Amusements in Mathematics Dudeney by Henry E. Dudeney

By Henry E. Dudeney

In issuing this quantity of my Mathematical Puzzles, of which a few have seemed in periodicals and others are given the following for the 1st time, i have to recognize the encouragement that i've got acquired from many unknown correspondents, at domestic and in a foreign country, who've expressed a wish to have the issues in a accumulated shape, with a number of the options given at larger size than is feasible in magazines and newspapers. even though i've got incorporated a number of outdated puzzles that experience the area for generations, the place I felt that there has been whatever new to be acknowledged approximately them, the issues are ordinarily unique. it truly is real that a few of these became well known throughout the press, and it really is attainable that the reader will be pleased to grasp their source.
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The properties of terminating fractions shown above, which also hold for infinite continued fractions, permit the easy finding of rational numbers as close as desired to a, provided sufficient partial denominators are known. Indeed, on the basis of the second property of convergent fractions, a is enclosed between any neighbouring P convergent fractions -P k ~— and —- But the absolute value OA-I Qk of their difference equals (third property). ] we have k 0 1 2 3 4... qk 2 4 4 4 4... Pk 2 9 38 161 682 ...

K4 check! 2JT radicals it is even possible to express any natural number n by means of three 4's. 36 § 7. Numerical Tricks One sometimes comes across people who carry out mentally and with phenomenal speed, operations with multi-digit numbers. Such calculators, by utilizing a number of numerical tricks and various artificial steps, astonish audiences with one or two spectacular acts. Of course, even when knowing the secret of this or that ''act", not everybody can be successful as a performer.

Rapid Calculations The desire to simplify operations with many-figure numbers brought about the invention, in the 17th century, of tables of logarithms, and thereafter of the logarithmic ruler. In the 19th century arithmometers made their appearance, and the beginning of the 20th century saw the advent of automatic calculating machines; about fifteen years ago there appeared electronic computers which are capable of solving problems, requiring millions of operations with multidigit numbers, within a matter of hours.

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