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Buffon’s needle problem 157 The circle can be approximated by polygons. Just imagine that together with the circular needle C we are dropping an inscribed polygon P n , as well as a circumscribed polygon P n . Every line that intersects P n will also intersect C, and if a line intersects C then it also hits P n . Thus the expected numbers of intersections satisfy E(P n ) ≤ E(C) ≤ E(P n ). Now both P n and P n are polygons, so the number of crossings that we may expect is “c times length” for both of them, while for C it is 2, whence c l(P n ) ≤ 2 ≤ c l(P n ). (1) Both P n and P n approximate C for n −→ ∞. In particular, lim l(P n) = dπ = n→∞ and thus for n −→ ∞ we infer from (1) that which gives c = 2 π 1 d . c dπ ≤ 2 ≤ c dπ, lim l(P n ), n→∞ But we could also have done it by calculus! The trick to obtain an “easy” integral is to first consider the slope of the needle; let’s say it drops to lie with an angle of α away from horizontal, where α will be in the range 0 ≤ α ≤ π 2 . (We will ignore the case where the needle comes to lie with negative slope, since that case is symmetric to the case of positive slope, and produces the same probability.) A needle that lies with angle α has height l sinα, and the probability that such a needle crosses one of the horizontal lines of distance d is l sin α d . Thus we get the probability by averaging over the possible angles α, as p = 2 π ∫π/2 0 l sin α d dα = 2 π l[ ] π/2 − cosα d 0 = 2 π For a long needle, we get the same probability l sin α d as long as l sinα ≤ d, that is, in the range 0 ≤ α ≤ arcsin d l . However, for larger angles α the needle must cross a line, so the probability is 1. Hence we compute p = 2 π ( ∫ arcsin(d/l) 0 l sinα d dα + ∫π/2 arcsin(d/l) l d . ) 1 dα = 2 ( l ] arcsin(d/l) ( π − cosα + π d[ 0 2 − arcsin d )) l = 1 + 2 √ ( l ) 1 − 1 − π d( d2 l 2 − arcsin d ) l for l ≥ d. So the answer isn’t that pretty for a longer needle, but it provides us with a nice exercise: Show (“just for safety”) that the formula yields 2 π for l = d, that it is strictly increasing in l, and that it tends to 1 for l −→ ∞. □ P n P n α
158 Buffon’s needle problem References [1] E. BARBIER: Note sur le problème de l’aiguille et le jeu du joint couvert, J. Mathématiques Pures et Appliquées (2) 5 (1860), 273-286. [2] L. BERGGREN, J. BORWEIN & P. BORWEIN, EDS.: Pi: A Source Book, Springer-Verlag, New York 1997. [3] G. L. LECLERC, COMTE DE BUFFON: Essai d’arithmétique morale, Appendix to “Histoire naturelle générale et particulière,” Vol. 4, 1777. [4] D. A. KLAIN & G.-C. ROTA: Introduction to Geometric Probability, “Lezioni Lincee,” Cambridge University Press 1997. [5] T. H. O’BEIRNE: Puzzles and Paradoxes, Oxford University Press, London 1965. “Got a problem?”
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Martin Aigner Günter M. Ziegler Pr
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Prof. Dr. Martin Aigner FB Mathemat
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VI Preface to the Fourth Edition Wh
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VIII Table of Contents 21. A theore
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Six proofs of the infinity of prime
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Six proofs of the infinity of prime
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Bertrand’s postulate Chapter 2 We
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Bertrand’s postulate 9 times. Her
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Bertrand’s postulate 11 by compar
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Binomial coefficients are (almost)
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Binomial coefficients are (almost)
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Representing numbers as sums of two
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The law of quadratic reciprocity Ch
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The law of quadratic reciprocity 25
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Every finite division ring is a fie
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Every finite division ring is a fie
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Some irrational numbers Chapter 7
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Some irrational numbers 37 and this
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Some irrational numbers 39 F(x) may
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Some irrational numbers 41 Now assu
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44 Three times π 2 /6 v 1 1 2 y v
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46 Three times π 2 /6 For m = 1, 2
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48 Three times π 2 /6 1 f(t) = 1 t
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50 Three times π 2 /6 (3) It has b
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Hilbert’s third problem: decompos
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64 Lines in the plane, and decompos
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66 Lines in the plane, and decompos
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The slope problem Chapter 11 Try fo
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The slope problem 71 • Every move
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The slope problem 73 For the second
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76 Three applications of Euler’s
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82 Cauchy’s rigidity theorem Q: Q
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84 Cauchy’s rigidity theorem A be
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86 Touching simplices l l l ′ Pr
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88 Touching simplices For our examp
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90 Every large point set has an obt
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96 Borsuk’s conjecture Theorem. L
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98 Borsuk’s conjecture On the oth
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100 Borsuk’s conjecture To obtain
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Sets, functions, and the continuum
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Sets, functions, and the continuum
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Page 128 and 129: In praise of inequalities Chapter 1
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Page 148 and 149: A theorem of Pólya on polynomials
Page 154 and 155: On a lemma of Littlewood and Offord
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Page 158 and 159: Cotangent and the Herglotz trick Ch
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Page 168: Combinatorics 25 Pigeon-hole and do
Page 171 and 172: 162 Pigeon-hole and double counting
Page 182 and 183: Tiling rectangles Chapter 26 Some m
Page 184 and 185: Tiling rectangles 175 Third proof.
Page 186: Tiling rectangles 177 References [1
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Page 191 and 192: 182 Three famous theorems on finite
Page 194 and 195: Shuffling cards Chapter 28 How ofte
Page 196 and 197: Shuffling cards 187 Indeed, if A i
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Identities versus bijections Chapte
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Identities versus bijections 209 Pr
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Identities versus bijections 211 As
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Completing Latin squares Chapter 32
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Completing Latin squares 215 Proof
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Completing Latin squares 217 be dis
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Graph Theory 33 The Dinitz problem
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222 The Dinitz problem In 1976 Vizi
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224 The Dinitz problem X A bipartit
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226 The Dinitz problem G : L(G) : a
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228 Five-coloring plane graphs From
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230 Five-coloring plane graphs is s
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232 How to guard a museum The follo
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234 How to guard a museum “Museum
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236 Turán’s graph theorem Let us
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238 Turán’s graph theorem Thus b
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Communicating without errors Chapte
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Communicating without errors 243 cl
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Communicating without errors 245 Le
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Communicating without errors 247 On
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Communicating without errors 249 No
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The chromatic number of Kneser grap
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258 Of friends and politicians w 1
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Probability makes counting (sometim
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Proofs from THE BOOK 269
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Index acyclic directed graph, 196 a
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Index 273 matrix-tree theorem, 203
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