By Martin Gardner
Aha! perception demanding situations the reader's reasoning strength and instinct whereas encouraging the improvement of 'aha! reactions'.
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Extra info for Aha! Insight
For readers who care to work through the solution of the equation for square stars, 6n(n - 1) 1 = m2,I shall say only that this reduces to a solution of 2 x 2 1 = 3y2,where x is the square root of a square star. It can be solved by finding alternate convergents of the square root of 4. The extract nonrecursive formula for the nth square star is + + One of the most remarkable properties of square stars is that they provide a simple algorithm for producing every number that can be expressed as the sum of two consecutive squares and also as the sum of three consecutive squares.
The smallest square star greater than 1 is 121. Three times 121, plus 2, is 365. The next square star, + + + HEXES A N D STARS + 23 11881, leads to the number 35645, which equals 1332 1342and also equals 10g2 1092 1102. The third case is 3(1164241) 2 = 3492725 = 13212 13222 = 10782 10792 10802. In each case, the middle term of the triplet of consecutive squares is the original square star. It is a pleasant exercise, demanding no special skills in number theory, to show that the algorithm always works.
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. 40 -42. The set of central figurate numbers is the "other alternative" to the vertex figurate numbers. THREE Tangrams, Part 1 "The seven Books of Tan .
Aha! Insight by Martin Gardner