wiley mathematical journeys

[...]... I was able to track it down to the 1901 Hungarian E¨ tv¨ s Competition, named after the o o Mathematical Journeys, by Peter D Schumer ISBN 0-471-22066-3 Copyright c 2004 John Wiley & Sons, Inc 13 14 MATHEMATICAL JOURNEYS Middlebury and Williams Colleges’ Green Chicken trophy founder and first president of the Mathematical and Physical Society of Hungary Maybe the problem is even older Problem #1 (1978... eliminated Figure 3.1 shows the elimination procedure with person number 7 being the last one to go The order of elimination is 2, 4, 6, 1, 5, 3, 7 Mathematical Journeys, by Peter D Schumer ISBN 0-471-22066-3 Copyright c 2004 John Wiley & Sons, Inc 23 24 MATHEMATICAL JOURNEYS 4th 7 3rd 1 1st 2 6 5 3 6th 5th 4 2nd Figure 3.1 n−1 Elimination order for seven people n 1st 1 2 3 2nd 4 5 6 3rd Figure 3.2 Elimination... )/2 , which is the formula for tn+1 By mathematical induction, the formula holds for all n ≥ 1 Make sure to double-check the little bit of algebra in the proof above and be certain that you are comfortable with the logic behind mathematical induction If so, you are well on your way toward thinking like a real mathematician Now LET’S GET COOKING: A VARIETY OF MATHEMATICAL INGREDIENTS 5 if you need...2 MATHEMATICAL JOURNEYS the ancient Greek geometers more than 2000 years ago Others still have not been answered fully Let’s get right to it and answer the first question, which is perhaps the most fundamental... for mathematical discovery This is the essence of mathematics.) No doubt you have noticed that the answer is always 1 or −1 In fact, if we pick an odd-indexed Fibonacci number we get +1 and if we choose an evenindexed Fibonacci number we obtain −1 This leads us to the conjecture that 2 Fn − Fn−1 · Fn+1 = (−1)n−1 In fact, this is a theorem that we now prove in two different ways, each making use of mathematical. .. 2 = Fn+1 Fn + Fn−1 Fn+1 + (Fn − Fn ) (mathematicians love to add zero to equations since nothing gets disturbed) 2 2 = Fn + Fn+1 Fn − (Fn − Fn−1 Fn+1 ) 2 = Fn (Fn + Fn+1 ) − (Fn − Fn−1 Fn+1 ) 6 MATHEMATICAL JOURNEYS 2 = Fn Fn+2 − (Fn − Fn−1 Fn+1 ) by the definition of Fn+2 = Fn Fn+2 − (−1 )n−1 by the inductive assumption 2 Hence Fn+1 − Fn Fn+2 = (−1 )n and the result follows Theorem 1.4 also affords... visual paradox makes use of the fact that the ratio Fn−1 /Fn+1 converges fairly quickly so that it’s difficult to see that there are actually two broken diagonal lines rather than a single diagonal 8 MATHEMATICAL JOURNEYS 8 3 3 5 3 5 5 3 5 5 8 3 5 5 3 8 Figure 1.1 5 Apparent rearrangement of four figures results in different areas Another interesting application of Fibonacci numbers is the representation... forming an L-shape Fill that piece with an L-shaped domino and we are done (Fig 1.3)! In the mathematical literature such L-shaped dominoes are often called right trominoes From there it’s a small step to objects like tetraminoes and pentominoes and a whole host of new and interesting questions Such is the nature of mathematical generalization! Finally, please try your hand at the following cute problem... nature of mathematical generalization! Finally, please try your hand at the following cute problem Show that a given geometric square can be decomposed into n squares, not necessarily all the same 10 MATHEMATICAL JOURNEYS Figure 1.4 Four squares Figure 1.5 Six squares size, for n = 4 and for all n ≥ 6 It’s usually a good idea to check out some initial cases by hand Then try to understand why the assertion... primes Let PA and PB be the product of the elements of A and B, respectively Consider the factorization of PA + PB (For fun and insight, carry out this process on some of your own examples.) 12 MATHEMATICAL JOURNEYS 4 Explicitly work through the construction of six consecutive composites in the proof of Theorem 1.2 Find a smaller example of six consecutive composites 5 (a) Let tn be the nth triangular . MATHEMATICAL JOURNEYS Peter D. Schumer Middlebury College A JOHN WILEY & SONS, INC., PUBLICATION Copyright c  2004 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley. these questions were answered by Mathematical Journeys, by Peter D. Schumer ISBN 0-471-22066-3 Copyright c  2004 John Wiley & Sons, Inc. 1 2 MATHEMATICAL JOURNEYS the ancient Greek geometers. of Congress Cataloging-in-Publication Data: Schumer, Peter D., 1954– Mathematical journeys / Peter D. Schumer. p. cm. “A Wiley- Interscience publication.” Includes bibliographical references and