famous puzzles of great mathematicians
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-f Great Mathematicians
Miodrag S. Petkovic
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FAMOUS PUZZLES
ofGreat Mathematicians
Miodrag S. Petkovic
@AMSAMERICAN MATHEMATICAL S O C IE T Y
Pro\"ide:nce:, Rhode: Island
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2000 Matlwrwllic." Subjed C/a.'!1iJication. Primary OOA08. 97A20. 01A05, OIA 70, 05A05,
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CONTENTS
Preface
Art and Photo Credits
1. RECREATIONAL MATHEMATICS
2. ARITHMETICS
DiuphanllUi' ug(! (Dio/Jlw,ntut/)
Number of arrows (A'lahiitrim)
How many rabbit.s? (Fiborwcci. p. JJ)
Fibunacci s S ( ~ q u c I l C ( ~Chessboard parndox
Square 1l1lmbcl'8 problem (Fibonacc'i)
Moncy in It. pile (Fibollacci)
Magic configurntioll (Hui)
Triangle with integral sidt!s (Bachct.)
\¥cighlS problem (Fibo'IIacC'i. Tartaglia. Bachet)
MllcMahon ,s gencralllppronch
Division oC 17 hors(!S (TartU!llia. p. 2.'1)
\Vinc and w l l t ( ~ r (Tartaglia)
CUiJL"'t in hands (Reco1'tle)
Sides or two cubes ( i l ~ t e . p. 25)
Animnls on a field (Ncwton)
Polya's solution
Dorrie's variant
Olher variantsv
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2·125
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vi CONTENTS
Gathering an army (Alcuin of York)
Answers to Problems
3. NUMBER THEORY
Cattle problem (A rchimedes, p. 38)
Dividing the square (Dioplaantu.s)
Wine problem (Diophantus)
Amicable numbers (ilm Qorra)
Qorra'8 formula
Euler's rule
How many soldiers? (Bhaskara)
Horses and bulls - a Diophantine equation (Euler)
The sailors, the coconuts, and the monkey (P. Dime)
Dirac's solution
General solution
Unknown a d d r ~ (Ramanujan)
Stamp combinations (Frobeni'Us, Sylvester, p. 60)
Generalized problem
Answers to Problems
4. GEOMETRY
Arbelos problem (Archimedes)
Archimedean circles
Perpendicular distance
Two touching circles
~ H n i m a l distance (Heron)
A fly and a drop of honey
Peninsula problem
The same distance of traversed paths (Bmhmagupta)
Broken bamboo (Brahmagupta)
Height of a suspended string (klah«vira)
The diameter of the material sphere (ibn Qonu)
Dissection of three squares (Abu'l- Wafa)
Dudeney's six-cut dissection
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56
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67
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CONTENTS
Dissection of four trianglt-s (Abu '1- Wafa)
Billiard problem (AU,.azen)
Distance of the optimal viewpoint (Regiomontallus)Rugby football I)roblem
Saturn problem
The minimal sum of distances in a triangle
( S t e i f l c r ~ Fcnnat t Torricelli, Cavaliet-i)
Volumes of cylinders and spheres (Kepler, p. 93)
Dido's problem (Ste'ine1", p. 95)
Di\rision of space by p l a n ~ (Steiner)
Division of plane
Number of hounded regions
Road s)'stem ill a S<llUlfC (Steiner-)
Kissing circles (Soddy, Deticllrtes . Kowa)
Kazayuki's problemF i v ( ~ kissing s p h e r ~n-dimensional kissing spheres
T h ( ~ shortesl b i S L ~ t i t l g arc of area (Polya)
Answers to Problems
5. TILING AND PACKING
r..·josaics (Kel'ler)
Escher's mosaics
NOllperiudic tiling (Pellrose. Conway)
Penrose's kite-dart t.iling
Conway's p i n w l l ( ~ ~ l tiling
l\·laximulDa r t ~ a
by pentaminoes (Klntth)Pentamillo tilings
Squaring t.he S<luarc
Kissing spheres (D. Gregory, Newton)
The densest. s J > h c r t ~ packing (Keple,., Gauss)
Kepler's conjecture
Circle packing
Hales' solul.ioll
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108109
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141
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viii CONTENTS
Cube-packing puzzles (Coll'way) 142
Answers to Problems 144
6. PHYSICS 151
The gold crown of King Hi("ro (Archimedes) 152
The I<mgth uf traveled t.rip ( O , · e . ~ m c ) 153
~ I e e l . i n g of shillS (Lucas. p. 155) 156
A girl and a bil'd ( \ l01l NC·lI.marl1l. I'. 157) 158
Von Neumann's ( ~ n s y S l ~ r i e s 158
l\Iure difficult. p r ( ) h l ( ~ m s 160
The lion and the man ( L U t . l ( ~ u : o o d . Besico'uitc/a, R. Rado) 161
Answers to Problems 166
T. COMBINATORICS 171
Combination with flavors (AJahiiviro) 172
Married couples c r ( ) s . ~ the river (Bachet) 173
Four married c o n p l l ~ cross tbe river 175
.Josephus I>roblem (Bachet. an.d otller's) 176
Dudeney's version 177
Japanl'SC version 177
Graham-Knuth-Patasnik's v ( ~ r s i u n 178
Rings puzzle (Cardano, p. 18lJ) 181G r a ~ r - c o d e numbers 182
The problem of the misaddre.sSt.>(lletters (N. (/I) Dcnl.oulli. E'ulcr) 184
Eulerian squares (Euler') 186
Lat.in squares 187
Gracco- Latin squares 187
Euler's officers problem 188
Euler's conjecture 188
Parker's square of order 10 189
Kirkman's schoolgirls problem
(Kirkman.. Steiner. Sylvester. Cayley) 189
Steiner tri pIe system 190
General problem 192
Sylvester's problmn 192
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C O ~ T E N T S ix
Counting prublem ( C ( l . ' l l ( ~ ! I ) 19:1
Rn<:c.os with t.it,S 19-1
Thl' Towel' uf Hanui (Lunl,'j) 196
D(,IUU'('S l('>llIpl(> 19X
Iliterdianging thf' d t e c k ( ~ r s (I) (Lur.w,) 199
Int,l'rclinngillg lhe dwck('I1; (II) (LIlCCl,'i) 20()
Shullt.ing pl'ohl(,11l (/ . ucfI.4j) 200
Pruhlt'1II uf marriNI cuuples (problellll' dC!i mellages) (LIICU,li) 201
TIl(' t.ree plcllltillg prohlc.'1lI (Syi-lJfslcr) 202D u d e n e ~ " s militnry puzzle 203
Dudency's fuul' pluJlting problem 204
DHfl'relll dist.anct-s (E,'dos, p, 20,,) 2U6
A l1swers to Problems 206
8. PROBABILITY 209
The prohlem of til(' point.s ( f ( ~ r ' m a t , PaM:al, I'. 21 I) 212
Paseal-Fl'I'nult's general prohlem 213
C a l l 1 b l i l l ~ gallic \villa dkl' (HuYfJf'Il,'1) 215
Gamhl('r"s ruill problem (PfMr.lll, FC"mat, Hl&u,qcru;) 21 i
C('lu'I'al pruhll>11l 218
Thl' P('ler!ihllrg paradox (N. (III) BenuJlllli. D, B ( ~ n l ( m l l i . Cnmw1') 220Derlloulli's moral expect.at.ion
C .. , t·mlller s varian.
The probability prohlel1l with t.he misaddressed lett.ers
(N. (II) 8e1'1wulli. EfllfT)
~ I a t c h h o x p r ( ) h l ( ~ 1 1 1 (BfU1fU:h)
Comhinatorial Sllm
A l1swcrs to Problems
9. GRAPHS
The prohlt!l1l of I(c)lligsherg's hridges (Eliler)
Diugnun- t.racing p1lzzle
Eul('r's thror(,111
Euler's pat.h
22l
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225
225
229
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