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 maths, Martin Gardner. For more information about the meeting, see http://ludicum.org/ev/rm/19 .

Continue reading ‘Our Dearest Problems’## Posts Tagged 'Puzzles'

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.

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 ClosedTags: 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 ClosedTags: 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 ClosedTags: 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 ClosedTags: 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 ]

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.### Speed Cubing & Group Theory

Published February 13, 2014 Irish Times ClosedTags: Algebra, Algorithms, Group Theory, Puzzles

The article in this week’s *That’s Maths* column in the* Irish Times* ( TM038 ) is about Rubik’s Cube and the Group Theory that underlies its solution.

### The Watermelon Puzzle

Published November 14, 2013 Occasional ClosedTags: Algebra, Puzzles, Recreational Maths

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. Continue reading ‘The Watermelon Puzzle’

### Chess Harmony

Published January 30, 2013 Occasional ClosedTags: Games, Number Theory, Puzzles, Recreational Maths

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 foot-soldiers. Continue reading ‘Chess Harmony’

Four friends, exhausted after a long hike, stagger into a pub to slake their thirst. But, pooling their funds, they have enough money for only one pint.

*Annie* drinks first, until the surface of the beer is half way down the side (Fig. 1(A)). Then *Barry* drinks until the surface touches the bottom corner (Fig. 1(B)). *Cathy* then takes a sup, leaving the level as in Fig. 1(C), with the surface through the centre of the bottom. Finally, *Danny* empties the glass.

**Question**: Do all four friends drink the same amount? If not, who gets most and who gets least? Continue reading ‘Sharing a Pint’