AMUSEMENTS IN MATHEMATICS.

ROLCAM · Posted: Thu May 24, 2007 1:42 pm Post subject: AMUSEMENTS IN MATHEMATICS.

AMUSEMENTS IN MATHEMATICS

by

HENRY ERNEST DUDENEY

In Mathematicks he was greater
Than Tycho Brahe or Erra Pater:
For he, by geometrick scale,
Could take the size of pots of ale;
Resolve, by sines and tangents, straight,
If bread or butter wanted weight;
And wisely tell what hour o' th' day
The clock does strike by algebra.

BUTLER'S _Hudibras_.

1917
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Moderator

Sydney , Australia.

kheyanne · Posted: Fri Jun 08, 2007 1:47 am Post subject:

If this one is a riddle, I certainly cannot answer it. But it's a good read, anyway.
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ROLCAM · Posted: Sat Jun 09, 2007 1:09 am Post subject: Mathematics - Explanation.

Preface

1

In Mathematicks he was greater

Than Tycho Brahe or Erra Pater:

For he, by geometrick scale,

Could take the size of pots of ale;

Resolve, by sines and tangents, straight,

If bread or butter wanted weight;

And wisely tell what hour o' th' day

The clock does strike by algebra.

BUTLER'S Hudibras.

--------------------------------------------------------------------------------

In issuing this volume of my Mathematical Puzzles, of which some have appeared in periodicals and others are given here for the first time, I must acknowledge the encouragement that I have received from many unknown correspondents, at home and abroad, who have expressed a desire to have the problems in a collected form, with some of the solutions given at greater length than is possible in magazines and newspapers. Though I have included a few old puzzles that have interested the world for generations, where I felt that there was something new to be said about them, the problems are in the main original. It is true that some of these have become widely known through the press, and it is possible that the reader may be glad to know their source.

On the question of Mathematical Puzzles in general there is, perhaps, little more to be said than I have written elsewhere. The history of the subject entails nothing short of the actual story of the beginnings and development of exact thinking in man. The historian must start from the time when man first succeeded in counting his ten fingers and in dividing an apple into two approximately equal parts. Every puzzle that is worthy of consideration can be referred to mathematics and logic. Every man, woman, and child who tries to "reason out" the answer to the simplest puzzle is working, though not of necessity consciously, on mathematical lines. Even those puzzles that we have no way of attacking except by haphazard attempts can be brought under a method of what has been called "glorified trial"—a system of shortening our labours by avoiding or eliminating what our reason tells us is useless. It is, in fact, not easy to say sometimes where the "empirical" begins and where it ends.

When a man says, "I have never solved a puzzle in my life," it is difficult to know exactly what he means, for every intelligent individual is doing it every day. The unfortunate inmates of our lunatic asylums are sent there expressly because they cannot solve puzzles—because they have lost their powers of reason. If there were no puzzles to solve, there would be no questions to ask; and if there were no questions to be asked, what a world it would be! We should all be equally omniscient, and conversation would be useless and idle.

It is possible that some few exceedingly sober-minded mathematicians, who are impatient of any terminology in their favourite science but the academic, and who object to the elusive x and y appearing under any other names, will have wished that various problems had been presented in a less popular dress and introduced with a less flippant phraseology. I can only refer them to the first word of my title and remind them that we are primarily out to be amused—not, it is true, without some hope of picking up morsels of knowledge by the way. If the manner is light, I can only say, in the words of Touchstone, that it is "an ill-favoured thing, sir, but my own; a poor humour of mine, sir."

As for the question of difficulty, some of the puzzles, especially in the Arithmetical and Algebraical category, are quite easy. Yet some of those examples that look the simplest should not be passed over without a little consideration, for now and again it will be found that there is some more or less subtle pitfall or trap into which the reader may be apt to fall. It is good exercise to cultivate the habit of being very wary over the exact wording of a puzzle. It teaches exactitude and caution. But some of the problems are very hard nuts indeed, and not unworthy of the attention of the advanced mathematician. Readers will doubtless select according to their individual tastes.

In many cases only the mere answers are given. This leaves the beginner something to do on his own behalf in working out the method of solution, and saves space that would be wasted from the point of view of the advanced student. On the other hand, in particular cases where it seemed likely to interest, I have given rather extensive solutions and treated problems in a general manner. It will often be found that the notes on one problem will serve to elucidate a good many others in the book; so that the reader's difficulties will sometimes be found cleared up as he advances. Where it is possible to say a thing in a manner that may be "understanded of the people" generally, I prefer to use this simple phraseology, and so engage the attention and interest of a larger public. The mathematician will in such cases have no difficulty in expressing the matter under consideration in terms of his familiar symbols.

I have taken the greatest care in reading the proofs, and trust that any errors that may have crept in are very few. If any such should occur, I can only plead, in the words of Horace, that "good Homer sometimes nods," or, as the bishop put it, "Not even the youngest curate in my diocese is infallible."

I have to express my thanks in particular to the proprietors of The Strand Magazine, Cassell's Magazine, The Queen, Tit-Bits, and The Weekly Dispatch for their courtesy in allowing me to reprint some of the puzzles that have appeared in their pages.

THE AUTHORS' CLUB
March 25, 1917

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Roland Camilleri

Moderator

Sydney , Australia.

ROLCAM · Posted: Sat Jun 09, 2007 1:22 am Post subject: Brahe, Tycho, (1546-1601),

Brahe, Tycho, (1546-1601), was a Danish astronomer. Brahe developed a systematic approach for observing the planets and stars. He stressed the importance of making such observations on a regular basis. The telescope had not yet been invented, and so Brahe used his eyesight and such instruments as astrolabes and quadrants to estimate the positions of celestial objects. His observations were far more precise than those of any earlier astronomer.

Brahe's observations of planetary motion revealed that the tables then in use to predict the positions of the planets were inaccurate. His sighting of a supernova (type of exploding star) in 1572 helped disprove the ancient idea that no change could occur in the heavens beyond the orbit of the moon.

Like many astronomers of his time, Brahe refused to accept the Copernican theory of the solar system. According to this theory, the earth and the other planets move around the sun. Brahe reasoned that if the earth revolved around the sun, he should have been able to measure changes in the positions of the stars resulting from the earth's movement. He did not realize that such changes were too small for his instruments to detect. However, Brahe's observational data later enabled Johannes Kepler, a German astronomer and mathematician, to confirm the Copernican theory.

Brahe was born in Knudstrup (then a Danish city but now in Sweden), near Malmo. As a member of the nobility, he attended universities in Denmark, Germany, and Switzerland. Brahe built an elaborate observatory on the island of Hven (now called Ven), where he made many of his observations.
_________________
Roland Camilleri

Moderator

Sydney , Australia.