Problems for Metagrobologists
David Singmaster is a retired mathematics professor from London South Bank University. He has published several books on Rubik's cube and a collection of mathematical puzzles. He regularly contributes to puzzle sections of several magazines. This is yet another collection of his puzzles with their solutions. They are truly mathematical as the subtitle states: "A collection of puzzles with real mathematical, logical or scientific content". Speaking of the title, for the less geeky readers it may need some explanation what "metagrobologists" really means. Singmaster realizes that his catchy title may raise question marks in the eyes of the potential buyer. So the first thing he does in his introduction is explain the title. It seems that Rabelais was the first to use metagraboulizer in his Gargantua (1534). It is a humorous version derived from the French grabeler which means to sieve fragments out of a medical substance. It has been translated as 'to puzzle' or 'to make a dunce of somebody', meaning 'to confuse somebody'. The English form metagrobolize was used in the translation of the Gargantua by Thomas Urquhart in 1653. Metagrobolize is in the Oxford English Dictionary described as 'to puzzle' or 'mystify'. The word was popularized among puzzlers since Rick Irby used it in 1981 in the Wall Street Journal in a tongue-in-cheek kind of way. Nowadays Singmaster is one of the best known self-declared metagrobologists.
Some of the puzzles from the book belong to the puzzler's public folklore, so that the origin is not always clear, but the collection here contains 221 puzzles that were at some point proposed by Singmaster, so that most of them can safely be considered to be his invention or his own variant or generalization of a classical one. Where the origin is recalled, the original is acknowledged together with the generalization.
The puzzles are not simple. Some allow an easy answer provided the proper insight is used. Others do require some longer calculations (the solution part of the book is thicker than the part where the puzzles are formulated). Many of them also have an additional teaser: asking to consider a generalization or when one solution is found, asking for (at least) one other solution or for all possible solutions if there is some freedom left. Some of these additional problems are still open.
The puzzles are grouped by theme: arithmetic, digits, geometry, geography, sequences, monetary, clocks, calendar, combinatorics, word puzzles etc. Usually the puzzle is formulated as a short story or a dialogue, for example with characters confronted with a problem and the reader is asked to help them out.
Let me give some distilled bare-bones examples (after evaporating the story) to provide an idea of the level.
From the digits chapter: 14 x 926 = 12964 is special because the 5 digits appearing in the product of the left-hand side are just the digits that appear in the result of the right-hand side. Twelve such examples were found, but Singmaster's contribution is that there are three examples with only three digits are missing from the list. The problem is to find these.
From the physics chapter: why is there no differential gear on a railroad car while it can still go around the bend?
A geographical problem: How high above the earth should one be to see one third of its surface? Assuming of course that the earth is a perfect ball.
The problems are rather hard and it may need some puzzle experience to find some of the solutions without looking up the answer. If you are patient enough to solve them all by yourself, there is material for many hours of cheap entertainment here.