Suppose a light is switched on for a half-minute, off for a quarter minute, on for one eighth of a minute and so on until precisely one minute has elapsed. Is the light on or off at the end of this (infinite) process? Representing the two states “on” and “off” by and , the sequence of states over the first minute is . But how do we ascertain the final state from this sequence? This question is sometimes known as Thomson’s Lamp Puzzle.

### Leopold Bloom’s Arithmetical Adventures

Published June 7, 2018 Irish Times Leave a CommentTags: Arithmetic, Puzzles

As Bloomsday approaches, we reflect on James Joyce and mathematics. Joyce entered UCD in September 1898. His examination marks are recorded in the archives of the National University of Ireland, and summarized in a table in Richard Ellmann’s biography of Joyce (reproduced below) [TM140 or search for “thatsmaths” at irishtimes.com].

### Lateral Thinking in Mathematics

Published July 7, 2016 Irish Times 4 CommentsTags: Algorithms, Puzzles, Recreational Maths

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” at irishtimes.com].

### The Ubiquitous Cycloid

Published September 17, 2015 Irish Times Leave a CommentTags: Geometry, History, Puzzles

**Puzzle:** *However fast a train is travelling, part of it is moving backwards. Which part?*

For the answer, see the end of this post.

Imagine a small light fixed to the rim of a bicycle wheel. As the bike moves, the light rises and falls in a series of arches. A long-exposure nocturnal photograph would show a *cycloid*, the curve traced out by a point on a circle as it rolls along a straight line. A light at the wheel-hub traces out a straight line. If the light is at the mid-point of a spoke, the curve it follows is a curtate cycloid. A point outside the rim traces out a prolate cycloid, with a backward loop. [TM076; or search for “thatsmaths” at irishtimes.com ]

### Eccentric Pizza Slices

Published May 14, 2015 Occasional Leave a CommentTags: Puzzles, Recreational Maths

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 are vegetarians. How can we cut the pizza into slices of identical shape and size, some of them not including the central region?

Have a think about this before reading on. There is more than one solution.