Posts Tagged 'Puzzles'

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” at].

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The Ubiquitous Cycloid

Puzzle: However fast a train is travelling, part of it is moving backwards. Which part?
For the answer, see the end of this post.

Timelapse image of bike with two lights on the wheel-rims. Photo from Webpage of Alexandre Wagemakers.

Timelapse image of bike with lights on the wheel-rims. [Photo from Website of Alexandre Wagemakers, with thanks]

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 ]

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

A pizza with various toppings. Image: Pizza Masetti Craiova, Romania (Flickr)  [CC BY 2.0 (], via Wikimedia Commons.

A pizza with various toppings. Image: Pizza Masetti Craiova, Romania (Flickr) [CC BY 2.0 (, via Wikimedia Commons.

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

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Speed Cubing & Group Theory

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.

Rubik's Cube, invented in 1974 by Hungarian professor of architecture Ernő Rubik.

Rubik’s Cube, invented in 1974 by Hungarian professor of architecture Ernő Rubik.

Continue reading ‘Speed Cubing & Group Theory’

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. Continue reading ‘The Watermelon Puzzle’

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 foot-soldiers. Continue reading ‘Chess Harmony’

Sharing a Pint

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’

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