Transcript Triumph der Mathematik

TRIUMPH DER MATHEMATIK
100 Great Problems of Elementary Mathematics
By
Heinrich Dörrie
SOME BACKGROUND

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 1965


http://www.washjeff.edu/users/MWoltermann/Dor
rie/DorrieContents.htm
FROM THE PREFACE
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.

FROM A REVIEW AT AMAZON.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.
FROM M.W. (SPRING 2010)
A lot of things have changed since 1965.
 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.

WHY?



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
http://www.washjeff.edu/users/MWoltermann/Dor
rie/DorrieContents.htm
Some of the topics are suitable for Junior and
Senior MathTalks (MTH320, MTH420).
TYPES OF PROBLEMS IN TRIUMPH…

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

19. THE FERMAT-EULER PRIME
NUMBER THEOREM

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.

20. THE FERMAT EQUATION
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=4,098
 x=16,793,602

y=640
y=2,622,720
13. NEWTON'S EXPONENTIAL SERIES
Find the power series representation for e x .
 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 x c and e x .
 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.

68. EULER'S TETRAHEDRON PROBLEM
EXPRESS THE VOLUME OF A TETRAHEDRON IN
TERMS OF IT SIX EDGES.
C
a

B
r
pxq
q
b

c
F
O
p
A
VOLUME =
0 p2 q2 r 2 1
p2
1
288
0 c2 b2 1
q2 c2
0 a2 1
r 2 b2 a2
0 1
1
1 0
1
1
.
69. THE SHORTEST DISTANCE BETWEEN
SKEW LINES
FIND THE ANGLE AND DISTANCE BETWEEN TWO
GIVEN SKEW LINES. (SKEW LINES ARE NONPARALLEL NON-INTERSECTING LINES.)
7. EULER'S PROBLEM OF POLYGON
DIVISION



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:
2610 4n10
En
n1 !
FOR EXAMPLE, WITH N=4.
8. LUCAS' PROBLEM OF THE MARRIED
COUPLES

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?
M1
F4
F3
M4
M2
F2
F5
M3
F1
M5
5. KIRKMAN'S SCHOOLGIRL PROBLEM


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.
67. STEINER'S DIVISION OF SPACE BY
PLANES



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
n2  n  2
.
2
The maximum number of parts into which space
can be divided by n planes is
n 3  5n  6
.
6
1. ARCHIMEDES' PROBLEMA BOVINUM




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.
X  10366482g
0 1 0
0
0
0
X
1 209 1 0
0
0
0
Y
1 56
0
1342 0
1 1 0
0 127 0
0
0
0 209 0
0
1342
0
0
0
0
0
0
1 127 0
0
0
0
1 209 0
1130
0
1130
0
1342
0
0
1
0
1
Z
T
x
y
z
t
0
Y  7460514g
0
Z  7358060g
0
T  4149387g
 0 .
0
x  7206360g
0
y  4893246g
0
z  3515820g
t  5439213g
3. NEWTON'S PROBLEM OF THE FIELDS AND
COWS
a
cows graze b
fields bare in c
days,
a  cows graze b  fields bare in c  days,
a  cows graze b  fields bare in c  days.
What relation exists between the 9 (positive
integer) quantities a, b, c, . . . , a , b , c ?
b
cb
ca
b  c b  c a 
b  c b  c a 
0
6. THE BERNOULLI-EULER PROBLEM OF
THE MISADDRESSED LETTERS


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?
18. BUFFON'S NEEDLE PROBLEM

Parallel lines a distance of d apart are drawn on
a table . A needle of length ℓ<d is thrown at
random on the table. What is the probability
that the needle will touch one of the parallel
lines?
/2
0
cos d
d
2


d
2

2
.
d
32. THE TANGENCY PROBLEM OF
APOLLONIUS.
CONSTRUCT ALL CIRCLES TANGENT TO THREE
GIVEN CIRCLES.
a
b
c
4 CIRCLES
a
a
b
b
c
c
AND 4 MORE OF THEM
a
b
a
b
c
c
34. STEINER'S STRAIGHT-EDGE
PROBLEM
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.)

BEWEIS EINFACH. (!)
(TR. “A SIMPLE PROOF”)
51. A PARABOLA AS AN ENVELOPE
A=1 2
1
2
3
4
5
7
6
8
9
S
12 11
4
3
2
B=1
10
11
55. THE CURVATURE OF CONIC
SECTIONS
directrix
tangent
P
n
axis
F
p2
The curvature at P is   3 .
n
60. STEINER'S DOUBLE ELEMENT
CONSTRUCTION

Construct the double elements of a superposed
projectivity given by three pairs of corresponding
elements.
axis of perspectivity
A'
H
C'
C
c
B'
A
K
B
29. CASTILLON'S PROBLEM (V2)

Inscribe a triangle in a given circle, the sides of
which pass through three given points.
X2
k
Zr
X'1
k
X1
X'2
X3
Ys
Zs
C
X'3
C
Yr
Xr
Xr
Xs
Xs
A
A
B
B
72. A QUADRILATERAL AS AN IMAGE OF
A SQUARE
C0
D0
O0
B0
A
S
K
M
N
H
axis 
B
O
D
C
P
Q
vanishing line
75. HIPPARCHUS' STEREOGRAPHIC
PROJECTION AND
76. THE MERCATOR PROJECTION


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
77. THE PROBLEM OF THE LOXODROME

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.
0o
80o N
P2
6 0o N
4 0o N
20 o N
20 o S
40 o S

60 o S
loxo drom e
P1
80 o S
94. REGIOMONTANUS' MAXIMUM
PROBLEM

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?)
A
rod
B

ground
F
P
l


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.
99. STEINER'S CIRCLE PROBLEM


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.
ANY