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T RIUMPH DER M ATHEMATIK 100 Great Problems of Elementary Mathematics By Heinrich Dörrie

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S OME B ACKGROUND Heinrich Dörrie Ph. D. Georg-August-Universität Göttingen 1898 Dissertation Das quadratische Reziprozitätsgesetz im quadratischen Zahlkörper mit der Klassenzahl 1. Advisor David Hilbert Triumph der Mathematik German editions 1932, 1940 Dover (English) edition rie/DorrieContents.htm

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F ROM THE P REFACE For a long time, I (H. Dörrie) have considered it a necessary and appealing task to write a book of celebrated problems of elementary mathematics, their origins, and above all brief, clear and understandable solutions to them. … The present work contains many pearls of mathematics from Gauss, Euler, Steiner and others. So then, let this book do its part to awaken and spread interest and pleasure in mathematical thought.

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F ROM A R EVIEW AT A MAZON. COM The selection of problems is outstanding and lives up to the book's original title. The proofs are concise, clever, elegant, often extremely difficult and not particularly enlightening. To say that this book requires a background in college math is like saying that playing chess requires a background in how to move the pieces; it also requires a lot of thought and, preferably, a lot of experience.

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F ROM M.W. ( SPRING 2010) A lot of things have changed since For example, terminology has changed, people are not as knowledgeable about some areas of mathematics (especially geometry) as they once were, but more knowledgeable about others (e.g. calculus). A straightforward translation would not necessarily shed more light on the problems in this book. What was required was in some cases more (or less) mathematical background, current terminology and notation to bring Triumph der Mathematik into the twenty first century.

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W HY ? I wanted to know the solutions to a lot of these problems, And because the book is still sited as a reference today, share the results with others, e.g. at rie/DorrieContents.htm rie/DorrieContents.htm Some of the topics are suitable for Junior and Senior MathTalks (MTH320, MTH420).

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T YPES OF P ROBLEMS IN T RIUMPH … Arithmetical Problems Number Theory (MTH311) Calculus (MTH151,152) Combinatorics (MTH361) Linear Algebra (MTH217) Probability (MTH225, MTH305) Problems from Plane Geometry Problems about Conic Sections and Cycloids Problems from Solid Geometry (MTH208,217) Nautical and Astronomical Problems Max/Min Problems

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19. T HE F ERMAT -E ULER P RIME N UMBER T HEOREM Every prime number of the form 4n+1 can be written as a sum of two squares in only one way (aside from the order of the summands). 5=1+4 13=4+9 17=1+16 Today there are several proofs of the theorem. The following one is noted for its simplicity. It does however use a fair number of results from number theory, some of which will be need in No. 22 as well.

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20. T HE F ERMAT E QUATION Find all integer solutions of x²-dy²=1, where d is a positive whole number but not a square. We will examine a somewhat modified and more general equation X²-DY²=4, which includes the original Fermat equation. Indeed, in order to solve x²-dy²=1, we need only solve 4x²-4dy²=4. Example x²-41y²=4. x=4,098y=640 x=16,793,602y=2,622,720

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13. N EWTON ' S E XPONENTIAL S ERIES Find the power series representation for Newton's derivation of the exponential series, is however, not rigorous and rather complicated. The following derivation is based on the mean values of the functions and Today, it is common to use Maclaurin's formula to find the power series expansion. The derivation by average or mean values is rather clever and not as mysterious as using Maclaurin’s formula. The same technique is used in problems 14 through 17.

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68. E ULER ' S T ETRAHEDRON P ROBLEM E XPRESS THE VOLUME OF A TETRAHEDRON IN TERMS OF IT SIX EDGES.

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V OLUME =

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69. T HE S HORTEST D ISTANCE B ETWEEN S KEW L INES F IND THE ANGLE AND DISTANCE BETWEEN TWO GIVEN SKEW LINES. (S KEW LINES ARE NON - PARALLEL NON - INTERSECTING LINES.)

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7. E ULER ' S P ROBLEM OF P OLYGON D IVISION In how many ways can a plane convex polygon of n sides be divided into triangles by diagonals? Leonhard Euler posed this problem in 1751 to the mathematician Christian Goldbach. Euler found the following formula for the number of possible divisions:

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F OR E XAMPLE, WITH N =4.

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8. L UCAS ' P ROBLEM OF THE M ARRIED C OUPLES In how many ways can n married couples be seated about a round table in such a way that there is always one man between two women but no man is ever seated next to his own wife?

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5. K IRKMAN ' S S CHOOLGIRL P ROBLEM In a boarding school there are fifteen schoolgirls who always take their daily walks in row of threes. How can it be arranged so that each schoolgirl walks in the same row with every other schoolgirl exactly once a week (7 days)? Kirkman's schoolgirl problem is an example of a problem in combinatorial design theory. The solution is an example of a resolvable (35,15,7,3,1) design.

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67. S TEINER ' S D IVISION OF S PACE BY P LANES What is the maximum number of parts into which space can be divided by n planes? The maximum number of parts into which a plane can be divided by n lines is The maximum number of parts into which space can be divided by n planes is

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1. A RCHIMEDES ' P ROBLEMA B OVINUM This problem deals with finding the number of black, white, spotted and brown bulls and cows subject to numerous conditions. It leads to a system of 7 equations in 8 unkowns. Dörrie spends a fair amount of time doing algebra to solve the system. Today, most people use computer software to solve such systems of equations.

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3. N EWTON ' S P ROBLEM OF THE F IELDS AND C OWS

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6. T HE B ERNOULLI -E ULER P ROBLEM OF THE M ISADDRESSED L ETTERS To determine the number of permutations of n elements in which no element occupies it natural place. OR Someone writes n letters and writes the corresponding addresses on n envelopes. How many different ways are there of placing all the letters in the wrong envelopes?

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18. B UFFON ' S N EEDLE P ROBLEM Parallel lines a distance of d apart are drawn on a table. A needle of length ℓ

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32. T HE T ANGENCY P ROBLEM OF A POLLONIUS. C ONSTRUCT ALL CIRCLES TANGENT TO THREE GIVEN CIRCLES.

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4 C IRCLES

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A ND 4 MORE OF THEM

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34. S TEINER ' S S TRAIGHT - EDGE P ROBLEM Prove that every construction that can be done with compass and straight-edge can be done with straight-edge alone given a fixed circle in the plane. One of 5 preliminary problems: construct a line through point a P parallel to the line through two points A and B if the midpoint M of segment AB is given. (Draw AP and let S be a point on AP extended. Connect S with M and B. Let O be the intersection point of BP and MS. Finally let line AO meet BS at Q. PQ is the desired line.)

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B EWEIS E INFACH. (!) ( TR. “A SIMPLE P ROOF ”)

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51. A P ARABOLA AS AN E NVELOPE

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55. T HE C URVATURE OF C ONIC S ECTIONS

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60. S TEINER ' S D OUBLE E LEMENT C ONSTRUCTION Construct the double elements of a superposed projectivity given by three pairs of corresponding elements.

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29. C ASTILLON ' S P ROBLEM ( V 2) Inscribe a triangle in a given circle, the sides of which pass through three given points.

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72. A Q UADRILATERAL AS AN I MAGE OF A S QUARE

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75. H IPPARCHUS ' S TEREOGRAPHIC P ROJECTION AND 76. T HE M ERCATOR P ROJECTION Describe a conformal map projection that transforms circles on a globe (sphere) into circles of the map. Draw a conformal map (of the globe or sphere) on a rectangular grid

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77. T HE P ROBLEM OF THE L OXODROME A loxodrome is a "line" on the earth's surface that makes the same angle with all the meridians it cuts. It is a straight line on a Mercator map of the earth. As long as a ship does not alter its course, it is sailing on a loxodrome.

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94. R EGIOMONTANUS ' M AXIMUM P ROBLEM At what point on the earth's surface does a perpendicularly suspended rod appear longest? (i.e., at what point is the visual angle largest?)

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Johannes Müller, called Regiomantus after his birthplace of Königsberg, posed this problem in 1471 to professor Christian Roder of Erfurt. This problem, which in itself is not difficult, nevertheless is of note as being the first extremal problem in mathematics since antiquity. The following simple solution comes to us from A. Lorsch, who published it in vol. XXIII of the Zeitschrift für Mathematik und Physik. A Lorsch was a student. Thanks to John Henderson for finding this out.

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99. S TEINER ' S C IRCLE P ROBLEM Of all isometric plane regions (i.e., plane regions have equal perimeters) the circle has the greatest area. And Of all plane regions with equal area the circle has the smallest perimeter.

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ANY

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