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Martin Gardner's Aha! Insight PDF

By Martin Gardner

ISBN-10: 071671017X

ISBN-13: 9780716710172

Aha! perception demanding situations the reader's reasoning energy and instinct whereas encouraging the advance of 'aha! reactions'.

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Example text

The correspondent was puzzled by a mysterious inscription on the front page that he said he had tried to have translated, but no Chinese to whom he had shown it had been willing or able to read it. Dudeney reproduced the inscription and asked the reader for help. W e do not know what the response was to this request, but Read, who owns a copy of the same booklet, had no difficulty clearing up the mystery. The, inscription is nothing more TANGRAMS, PART 1 31 than a caption under the tangrams of two men.

In this game the winning relation is nontransitive. Occasionally in mathematics, particularly in probability theory and decision theory, one comes on a relation that one expects to be transitive but that actually is not. If the nontransitivity is so counterintuitive as to boggle the mind, we have what is called a nontransitive paradox. The oldest and best-known paradox of this type is a voting paradox sometimes called the Arrow paradox after Kenneth J. Arrow because of its crucial role in Arrow's "impossibility theorem," for which he shared a Nobel prize in economics in 1972.

1. How many different convex polygons can be formed with the seven tans? There must not be any "windows" in the figures. Rotations and reflections are not, as is customary, considered to be different. Because all three-sided polygons are convex, and no nonconvex polygon of four sides can be made with all seven tans, answering this question also gives the number of three- and four-sided polygons. It is easy to see that only one triangle is possible (since corners must be multiples of 45 degrees, the triangle must be a right isosceles one), but finding all the higher convex polygons is a bit tricky.

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Aha! Insight by Martin Gardner

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