By Martin Gardner
Aha! perception demanding situations the reader's reasoning strength and instinct whereas encouraging the advance of 'aha! reactions'.
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Extra info for Aha! Insight
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.
Recreations in the Theory ofNumbers. Albert H. Beiler. Dover, 1964. Diophantine Equations. Louis J . Mordell. Academic Press, 1969. A Handbook of Integer Sequences. N. J . A. Sloane. Academic Press, 1973. " Martin Gardner, in Wheels, Life,and Other Mathematical Amusements. W . H. Freeman and Company, 1983. " Harvey J. Hindin, in Journal ofRecreationa1 Mathematics 16, 1983- 1984, pp. 191- 193. " Allan Whitcombe, in Mathematics in School (a publication of England's Mathematical Association), September 1986, pp.
FIVE Nontransitive Paradoxes "I have just so much logic, as to be able to see . . " -ELIZABETHBARRETT, in a letter to Robert Browning. Whenever a relation R that applies to xRy and yRz also applies to xRz, the relation is said to be transitive. For example, "less than" is transitive among all real numbers. If 2 is less than 71, and the square root of 3 is less than 2, we can be certain that the square root of 3 is less than n. Equality also is transitive: ifa = b and b = c, then a = c. In everyday life such relations as "earlier than," "heavier than," "taller than," "inside of," and hundreds of others are transitive.
Aha! Insight by Martin Gardner