6261:
5696:
2760:
6256:{\displaystyle {\begin{aligned}\operatorname {Var} (x_{n})=\operatorname {Var} (y_{n})&=\sum _{i=1}^{2}p_{i}{\bigl (}x_{i}-\mathbb {E} (x_{i}){\bigr )}^{2}\\&=P(x_{i}=1)\left(1-{\frac {1}{\pi }}\right)^{2}+P(x_{i}=0)\left(0-{\frac {1}{\pi }}\right)^{2}\\&={\frac {1}{\pi }}\left(1-{\frac {1}{\pi }}\right)^{2}+\left(1-{\frac {1}{\pi }}\right)\left(-{\frac {1}{\pi }}\right)^{2}={\frac {1}{\pi }}\left(1-{\frac {1}{\pi }}\right).\\\operatorname {Cov} (x_{n},y_{n})&=\mathbb {E} (x_{n}y_{n})-\mathbb {E} (x_{n})\mathbb {E} (y_{n})\end{aligned}}}
2862:
1863:
2483:
2016:
1546:
20:
2755:{\displaystyle {\begin{aligned}P&={\frac {\displaystyle \int _{0}^{\frac {\pi }{2}}l\cos \theta \,d\theta }{\displaystyle \int _{0}^{\frac {\pi }{2}}t\,d\theta }}\\&={\frac {l}{t}}\cdot {\frac {\displaystyle \int _{0}^{\frac {\pi }{2}}\cos \theta \,d\theta }{\displaystyle \int _{0}^{\frac {\pi }{2}}d\theta }}\\&={\frac {l}{t}}\cdot {\frac {1}{\,{\frac {\pi }{2}}\,}}\\&={\frac {2l}{t\pi }}.\end{aligned}}}
2802:
1858:{\displaystyle {\begin{aligned}P&=\left(\int _{\theta =0}^{\arcsin {\frac {t}{l}}}\int _{x=0}^{{\frac {l}{2}}\sin \theta }{\frac {4}{t\pi }}dxd{\theta }\right)+\left(\int _{\arcsin {\frac {t}{l}}}^{\frac {\pi }{2}}{\frac {2}{\pi }}d{\theta }\right)\\&={\frac {2l}{t\pi }}-{\frac {2}{t\pi }}\left({\sqrt {l^{2}-t^{2}}}+t\arcsin {\frac {t}{l}}\right)+1\end{aligned}}}
6708:
2332:
7100:
740:
3419:
Dutch science journalist Hans van Maanen argues, however, that
Lazzarini's article was never meant to be taken too seriously as it would have been pretty obvious for the readers of the magazine (aimed at school teachers) that the apparatus that Lazzarini said to have built cannot possibly work as
1427:
519:
3415:
The above description of strategy might even be considered charitable to
Lazzarini. A statistical analysis of intermediate results he reported for fewer tosses leads to a very low probability of achieving such close agreement to the expected value all through the experiment. This makes it very
6503:
3561:
To solve such a problem, we first compute the probability that the needle crosses no lines, and then we take its compliment. We compute this first probability by determining the volume of the domain where the needle crosses no lines and then divide that by the volume of all possibilities,
956:
6492:
5685:
5183:
3980:
1432:
In the second expression, the first term represents the probability of the angle of the needle being such that it will always cross at least one line. The right term represents the probability that the needle falls at an angle where its position matters, and it crosses the line.
6839:
2140:
340:
4979:
6948:
2419:
There is an even more elegant and simple method of calculating the "short needle case". The end of the needle farthest away from any one of the two lines bordering its region must be located within a horizontal (perpendicular to the bordering lines) distance of
1292:
583:
5360:
4569:
4487:
4117:
4286:
1092:
3428:
Now consider the case where the plane contains two sets of parallel lines orthogonal to one another, creating a standard perpendicular grid. We aim to find the probability that the needle intersects at least one line on the grid. Let
3670:
5510:
1303:
409:
6703:{\displaystyle \operatorname {Var} (x_{n}+y_{n})={\frac {1}{\pi }}\left(1-{\frac {1}{\pi }}\right)+{\frac {1}{\pi }}\left(1-{\frac {1}{\pi }}\right)+2\left({\frac {\pi -4}{4\pi ^{2}}}\right)={\frac {5\pi -8}{2\pi ^{2}}}}
820:
5701:
2488:
3298:
6362:
5544:
7161:
4765:
3800:
6351:
2145:
2327:{\displaystyle {\begin{aligned}P_{2}&=\int _{0}^{1}{\frac {2\theta (x)}{\pi }}\,dx\\&={\frac {2}{\pi }}\int _{0}^{1}\cos ^{-1}(x)\,dx\\&={\frac {2}{\pi }}\cdot 1={\frac {2}{\pi }}.\end{aligned}}}
3212:
6937:
6723:
5021:
1551:
5016:
2434:
is the angle between the needle and the horizontal) from this line in order for the needle to cross it. The farthest this end of the needle can move away from this line horizontally in its region is
230:
145:
2809:. Matches with the length of 9 squares have been thrown 17 times between rows with the width of 9 squares. 11 of the matches have landed at random across the drawn lines marked by the green points.
3057:
7095:{\displaystyle \operatorname {Var} ({\hat {q}})={\frac {1}{M^{2}}}(M)\operatorname {Var} (x_{m})={\frac {1}{M}}\cdot {\frac {1}{\pi }}\left(1-{\frac {1}{\pi }}\right)\approx {\frac {0.217}{M}}}
4392:
shows up in that answer as well. The following question then naturally arises and is discussed by E.F. Schuster in 1974. Is Buffon's experiment or
Laplace's a better estimator of the value of
4370:
791:
7198:
are negatively correlated random variables, they act to reduce the total variance in the estimator that is an average of the two of them. This method of variance reduction is known as the
735:{\displaystyle f_{X,\Theta }(x,\theta )={\begin{cases}{\dfrac {4}{t\pi }}&:\ 0\leq x\leq {\dfrac {t}{2}},\ 0\leq \theta \leq {\dfrac {\pi }{2}}\\0&:{\text{elsewhere.}}\end{cases}}}
4791:
1171:
376:
represents a needle that is perfectly centered between two lines. The uniform PDF assumes the needle is equally likely to fall anywhere in this range, but could not fall outside of it.
5231:
3416:
possible that the "experiment" itself was never physically performed, but based on numbers concocted from imagination to match statistical expectations, but too well, as it turns out.
4506:
4424:
3345:
2954:
3991:
4128:
992:
7166:
Thus, it takes 222 drops with only parallel lines to have the same certainty as 100 drops in
Laplace's case. This isn't actually surprising because of the observation that
3408:
accurate to six decimal places. If not, one can just do 213 more trials and hope for a total of 226 successes; if not, just repeat as necessary. Lazzarini performed
2071:
3596:
5420:
1422:{\displaystyle P={\frac {2}{\pi }}\arccos {\frac {t}{l}}+{\frac {2}{\pi }}\cdot {\frac {l}{t}}\left(1-{\sqrt {1-\left({\frac {t}{l}}\right)^{2}}}\right).}
514:{\displaystyle f_{\Theta }(\theta )={\begin{cases}{\dfrac {2}{\pi }}&:\ 0\leq \theta \leq {\dfrac {\pi }{2}}\\0&:{\text{elsewhere.}}\end{cases}}}
2030:
on each side, so it crosses the vertical line; the blue one does not. The proportion of the circle that is gray is what we integrate as the center
951:{\displaystyle P=\int _{\theta =0}^{\frac {\pi }{2}}\int _{x=0}^{{\frac {l}{2}}\sin \theta }{\frac {4}{t\pi }}\,dx\,d\theta ={\frac {2l}{t\pi }}.}
1873:
The following solution for the "short needle" case, while equivalent to the one above, has a more visual flavor, and avoids iterated integrals.
6487:{\displaystyle \operatorname {Cov} (x_{n},y_{n})={\frac {1}{4\pi }}-{\frac {1}{\pi }}\cdot {\frac {1}{\pi }}={\frac {\pi -4}{4\pi ^{2}}}<0}
3062:
In 1901, Italian mathematician Mario
Lazzarini performed Buffon's needle experiment. Tossing a needle 3,408 times, he obtained the well-known
5680:{\displaystyle \operatorname {Var} (x_{n}+y_{n})=\operatorname {Var} (x_{n})+\operatorname {Var} (y_{n})+2\operatorname {Cov} (x_{n},y_{n}).}
3247:
3975:{\displaystyle F(\varphi )=(a-l\cos \varphi )(b-l\sin \varphi )=ab-bl\cos \varphi -al|\sin \varphi |+{\tfrac {1}{2}}l^{2}|\sin 2\varphi |.}
7115:
166:
in this expression occurs because the underlying probability distribution function for the needle orientation is rotationally symmetric.
1921:
Looking at the illustration in the above section, it is apparent that the needle can cross a line if the center of the needle is within
6272:
6834:{\displaystyle \operatorname {Var} ({\hat {p}})={\frac {1}{200^{2}}}(100)\left({\frac {5\pi -8}{2\pi ^{2}}}\right)\approx 0.000\,976.}
3794:
is positive or negative. Taking the positive case and then adding the absolute value signs in the final answer for generality, we get
6855:
then we can conclude that the setup with only parallel lines is more efficient than the case with perpendicular lines. Conversely if
7430:
5516:
We are interested in the variance of such an estimator to understand the usefulness or efficiency of it. To compute the variance of
4295:, that the needle does intersect at least one line, we need to subtract the above result from 1 to compute its compliment, yielding
3171:
6869:
5178:{\displaystyle {\begin{aligned}P(A)&={\frac {1}{\pi }}=P(AB)+P(AB')\\P(B)&={\frac {1}{\pi }}=P(AB)+P(A'B).\end{aligned}}}
558:
represents a needle that is perpendicular to the marked lines. Any angle within this range is assumed an equally likely outcome.
335:{\displaystyle f_{X}(x)={\begin{cases}{\dfrac {2}{t}}&:\ 0\leq x\leq {\dfrac {t}{2}}\\0&:{\text{elsewhere.}}\end{cases}}}
44:
101:
7533:
7396:
7659:
3105:(in fact, there is no better rational approximation with fewer than five digits in the numerator and denominator, see also
7256:"Nineteenth-Century Developments in Geometric Probability: J. J. Sylvester, M. W. Crofton, J.-É. Barbier, and J. Bertrand"
3010:
4649:
for
Laplace's result, that is, the probability the needle intersects both a horizontal and a vertical line. We know that
4301:
1909:
is the probability that the center of the needle falls close enough to a line for the needle to possibly cross it, and
751:
3404:
successes. If one drops 213 needles and happens to get 113 successes, then one can triumphantly report an estimate of
7310:
4974:{\displaystyle P(A'B')=1-{\frac {2l(a+b)-l^{2}}{\pi ab}}=1-{\frac {2l(4l)-l^{2}}{4l^{2}\pi }}=1-{\frac {7}{4\pi }}.}
4655:
1287:{\displaystyle P={\frac {2l}{t\pi }}-{\frac {2}{t\pi }}\left({\sqrt {l^{2}-t^{2}}}+t\arcsin {\frac {t}{l}}\right)+1}
5355:{\displaystyle P(AB)=1-2\left({\frac {1}{\pi }}-P(AB)\right)-\left(1-{\frac {7}{4\pi }}\right)={\frac {1}{4\pi }}}
7334:
4564:{\displaystyle y={\begin{cases}1&:{\text{intersection occurs}}\\0&:{\text{no intersection}}\end{cases}}}
4482:{\displaystyle x={\begin{cases}1&:{\text{intersection occurs}}\\0&:{\text{no intersection}}\end{cases}}}
7211:
574:
4112:{\displaystyle V^{*}=\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}F(\varphi )\,d\varphi =\pi ab-2bl-2al+l^{2}.}
160:, although that was not the original motivation for de Buffon's question. The seemingly unusual appearance of
7707:
7543:
Dell, Zachary; Franklin, Scott V. (September 2009). "The Buffon-Laplace needle problem in three dimensions".
2866:
4281:{\displaystyle {\frac {V^{*}}{V}}={\frac {\pi ab-2bl-2al+l^{2}}{\pi ab}}=1-{\frac {2l(a+b)-l^{2}}{\pi ab}}.}
3701:, let's first look at the case for the horizontal edges of the bounding rectangle. The total side length is
3309:
3132:
of the width of the strips of wood. In this case, the probability that the needles will cross the lines is
2913:
2770:
The short-needle problem can also be solved without any integration, in a way that explains the formula for
1087:{\displaystyle \int _{\theta =0}^{\frac {\pi }{2}}\int _{x=0}^{m(\theta )}{\frac {4}{t\pi }}\,dx\,d\theta ,}
814:
Integrating the joint probability density function gives the probability that the needle will cross a line:
7697:
626:
440:
261:
197:
5406:. With the values above, we are now able to determine which of these estimators is a better estimator for
7629:
7579:
Schroeder, L. (1974). "Buffon's needle problem: An exciting application of many mathematical concepts".
7702:
6859:
is equal to or more than 200, than Buffon's experiment is equally or less efficient, respectively. Let
7364:"Un'applicazione del calcolo della probabilità alla ricerca sperimentale di un valore approssimato di
1918:
is the probability that the needle actually crosses the line, given that the center is within reach.
4521:
4439:
7642:
7363:
2783:
3063:
7237:
7231:
76:
7564:
7552:
7370:[An Application of Probability Theory to Experimental Research of an Approximation of
3586:
be the volume of possibilities where the needle does not intersect any line. Developed by
3412:
trials, making it seem likely that this is the strategy he used to obtain his "estimate".
182:
units apart, what is the probability that the needle will lie across a line upon landing?
8:
7199:
2782:
strips always (i.e. with probability 1) in exactly two spots. This solution was given by
2448:
away from the line (and thus that the needle crosses the line) out of the total distance
2438:. The probability that the farthest end of the needle is located no more than a distance
2050:
56:
7556:
3729:
of either endpoint of the edge. Thus, the total allowable length for no intersection is
3665:{\displaystyle V^{*}=\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}F(\varphi )\,d\varphi }
7596:
7568:
7510:
7481:
7291:
7283:
3437:
be the sides of the rectangle that contains the midpoint of the needle whose length is
2787:
2038:
Now, we assume that the center is within reach of the edge of the strip, and calculate
151:
36:
7638:
2861:
7529:
7295:
7275:
3086:
80:
7614:
7572:
5505:{\displaystyle {\hat {p}}={\frac {1}{100}}\sum _{n=1}^{100}{\frac {x_{n}+y_{n}}{2}}}
189:
be the distance from the center of the needle to the closest parallel line, and let
7605:
7560:
7502:
7473:
7267:
562:
1481:, the probability of crossing is the same as in the short needle case. However if
7523:
5414:
be the estimator for the probability that there is a line intersection such that
7673:
7619:
6848:, needed to achieve the same variance as 100 drops over perpendicular lines. If
4396:? Since in Laplace's extension there are two sets of parallel lines, we compare
7646:
2099:
radians of possible orientations. This represents the gray area to the left of
986:. In this case, integrating the joint probability density function, we obtain:
2015:
7691:
7651:
7279:
7251:
6713:
Returning to the original problem of this section, the variance of estimator
3587:
64:
3085:, accurate to six decimal places. Lazzarini's "experiment" is an example of
7609:
7600:
7586:
Uspensky, James Victor. "Introduction to mathematical probability." (1937).
4414:
be the event that the needle intersects a horizontal line (parallel to the
2903:
In the first, simpler case above, the formula obtained for the probability
7271:
3686:
is the region where the needle does not intersect any line given an angle
7669:
7664:
4496:
be the event that the needle intersects a vertical line (parallel to the
68:
7351:(2nd ed.). Springer Science & Business Media. pp. 189–192.
7287:
7255:
3089:, as it was set up to replicate the already well-known approximation of
2088:, the needle will cross the vertical axis if it falls within a range of
2084:
be as in the illustration in this section. Placing a needle's center at
7633:
7514:
7485:
7443:
4384:
As mentioned above, Buffon's needle experiment can be used to estimate
2870:
5365:
Although not necessary to the problem, it is now possible to see that
3293:{\displaystyle {\frac {355}{113}}={\frac {5}{3}}\cdot {\frac {n}{x}},}
4122:
Thus, the probability that the needle does not intersect any line is
193:
be the acute angle between the needle and one of the parallel lines.
7506:
7477:
174:
The problem in more mathematical terms is: Given a needle of length
7156:{\displaystyle {\frac {0.217}{M}}=0.000\,976\implies M\approx 222.}
2023:. The red one falls within the gray area, contained by an angle of
7624:
1165:, the probability that the needle will cross at least one line is
6346:{\displaystyle \mathbb {E} (x_{n}y_{n})=P(AB)={\frac {1}{4\pi }}}
7417:
3106:
2801:
19:
555:
7460:
3207:{\displaystyle \pi \approx {\frac {5}{3}}\cdot {\frac {n}{x}}}
3113:
than would be expected from the number of trials, as follows:
531:
represents a needle that is parallel to the marked lines, and
6932:{\displaystyle {\hat {q}}={\frac {1}{M}}\sum _{m=1}^{M}x_{m}}
352:
represents a needle that is centered directly on a line, and
52:
4785:, or the probability that the needle intersects no lines is
3481:-axis. Similar to the examples described above, we consider
4557:
4475:
3493:
to be independent uniform random variables over the ranges
728:
507:
328:
60:
7493:
Ramaley, J. F. (October 1969). "Buffon's Noodle
Problem".
6863:
be the estimator for Buffon's original experiment. Then,
3423:
1155:
Thus, performing the above integration, we see that, when
71:
that the needle will lie across a line between two strips?
2873:
to sketch Buffon's needle experiment with the parameters
7333:
The problem formulation here avoids having to work with
140:{\displaystyle p={\frac {2}{\pi }}\cdot {\frac {l}{t}}.}
7545:
Journal of
Statistical Mechanics: Theory and Experiment
4574:
For simplicity in the algebraic formulation ahead, let
161:
155:
7542:
7311:"Buffon: Hat er Stöckchen geworfen oder hat er nicht?"
7250:
4291:
And finally, if we want to calculate the probability,
3926:
3473:
mark the coordinates of the needle's midpoint and let
7376:
Periodico di
Matematica per l'Insegnamento Secondario
7254:; Parshall, Karen Hunger; Jongmans, François (2001).
7118:
6951:
6872:
6726:
6506:
6365:
6275:
5699:
5547:
5423:
5234:
5019:
4794:
4658:
4592:
such that the original result in Buffon's problem is
4509:
4427:
4304:
4131:
3994:
3803:
3599:
3312:
3250:
3174:
3013:
2916:
2641:
2603:
2544:
2503:
2486:
2143:
2053:
1549:
1306:
1174:
995:
823:
754:
697:
667:
630:
586:
476:
444:
412:
297:
265:
233:
104:
7630:
Buffon's needle: fun and fundamentals (presentation)
4388:. This fact holds for Laplace's extension too since
3052:{\displaystyle \pi \approx {\frac {2l\cdot n}{th}}.}
7501:(8). Mathematical Association of America: 916–918.
5214:and plugging that into the original definition for
3772:. Equivalently, for the vertical edges with length
7155:
7094:
6931:
6833:
6702:
6486:
6345:
6255:
5679:
5504:
5354:
5177:
4973:
4759:
4563:
4481:
4364:
4280:
4111:
3974:
3664:
3339:
3292:
3206:
3051:
2948:
2774:from the geometric fact that a circle of diameter
2754:
2326:
2065:
2047:. To simplify the calculation, we can assume that
1857:
1421:
1286:
1086:
950:
785:
734:
513:
334:
139:
7472:(2). Mathematical Association of America: 83–91.
7446:, The American Mathematical Monthly, 1974, 29-29.
4365:{\displaystyle P={\frac {2l(a+b)-l^{2}}{\pi ab}}}
2895:-axis) approaching 3.14 as the number of tosses (
961:
786:{\displaystyle x\leq {\frac {l}{2}}\sin \theta .}
43:is a question first posed in the 18th century by
7689:
7639:Animations for the Simulation of Buffon's Needle
1979:from both sides and dividing by the whole width
1540:the probability is constant and is equal to 1.
799:
7418:'Het stokje van Lazzarini' (Lazzarini's stick)
4760:{\displaystyle P(AB)=1-P(AB')-P(A'B)-P(A'B').}
2959:Thus, if we conduct an experiment to estimate
2019:The red and blue needles are both centered at
7436:
7346:
5833:
5791:
178:dropped on a plane ruled with parallel lines
7391:
7389:
7238:Histoire naturelle, générale et particulière
4376:
3985:Now we can compute the following integral:
3477:mark the angle formed by the needle and the
3109:), yielding a more accurate "prediction" of
1868:
379:The uniform probability density function of
75:Buffon's needle was the earliest problem in
7347:Aigner, Martin; Ziegler, Günter M. (2013).
7528:. Newark: Gordon & Breach. p. 5.
7525:An Introduction to Geometrical Probability
7431:'Introduction to Mathematical Probability'
7410:
7143:
7139:
6844:Now let us calculate the number of drops,
4400:drops when there is a grid (Laplace), and
3217:So if Lazzarini was aiming for the result
1941:units of either side of the strip. Adding
83:. The solution for the sought probability
7386:
7361:
7335:Regular conditional probability densities
7135:
6827:
6277:
6229:
6208:
6174:
5810:
4050:
3655:
3116:Lazzarini chose needles whose length was
2963:, we will also have an estimate for
2711:
2700:
2633:
2568:
2536:
2271:
2204:
1074:
1067:
915:
908:
2978:of those needles are crossing lines, so
2860:
2800:
2014:
18:
7492:
7423:
6266:We know from the previous section that
4407:drops in Buffon's original experiment.
3441:. Since this is the short needle case,
3424:Laplace's extension (short needle case)
7690:
7521:
7459:
7232:Histoire de l'Acad. Roy. des. Sciences
3340:{\displaystyle x={\frac {113n}{213}}.}
2949:{\displaystyle \pi ={\frac {2l}{tP}}.}
87:, in the case where the needle length
45:Georges-Louis Leclerc, Comte de Buffon
7260:Archive for History of Exact Sciences
3790:. The ± accounts for the cases where
1880:as the product of two probabilities:
79:to be solved; it can be solved using
63:, each the same width, and we drop a
27:needle lies across a line, while the
7397:'Lazzarini's Lucky Approximation of
7308:
5690:Solving for each part individually,
3705:and the midpoint must not be within
2786:in 1860 and is also referred to as "
2765:
2337:Multiplying both results, we obtain
1436:Alternatively, notice that whenever
7657:
7625:Estimating PI Visualization (Flash)
3354:as a multiple of 213, because then
13:
2887:. Observe the calculated value of
2134:range from 0 to 1, and integrate:
598:
575:joint probability density function
418:
14:
7719:
7590:
7495:The American Mathematical Monthly
1876:We can calculate the probability
16:Question in geometric probability
7565:10.1088/1742-5468/2009/09/P09010
2982:is approximated by the fraction
7606:Math Surprises: Buffon's Noodle
7453:
5410:. For the Laplace variant, let
3379:needles, and hopes for exactly
2793:
7355:
7340:
7327:
7302:
7244:
7224:
7212:Bertrand paradox (probability)
7140:
7024:
7011:
7002:
6996:
6973:
6967:
6958:
6879:
6777:
6771:
6748:
6742:
6733:
6539:
6513:
6398:
6372:
6322:
6313:
6304:
6281:
6246:
6233:
6225:
6212:
6201:
6178:
6163:
6137:
5937:
5918:
5876:
5857:
5827:
5814:
5748:
5735:
5723:
5710:
5671:
5645:
5630:
5617:
5605:
5592:
5580:
5554:
5430:
5292:
5283:
5247:
5238:
5165:
5151:
5142:
5133:
5107:
5101:
5091:
5077:
5068:
5059:
5033:
5027:
4907:
4898:
4850:
4838:
4817:
4798:
4751:
4732:
4723:
4709:
4700:
4686:
4671:
4662:
4332:
4320:
4245:
4233:
4047:
4041:
3965:
3948:
3918:
3904:
3864:
3843:
3840:
3819:
3813:
3807:
3652:
3646:
3375:is an integer; one then drops
3161:crossings, one would estimate
2452:it can move in its region for
2268:
2262:
2195:
2189:
1047:
1041:
615:
603:
429:
423:
250:
244:
91:is not greater than the width
1:
7217:
3004:. This leads to the formula:
745:The needle crosses a line if
154:for approximating the number
150:This can be used to design a
5010:using the following method:
3350:To do this, one should pick
3153:. Thus if one were to drop
198:probability density function
67:onto the floor. What is the
7:
7620:Buffon's Needle Java Applet
7241:Supplément 4 (1777), p. 46.
7205:
2899:-axis) approaches infinity.
2103:in the figure. For a fixed
169:
10:
7724:
7660:"π Pi and Buffon's Needle"
7444:Buffon's Needle Experiment
5225:a few lines above, we get
573:, are independent, so the
3566:. We can easily see that
1869:Using elementary calculus
796:Now there are two cases.
4770:From the above section,
4377:Comparing estimators of
2778:will cross the distance
1454:, that is, in the range
7641:by Yihui Xie using the
7597:Buffon's Needle Problem
7420:, "Skepter" 31.3, 2018.
2869:based simulation using
1108:is the minimum between
41:Buffon's needle problem
7522:Mathai, A. M. (1999).
7362:Lazzarini, M. (1901).
7157:
7096:
6933:
6918:
6835:
6704:
6488:
6347:
6257:
5778:
5681:
5506:
5469:
5356:
5179:
4975:
4761:
4565:
4483:
4366:
4282:
4113:
3976:
3666:
3341:
3294:
3208:
3053:
2974:needles and find that
2950:
2900:
2858:
2805:An experiment to find
2756:
2328:
2067:
2035:
1859:
1440:has a value such that
1423:
1288:
1088:
952:
800:Case 1: Short needle (
787:
736:
515:
336:
141:
32:
7652:3D Physical Animation
7615:MSTE: Buffon's Needle
7272:10.1007/s004070100038
7158:
7097:
6934:
6898:
6836:
6705:
6489:
6348:
6258:
5758:
5682:
5507:
5449:
5357:
5180:
4976:
4762:
4566:
4484:
4367:
4283:
4114:
3977:
3667:
3342:
3295:
3209:
3054:
2951:
2907:can be rearranged to
2864:
2804:
2757:
2329:
2068:
2018:
1860:
1424:
1289:
1089:
962:Case 2: Long needle (
953:
788:
737:
516:
337:
142:
77:geometric probability
22:
7708:Probability problems
7654:by Jeffrey Ventrella
7466:Mathematics Magazine
7405:Mathematics Magazine
7349:Proofs from THE BOOK
7116:
6949:
6870:
6724:
6504:
6363:
6273:
5697:
5545:
5421:
5232:
5017:
4792:
4656:
4507:
4425:
4302:
4129:
3992:
3801:
3597:
3310:
3248:
3172:
3011:
2914:
2784:Joseph-Émile Barbier
2484:
2141:
2051:
1547:
1304:
1172:
993:
821:
752:
584:
410:
231:
102:
7698:Applied probability
7581:Mathematics Teacher
7557:2009JSMTE..09..010D
7309:Behrends, Ehrhard.
7200:antithetic variates
5520:, we first compute
4638:Now let us examine
4628:. Furthermore, let
4534:intersection occurs
4452:intersection occurs
4037:
3642:
2661:
2623:
2564:
2523:
2245:
2179:
2066:{\displaystyle l=2}
1714:
1639:
1602:
1051:
1021:
892:
855:
7153:
7092:
6929:
6831:
6700:
6484:
6343:
6253:
6251:
5677:
5502:
5352:
5175:
5173:
4971:
4757:
4561:
4556:
4479:
4474:
4362:
4278:
4109:
4008:
3972:
3935:
3662:
3613:
3337:
3290:
3204:
3049:
2946:
2901:
2859:
2752:
2750:
2668:
2642:
2640:
2604:
2575:
2545:
2543:
2504:
2324:
2322:
2231:
2165:
2063:
2036:
1855:
1853:
1682:
1603:
1569:
1419:
1284:
1084:
1022:
996:
948:
856:
830:
783:
732:
727:
706:
676:
644:
511:
506:
485:
453:
332:
327:
306:
274:
152:Monte Carlo method
137:
95:of the strips, is
51:Suppose we have a
37:probability theory
33:
7703:Integral geometry
7535:978-90-5699-681-9
7416:Hans van Maanen,
7127:
7090:
7072:
7051:
7038:
6994:
6970:
6896:
6882:
6815:
6769:
6745:
6698:
6658:
6613:
6592:
6574:
6553:
6476:
6443:
6430:
6417:
6341:
6117:
6096:
6072:
6048:
6013:
5991:
5960:
5899:
5500:
5447:
5433:
5350:
5327:
5275:
5125:
5051:
4984:We can solve for
4966:
4942:
4878:
4552:
4535:
4470:
4453:
4360:
4273:
4213:
4147:
4035:
4024:
3934:
3640:
3629:
3332:
3303:or equivalently,
3285:
3272:
3259:
3202:
3189:
3087:confirmation bias
3044:
2941:
2766:Without integrals
2743:
2713:
2709:
2689:
2669:
2659:
2621:
2596:
2576:
2562:
2521:
2315:
2296:
2229:
2202:
2130:. Now we can let
2111:as a function of
2107:, we can express
1838:
1816:
1784:
1766:
1723:
1712:
1701:
1653:
1627:
1599:
1409:
1397:
1363:
1350:
1337:
1321:
1271:
1249:
1217:
1199:
1065:
1019:
943:
906:
880:
853:
769:
723:
705:
683:
675:
653:
643:
502:
484:
462:
452:
323:
305:
283:
273:
132:
119:
81:integral geometry
7715:
7684:
7682:
7681:
7672:. Archived from
7583:, 67 (2), 183–6.
7576:
7539:
7518:
7489:
7463:
7447:
7442:E. F. Schuster,
7440:
7434:
7427:
7421:
7414:
7408:
7407:67, 1994, 83–91.
7400:
7393:
7384:
7383:
7373:
7367:
7359:
7353:
7352:
7344:
7338:
7331:
7325:
7324:
7322:
7320:
7315:
7306:
7300:
7299:
7248:
7242:
7228:
7197:
7190:
7183:
7162:
7160:
7159:
7154:
7128:
7120:
7108:
7101:
7099:
7098:
7093:
7091:
7083:
7078:
7074:
7073:
7065:
7052:
7044:
7039:
7031:
7023:
7022:
6995:
6993:
6992:
6980:
6972:
6971:
6963:
6938:
6936:
6935:
6930:
6928:
6927:
6917:
6912:
6897:
6889:
6884:
6883:
6875:
6862:
6858:
6854:
6847:
6840:
6838:
6837:
6832:
6820:
6816:
6814:
6813:
6812:
6799:
6785:
6770:
6768:
6767:
6755:
6747:
6746:
6738:
6716:
6709:
6707:
6706:
6701:
6699:
6697:
6696:
6695:
6682:
6668:
6663:
6659:
6657:
6656:
6655:
6642:
6631:
6619:
6615:
6614:
6606:
6593:
6585:
6580:
6576:
6575:
6567:
6554:
6546:
6538:
6537:
6525:
6524:
6493:
6491:
6490:
6485:
6477:
6475:
6474:
6473:
6460:
6449:
6444:
6436:
6431:
6423:
6418:
6416:
6405:
6397:
6396:
6384:
6383:
6352:
6350:
6349:
6344:
6342:
6340:
6329:
6303:
6302:
6293:
6292:
6280:
6262:
6260:
6259:
6254:
6252:
6245:
6244:
6232:
6224:
6223:
6211:
6200:
6199:
6190:
6189:
6177:
6162:
6161:
6149:
6148:
6123:
6119:
6118:
6110:
6097:
6089:
6084:
6083:
6078:
6074:
6073:
6065:
6054:
6050:
6049:
6041:
6025:
6024:
6019:
6015:
6014:
6006:
5992:
5984:
5976:
5972:
5971:
5966:
5962:
5961:
5953:
5930:
5929:
5911:
5910:
5905:
5901:
5900:
5892:
5869:
5868:
5847:
5843:
5842:
5837:
5836:
5826:
5825:
5813:
5805:
5804:
5795:
5794:
5788:
5787:
5777:
5772:
5747:
5746:
5722:
5721:
5686:
5684:
5683:
5678:
5670:
5669:
5657:
5656:
5629:
5628:
5604:
5603:
5579:
5578:
5566:
5565:
5537:
5519:
5511:
5509:
5508:
5503:
5501:
5496:
5495:
5494:
5482:
5481:
5471:
5468:
5463:
5448:
5440:
5435:
5434:
5426:
5413:
5409:
5405:
5404:
5402:
5401:
5395:
5392:
5361:
5359:
5358:
5353:
5351:
5349:
5338:
5333:
5329:
5328:
5326:
5315:
5299:
5295:
5276:
5268:
5224:
5213:
5202:
5184:
5182:
5181:
5176:
5174:
5161:
5126:
5118:
5090:
5052:
5044:
5009:
4998:
4980:
4978:
4977:
4972:
4967:
4965:
4954:
4943:
4941:
4937:
4936:
4923:
4922:
4921:
4890:
4879:
4877:
4866:
4865:
4864:
4830:
4816:
4808:
4784:
4766:
4764:
4763:
4758:
4750:
4742:
4719:
4699:
4648:
4634:
4627:
4626:
4624:
4623:
4618:
4615:
4591:
4570:
4568:
4567:
4562:
4560:
4559:
4553:
4550:
4536:
4533:
4499:
4495:
4488:
4486:
4485:
4480:
4478:
4477:
4471:
4468:
4454:
4451:
4417:
4413:
4406:
4399:
4395:
4391:
4387:
4380:
4371:
4369:
4368:
4363:
4361:
4359:
4348:
4347:
4346:
4312:
4294:
4287:
4285:
4284:
4279:
4274:
4272:
4261:
4260:
4259:
4225:
4214:
4212:
4201:
4200:
4199:
4153:
4148:
4143:
4142:
4133:
4118:
4116:
4115:
4110:
4105:
4104:
4036:
4028:
4026:
4025:
4017:
4004:
4003:
3981:
3979:
3978:
3973:
3968:
3951:
3946:
3945:
3936:
3927:
3921:
3907:
3793:
3789:
3775:
3771:
3757:
3751:
3749:
3748:
3745:
3742:
3728:
3723:
3721:
3720:
3717:
3714:
3704:
3700:
3689:
3685:
3671:
3669:
3668:
3663:
3641:
3633:
3631:
3630:
3622:
3609:
3608:
3585:
3575:
3565:
3557:
3556:
3554:
3553:
3550:
3547:
3534:
3532:
3531:
3528:
3525:
3514:
3503:
3492:
3488:
3484:
3480:
3476:
3472:
3460:
3450:
3440:
3436:
3432:
3411:
3410:3,408 = 213 × 16
3407:
3403:
3402:
3400:
3399:
3396:
3393:
3378:
3374:
3373:
3371:
3370:
3367:
3364:
3353:
3346:
3344:
3343:
3338:
3333:
3328:
3320:
3299:
3297:
3296:
3291:
3286:
3278:
3273:
3265:
3260:
3252:
3240:
3236:
3232:
3230:
3229:
3226:
3223:
3213:
3211:
3210:
3205:
3203:
3195:
3190:
3182:
3164:
3160:
3157:needles and get
3156:
3152:
3151:
3149:
3148:
3142:
3139:
3131:
3129:
3128:
3125:
3122:
3112:
3104:
3102:
3101:
3098:
3095:
3084:
3080:
3078:
3077:
3074:
3071:
3058:
3056:
3055:
3050:
3045:
3043:
3035:
3021:
3003:
3002:
3000:
2999:
2994:
2991:
2981:
2977:
2973:
2970:Suppose we drop
2966:
2962:
2955:
2953:
2952:
2947:
2942:
2940:
2932:
2924:
2906:
2898:
2894:
2890:
2886:
2879:
2856:
2855:
2851:
2849:
2848:
2845:
2842:
2835:
2833:
2832:
2827:
2824:
2808:
2797:
2781:
2777:
2773:
2761:
2759:
2758:
2753:
2751:
2744:
2742:
2734:
2726:
2718:
2714:
2712:
2710:
2702:
2695:
2690:
2682:
2674:
2670:
2660:
2652:
2650:
2622:
2614:
2612:
2602:
2597:
2589:
2581:
2577:
2563:
2555:
2553:
2522:
2514:
2512:
2502:
2476:
2475:
2473:
2472:
2469:
2466:
2451:
2447:
2437:
2433:
2429:
2415:
2414:
2412:
2411:
2406:
2403:
2393:
2391:
2390:
2385:
2382:
2375:
2373:
2372:
2367:
2364:
2333:
2331:
2330:
2325:
2323:
2316:
2308:
2297:
2289:
2281:
2258:
2257:
2244:
2239:
2230:
2222:
2214:
2203:
2198:
2181:
2178:
2173:
2157:
2156:
2133:
2129:
2114:
2110:
2106:
2102:
2098:
2095:radians, out of
2094:
2087:
2083:
2079:
2072:
2070:
2069:
2064:
2046:
2034:goes from 0 to 1
2033:
2029:
2022:
2011:
2010:
2008:
2007:
2002:
1999:
1982:
1978:
1977:
1975:
1974:
1971:
1968:
1959:
1957:
1956:
1953:
1950:
1940:
1939:
1937:
1936:
1933:
1930:
1917:
1908:
1899:
1879:
1864:
1862:
1861:
1856:
1854:
1844:
1840:
1839:
1831:
1817:
1815:
1814:
1802:
1801:
1792:
1785:
1783:
1772:
1767:
1765:
1757:
1749:
1741:
1737:
1733:
1732:
1724:
1716:
1713:
1705:
1703:
1702:
1694:
1673:
1669:
1668:
1654:
1652:
1641:
1638:
1628:
1620:
1617:
1601:
1600:
1592:
1583:
1539:
1538:
1536:
1535:
1532:
1529:
1516:
1514:
1513:
1508:
1505:
1494:
1480:
1479:
1477:
1476:
1471:
1468:
1453:
1439:
1428:
1426:
1425:
1420:
1415:
1411:
1410:
1408:
1407:
1402:
1398:
1390:
1377:
1364:
1356:
1351:
1343:
1338:
1330:
1322:
1314:
1293:
1291:
1290:
1285:
1277:
1273:
1272:
1264:
1250:
1248:
1247:
1235:
1234:
1225:
1218:
1216:
1205:
1200:
1198:
1190:
1182:
1164:
1151:
1150:
1148:
1147:
1144:
1141:
1131:
1126:
1124:
1123:
1120:
1117:
1107:
1093:
1091:
1090:
1085:
1066:
1064:
1053:
1050:
1036:
1020:
1012:
1010:
985:
971:
957:
955:
954:
949:
944:
942:
934:
926:
907:
905:
894:
891:
881:
873:
870:
854:
846:
844:
809:
792:
790:
789:
784:
770:
762:
741:
739:
738:
733:
731:
730:
724:
721:
707:
698:
681:
677:
668:
651:
645:
642:
631:
602:
601:
572:
568:
563:random variables
554:
553:
551:
550:
547:
544:
530:
520:
518:
517:
512:
510:
509:
503:
500:
486:
477:
460:
454:
445:
422:
421:
402:
401:
399:
398:
395:
392:
382:
375:
374:
372:
371:
368:
365:
351:
341:
339:
338:
333:
331:
330:
324:
321:
307:
298:
281:
275:
266:
243:
242:
223:
222:
220:
219:
216:
213:
203:
192:
188:
181:
177:
164:
158:
146:
144:
143:
138:
133:
125:
120:
112:
94:
90:
86:
31:needle does not.
30:
26:
7723:
7722:
7718:
7717:
7716:
7714:
7713:
7712:
7688:
7687:
7679:
7677:
7658:Padilla, Tony.
7593:
7536:
7507:10.2307/2317945
7478:10.2307/2690682
7461:
7456:
7451:
7450:
7441:
7437:
7429:J.V. Uspensky,
7428:
7424:
7415:
7411:
7398:
7394:
7387:
7371:
7365:
7360:
7356:
7345:
7341:
7332:
7328:
7318:
7316:
7313:
7307:
7303:
7249:
7245:
7235:(1733), 43–45;
7229:
7225:
7220:
7208:
7196:
7192:
7189:
7185:
7180:
7173:
7167:
7119:
7117:
7114:
7113:
7106:
7082:
7064:
7057:
7053:
7043:
7030:
7018:
7014:
6988:
6984:
6979:
6962:
6961:
6950:
6947:
6946:
6923:
6919:
6913:
6902:
6888:
6874:
6873:
6871:
6868:
6867:
6860:
6856:
6849:
6845:
6808:
6804:
6800:
6786:
6784:
6780:
6763:
6759:
6754:
6737:
6736:
6725:
6722:
6721:
6714:
6691:
6687:
6683:
6669:
6667:
6651:
6647:
6643:
6632:
6630:
6626:
6605:
6598:
6594:
6584:
6566:
6559:
6555:
6545:
6533:
6529:
6520:
6516:
6505:
6502:
6501:
6469:
6465:
6461:
6450:
6448:
6435:
6422:
6409:
6404:
6392:
6388:
6379:
6375:
6364:
6361:
6360:
6333:
6328:
6298:
6294:
6288:
6284:
6276:
6274:
6271:
6270:
6250:
6249:
6240:
6236:
6228:
6219:
6215:
6207:
6195:
6191:
6185:
6181:
6173:
6166:
6157:
6153:
6144:
6140:
6128:
6127:
6109:
6102:
6098:
6088:
6079:
6064:
6060:
6056:
6055:
6040:
6033:
6029:
6020:
6005:
5998:
5994:
5993:
5983:
5974:
5973:
5967:
5952:
5945:
5941:
5940:
5925:
5921:
5906:
5891:
5884:
5880:
5879:
5864:
5860:
5845:
5844:
5838:
5832:
5831:
5830:
5821:
5817:
5809:
5800:
5796:
5790:
5789:
5783:
5779:
5773:
5762:
5751:
5742:
5738:
5717:
5713:
5700:
5698:
5695:
5694:
5665:
5661:
5652:
5648:
5624:
5620:
5599:
5595:
5574:
5570:
5561:
5557:
5546:
5543:
5542:
5534:
5527:
5521:
5517:
5490:
5486:
5477:
5473:
5472:
5470:
5464:
5453:
5439:
5425:
5424:
5422:
5419:
5418:
5411:
5407:
5396:
5393:
5390:
5389:
5387:
5366:
5342:
5337:
5319:
5314:
5307:
5303:
5267:
5266:
5262:
5233:
5230:
5229:
5215:
5204:
5189:
5172:
5171:
5154:
5117:
5110:
5095:
5094:
5083:
5043:
5036:
5020:
5018:
5015:
5014:
5000:
4985:
4958:
4953:
4932:
4928:
4924:
4917:
4913:
4891:
4889:
4867:
4860:
4856:
4831:
4829:
4809:
4801:
4793:
4790:
4789:
4771:
4743:
4735:
4712:
4692:
4657:
4654:
4653:
4639:
4629:
4619:
4616:
4613:
4612:
4610:
4593:
4575:
4555:
4554:
4551:no intersection
4549:
4544:
4538:
4537:
4532:
4527:
4517:
4516:
4508:
4505:
4504:
4497:
4493:
4473:
4472:
4469:no intersection
4467:
4462:
4456:
4455:
4450:
4445:
4435:
4434:
4426:
4423:
4422:
4415:
4411:
4401:
4397:
4393:
4389:
4385:
4382:
4378:
4349:
4342:
4338:
4313:
4311:
4303:
4300:
4299:
4292:
4262:
4255:
4251:
4226:
4224:
4202:
4195:
4191:
4154:
4152:
4138:
4134:
4132:
4130:
4127:
4126:
4100:
4096:
4027:
4016:
4012:
3999:
3995:
3993:
3990:
3989:
3964:
3947:
3941:
3937:
3925:
3917:
3903:
3802:
3799:
3798:
3791:
3777:
3773:
3759:
3758:or simply just
3746:
3743:
3738:
3737:
3735:
3730:
3718:
3715:
3710:
3709:
3707:
3706:
3702:
3691:
3690:. To determine
3687:
3676:
3632:
3621:
3617:
3604:
3600:
3598:
3595:
3594:
3580:
3567:
3563:
3551:
3548:
3543:
3542:
3540:
3529:
3526:
3521:
3520:
3518:
3516:
3505:
3494:
3490:
3486:
3482:
3478:
3474:
3462:
3452:
3442:
3438:
3434:
3430:
3426:
3409:
3405:
3397:
3394:
3388:
3387:
3385:
3380:
3376:
3368:
3365:
3359:
3358:
3356:
3355:
3351:
3321:
3319:
3311:
3308:
3307:
3277:
3264:
3251:
3249:
3246:
3245:
3238:
3234:
3227:
3224:
3221:
3220:
3218:
3194:
3181:
3173:
3170:
3169:
3162:
3158:
3154:
3143:
3140:
3137:
3136:
3134:
3133:
3126:
3123:
3120:
3119:
3117:
3110:
3099:
3096:
3093:
3092:
3090:
3082:
3075:
3072:
3069:
3068:
3066:
3036:
3022:
3020:
3012:
3009:
3008:
2995:
2992:
2987:
2986:
2984:
2983:
2979:
2975:
2971:
2964:
2960:
2933:
2925:
2923:
2915:
2912:
2911:
2904:
2896:
2892:
2888:
2881:
2874:
2853:
2846:
2843:
2840:
2839:
2837:
2828:
2825:
2815:
2814:
2812:
2811:
2810:
2806:
2799:
2795:
2788:Buffon's noodle
2779:
2775:
2771:
2768:
2749:
2748:
2735:
2727:
2725:
2716:
2715:
2701:
2699:
2694:
2681:
2672:
2671:
2651:
2646:
2613:
2608:
2601:
2588:
2579:
2578:
2554:
2549:
2513:
2508:
2501:
2494:
2487:
2485:
2482:
2481:
2470:
2467:
2462:
2461:
2459:
2453:
2449:
2439:
2435:
2431:
2421:
2407:
2404:
2398:
2397:
2395:
2386:
2383:
2380:
2379:
2377:
2368:
2365:
2360:
2359:
2357:
2355:
2348:
2338:
2321:
2320:
2307:
2288:
2279:
2278:
2250:
2246:
2240:
2235:
2221:
2212:
2211:
2182:
2180:
2174:
2169:
2158:
2152:
2148:
2144:
2142:
2139:
2138:
2131:
2116:
2112:
2108:
2104:
2100:
2096:
2089:
2085:
2081:
2077:
2052:
2049:
2048:
2045:
2039:
2031:
2024:
2020:
2003:
2000:
1995:
1994:
1992:
1990:
1984:
1980:
1972:
1969:
1964:
1963:
1961:
1954:
1951:
1946:
1945:
1943:
1942:
1934:
1931:
1926:
1925:
1923:
1922:
1916:
1910:
1907:
1901:
1898:
1891:
1881:
1877:
1871:
1852:
1851:
1830:
1810:
1806:
1797:
1793:
1791:
1790:
1786:
1776:
1771:
1758:
1750:
1748:
1739:
1738:
1728:
1715:
1704:
1693:
1686:
1681:
1677:
1664:
1645:
1640:
1619:
1618:
1607:
1591:
1584:
1573:
1568:
1564:
1557:
1550:
1548:
1545:
1544:
1533:
1530:
1525:
1524:
1522:
1509:
1506:
1501:
1500:
1498:
1496:
1482:
1472:
1469:
1464:
1463:
1461:
1455:
1441:
1437:
1403:
1389:
1385:
1384:
1376:
1369:
1365:
1355:
1342:
1329:
1313:
1305:
1302:
1301:
1263:
1243:
1239:
1230:
1226:
1224:
1223:
1219:
1209:
1204:
1191:
1183:
1181:
1173:
1170:
1169:
1156:
1145:
1142:
1137:
1136:
1134:
1133:
1121:
1118:
1113:
1112:
1110:
1109:
1098:
1057:
1052:
1037:
1026:
1011:
1000:
994:
991:
990:
977:
974:
963:
935:
927:
925:
898:
893:
872:
871:
860:
845:
834:
822:
819:
818:
812:
801:
761:
753:
750:
749:
726:
725:
720:
715:
709:
708:
696:
666:
646:
635:
629:
622:
621:
591:
587:
585:
582:
581:
577:is the product
570:
566:
548:
545:
540:
539:
537:
532:
525:
505:
504:
499:
494:
488:
487:
475:
455:
443:
436:
435:
417:
413:
411:
408:
407:
396:
393:
388:
387:
385:
384:
380:
369:
366:
361:
360:
358:
353:
346:
326:
325:
320:
315:
309:
308:
296:
276:
264:
257:
256:
238:
234:
232:
229:
228:
217:
214:
209:
208:
206:
205:
201:
190:
186:
179:
175:
172:
162:
156:
124:
111:
103:
100:
99:
92:
88:
84:
28:
24:
17:
12:
11:
5:
7721:
7711:
7710:
7705:
7700:
7686:
7685:
7655:
7649:
7636:
7627:
7622:
7617:
7612:
7603:
7592:
7591:External links
7589:
7588:
7587:
7584:
7577:
7540:
7534:
7519:
7490:
7455:
7452:
7449:
7448:
7435:
7422:
7409:
7385:
7378:(in Italian).
7354:
7339:
7326:
7301:
7266:(6): 501–524.
7252:Seneta, Eugene
7243:
7222:
7221:
7219:
7216:
7215:
7214:
7207:
7204:
7194:
7187:
7178:
7171:
7164:
7163:
7152:
7149:
7146:
7142:
7138:
7134:
7131:
7126:
7123:
7103:
7102:
7089:
7086:
7081:
7077:
7071:
7068:
7063:
7060:
7056:
7050:
7047:
7042:
7037:
7034:
7029:
7026:
7021:
7017:
7013:
7010:
7007:
7004:
7001:
6998:
6991:
6987:
6983:
6978:
6975:
6969:
6966:
6960:
6957:
6954:
6940:
6939:
6926:
6922:
6916:
6911:
6908:
6905:
6901:
6895:
6892:
6887:
6881:
6878:
6842:
6841:
6830:
6826:
6823:
6819:
6811:
6807:
6803:
6798:
6795:
6792:
6789:
6783:
6779:
6776:
6773:
6766:
6762:
6758:
6753:
6750:
6744:
6741:
6735:
6732:
6729:
6711:
6710:
6694:
6690:
6686:
6681:
6678:
6675:
6672:
6666:
6662:
6654:
6650:
6646:
6641:
6638:
6635:
6629:
6625:
6622:
6618:
6612:
6609:
6604:
6601:
6597:
6591:
6588:
6583:
6579:
6573:
6570:
6565:
6562:
6558:
6552:
6549:
6544:
6541:
6536:
6532:
6528:
6523:
6519:
6515:
6512:
6509:
6495:
6494:
6483:
6480:
6472:
6468:
6464:
6459:
6456:
6453:
6447:
6442:
6439:
6434:
6429:
6426:
6421:
6415:
6412:
6408:
6403:
6400:
6395:
6391:
6387:
6382:
6378:
6374:
6371:
6368:
6354:
6353:
6339:
6336:
6332:
6327:
6324:
6321:
6318:
6315:
6312:
6309:
6306:
6301:
6297:
6291:
6287:
6283:
6279:
6264:
6263:
6248:
6243:
6239:
6235:
6231:
6227:
6222:
6218:
6214:
6210:
6206:
6203:
6198:
6194:
6188:
6184:
6180:
6176:
6172:
6169:
6167:
6165:
6160:
6156:
6152:
6147:
6143:
6139:
6136:
6133:
6130:
6129:
6126:
6122:
6116:
6113:
6108:
6105:
6101:
6095:
6092:
6087:
6082:
6077:
6071:
6068:
6063:
6059:
6053:
6047:
6044:
6039:
6036:
6032:
6028:
6023:
6018:
6012:
6009:
6004:
6001:
5997:
5990:
5987:
5982:
5979:
5977:
5975:
5970:
5965:
5959:
5956:
5951:
5948:
5944:
5939:
5936:
5933:
5928:
5924:
5920:
5917:
5914:
5909:
5904:
5898:
5895:
5890:
5887:
5883:
5878:
5875:
5872:
5867:
5863:
5859:
5856:
5853:
5850:
5848:
5846:
5841:
5835:
5829:
5824:
5820:
5816:
5812:
5808:
5803:
5799:
5793:
5786:
5782:
5776:
5771:
5768:
5765:
5761:
5757:
5754:
5752:
5750:
5745:
5741:
5737:
5734:
5731:
5728:
5725:
5720:
5716:
5712:
5709:
5706:
5703:
5702:
5688:
5687:
5676:
5673:
5668:
5664:
5660:
5655:
5651:
5647:
5644:
5641:
5638:
5635:
5632:
5627:
5623:
5619:
5616:
5613:
5610:
5607:
5602:
5598:
5594:
5591:
5588:
5585:
5582:
5577:
5573:
5569:
5564:
5560:
5556:
5553:
5550:
5532:
5525:
5514:
5513:
5499:
5493:
5489:
5485:
5480:
5476:
5467:
5462:
5459:
5456:
5452:
5446:
5443:
5438:
5432:
5429:
5363:
5362:
5348:
5345:
5341:
5336:
5332:
5325:
5322:
5318:
5313:
5310:
5306:
5302:
5298:
5294:
5291:
5288:
5285:
5282:
5279:
5274:
5271:
5265:
5261:
5258:
5255:
5252:
5249:
5246:
5243:
5240:
5237:
5186:
5185:
5170:
5167:
5164:
5160:
5157:
5153:
5150:
5147:
5144:
5141:
5138:
5135:
5132:
5129:
5124:
5121:
5116:
5113:
5111:
5109:
5106:
5103:
5100:
5097:
5096:
5093:
5089:
5086:
5082:
5079:
5076:
5073:
5070:
5067:
5064:
5061:
5058:
5055:
5050:
5047:
5042:
5039:
5037:
5035:
5032:
5029:
5026:
5023:
5022:
4982:
4981:
4970:
4964:
4961:
4957:
4952:
4949:
4946:
4940:
4935:
4931:
4927:
4920:
4916:
4912:
4909:
4906:
4903:
4900:
4897:
4894:
4888:
4885:
4882:
4876:
4873:
4870:
4863:
4859:
4855:
4852:
4849:
4846:
4843:
4840:
4837:
4834:
4828:
4825:
4822:
4819:
4815:
4812:
4807:
4804:
4800:
4797:
4768:
4767:
4756:
4753:
4749:
4746:
4741:
4738:
4734:
4731:
4728:
4725:
4722:
4718:
4715:
4711:
4708:
4705:
4702:
4698:
4695:
4691:
4688:
4685:
4682:
4679:
4676:
4673:
4670:
4667:
4664:
4661:
4572:
4571:
4558:
4548:
4545:
4543:
4540:
4539:
4531:
4528:
4526:
4523:
4522:
4520:
4515:
4512:
4490:
4489:
4476:
4466:
4463:
4461:
4458:
4457:
4449:
4446:
4444:
4441:
4440:
4438:
4433:
4430:
4381:
4375:
4374:
4373:
4358:
4355:
4352:
4345:
4341:
4337:
4334:
4331:
4328:
4325:
4322:
4319:
4316:
4310:
4307:
4289:
4288:
4277:
4271:
4268:
4265:
4258:
4254:
4250:
4247:
4244:
4241:
4238:
4235:
4232:
4229:
4223:
4220:
4217:
4211:
4208:
4205:
4198:
4194:
4190:
4187:
4184:
4181:
4178:
4175:
4172:
4169:
4166:
4163:
4160:
4157:
4151:
4146:
4141:
4137:
4120:
4119:
4108:
4103:
4099:
4095:
4092:
4089:
4086:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4053:
4049:
4046:
4043:
4040:
4034:
4031:
4023:
4020:
4015:
4011:
4007:
4002:
3998:
3983:
3982:
3971:
3967:
3963:
3960:
3957:
3954:
3950:
3944:
3940:
3933:
3930:
3924:
3920:
3916:
3913:
3910:
3906:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3833:
3830:
3827:
3824:
3821:
3818:
3815:
3812:
3809:
3806:
3673:
3672:
3661:
3658:
3654:
3651:
3648:
3645:
3639:
3636:
3628:
3625:
3620:
3616:
3612:
3607:
3603:
3425:
3422:
3348:
3347:
3336:
3331:
3327:
3324:
3318:
3315:
3301:
3300:
3289:
3284:
3281:
3276:
3271:
3268:
3263:
3258:
3255:
3215:
3214:
3201:
3198:
3193:
3188:
3185:
3180:
3177:
3060:
3059:
3048:
3042:
3039:
3034:
3031:
3028:
3025:
3019:
3016:
2957:
2956:
2945:
2939:
2936:
2931:
2928:
2922:
2919:
2798:
2792:
2767:
2764:
2763:
2762:
2747:
2741:
2738:
2733:
2730:
2724:
2721:
2719:
2717:
2708:
2705:
2698:
2693:
2688:
2685:
2680:
2677:
2675:
2673:
2667:
2664:
2658:
2655:
2649:
2645:
2639:
2636:
2632:
2629:
2626:
2620:
2617:
2611:
2607:
2600:
2595:
2592:
2587:
2584:
2582:
2580:
2574:
2571:
2567:
2561:
2558:
2552:
2548:
2542:
2539:
2535:
2532:
2529:
2526:
2520:
2517:
2511:
2507:
2500:
2497:
2495:
2493:
2490:
2489:
2353:
2346:
2335:
2334:
2319:
2314:
2311:
2306:
2303:
2300:
2295:
2292:
2287:
2284:
2282:
2280:
2277:
2274:
2270:
2267:
2264:
2261:
2256:
2253:
2249:
2243:
2238:
2234:
2228:
2225:
2220:
2217:
2215:
2213:
2210:
2207:
2201:
2197:
2194:
2191:
2188:
2185:
2177:
2172:
2168:
2164:
2161:
2159:
2155:
2151:
2147:
2146:
2062:
2059:
2056:
2043:
1988:
1914:
1905:
1896:
1889:
1870:
1867:
1866:
1865:
1850:
1847:
1843:
1837:
1834:
1829:
1826:
1823:
1820:
1813:
1809:
1805:
1800:
1796:
1789:
1782:
1779:
1775:
1770:
1764:
1761:
1756:
1753:
1747:
1744:
1742:
1740:
1736:
1731:
1727:
1722:
1719:
1711:
1708:
1700:
1697:
1692:
1689:
1685:
1680:
1676:
1672:
1667:
1663:
1660:
1657:
1651:
1648:
1644:
1637:
1634:
1631:
1626:
1623:
1616:
1613:
1610:
1606:
1598:
1595:
1590:
1587:
1582:
1579:
1576:
1572:
1567:
1563:
1560:
1558:
1556:
1553:
1552:
1430:
1429:
1418:
1414:
1406:
1401:
1396:
1393:
1388:
1383:
1380:
1375:
1372:
1368:
1362:
1359:
1354:
1349:
1346:
1341:
1336:
1333:
1328:
1325:
1320:
1317:
1312:
1309:
1295:
1294:
1283:
1280:
1276:
1270:
1267:
1262:
1259:
1256:
1253:
1246:
1242:
1238:
1233:
1229:
1222:
1215:
1212:
1208:
1203:
1197:
1194:
1189:
1186:
1180:
1177:
1095:
1094:
1083:
1080:
1077:
1073:
1070:
1063:
1060:
1056:
1049:
1046:
1043:
1040:
1035:
1032:
1029:
1025:
1018:
1015:
1009:
1006:
1003:
999:
973:
960:
959:
958:
947:
941:
938:
933:
930:
924:
921:
918:
914:
911:
904:
901:
897:
890:
887:
884:
879:
876:
869:
866:
863:
859:
852:
849:
843:
840:
837:
833:
829:
826:
811:
798:
794:
793:
782:
779:
776:
773:
768:
765:
760:
757:
743:
742:
729:
719:
716:
714:
711:
710:
704:
701:
695:
692:
689:
686:
680:
674:
671:
665:
662:
659:
656:
650:
647:
641:
638:
634:
628:
627:
625:
620:
617:
614:
611:
608:
605:
600:
597:
594:
590:
522:
521:
508:
498:
495:
493:
490:
489:
483:
480:
474:
471:
468:
465:
459:
456:
451:
448:
442:
441:
439:
434:
431:
428:
425:
420:
416:
383:between 0 and
343:
342:
329:
319:
316:
314:
311:
310:
304:
301:
295:
292:
289:
286:
280:
277:
272:
269:
263:
262:
260:
255:
252:
249:
246:
241:
237:
204:between 0 and
171:
168:
148:
147:
136:
131:
128:
123:
118:
115:
110:
107:
73:
72:
15:
9:
6:
4:
3:
2:
7720:
7709:
7706:
7704:
7701:
7699:
7696:
7695:
7693:
7676:on 2013-05-17
7675:
7671:
7667:
7666:
7661:
7656:
7653:
7650:
7648:
7644:
7640:
7637:
7635:
7631:
7628:
7626:
7623:
7621:
7618:
7616:
7613:
7611:
7607:
7604:
7602:
7598:
7595:
7594:
7585:
7582:
7578:
7574:
7570:
7566:
7562:
7558:
7554:
7550:
7546:
7541:
7537:
7531:
7527:
7526:
7520:
7516:
7512:
7508:
7504:
7500:
7496:
7491:
7487:
7483:
7479:
7475:
7471:
7467:
7458:
7457:
7445:
7439:
7432:
7426:
7419:
7413:
7406:
7402:
7392:
7390:
7381:
7377:
7369:
7358:
7350:
7343:
7336:
7330:
7312:
7305:
7297:
7293:
7289:
7285:
7281:
7277:
7273:
7269:
7265:
7261:
7257:
7253:
7247:
7240:
7239:
7234:
7233:
7227:
7223:
7213:
7210:
7209:
7203:
7201:
7181:
7174:
7150:
7147:
7144:
7136:
7132:
7129:
7124:
7121:
7112:
7111:
7110:
7087:
7084:
7079:
7075:
7069:
7066:
7061:
7058:
7054:
7048:
7045:
7040:
7035:
7032:
7027:
7019:
7015:
7008:
7005:
6999:
6989:
6985:
6981:
6976:
6964:
6955:
6952:
6945:
6944:
6943:
6924:
6920:
6914:
6909:
6906:
6903:
6899:
6893:
6890:
6885:
6876:
6866:
6865:
6864:
6852:
6828:
6824:
6821:
6817:
6809:
6805:
6801:
6796:
6793:
6790:
6787:
6781:
6774:
6764:
6760:
6756:
6751:
6739:
6730:
6727:
6720:
6719:
6718:
6692:
6688:
6684:
6679:
6676:
6673:
6670:
6664:
6660:
6652:
6648:
6644:
6639:
6636:
6633:
6627:
6623:
6620:
6616:
6610:
6607:
6602:
6599:
6595:
6589:
6586:
6581:
6577:
6571:
6568:
6563:
6560:
6556:
6550:
6547:
6542:
6534:
6530:
6526:
6521:
6517:
6510:
6507:
6500:
6499:
6498:
6481:
6478:
6470:
6466:
6462:
6457:
6454:
6451:
6445:
6440:
6437:
6432:
6427:
6424:
6419:
6413:
6410:
6406:
6401:
6393:
6389:
6385:
6380:
6376:
6369:
6366:
6359:
6358:
6357:
6337:
6334:
6330:
6325:
6319:
6316:
6310:
6307:
6299:
6295:
6289:
6285:
6269:
6268:
6267:
6241:
6237:
6220:
6216:
6204:
6196:
6192:
6186:
6182:
6170:
6168:
6158:
6154:
6150:
6145:
6141:
6134:
6131:
6124:
6120:
6114:
6111:
6106:
6103:
6099:
6093:
6090:
6085:
6080:
6075:
6069:
6066:
6061:
6057:
6051:
6045:
6042:
6037:
6034:
6030:
6026:
6021:
6016:
6010:
6007:
6002:
5999:
5995:
5988:
5985:
5980:
5978:
5968:
5963:
5957:
5954:
5949:
5946:
5942:
5934:
5931:
5926:
5922:
5915:
5912:
5907:
5902:
5896:
5893:
5888:
5885:
5881:
5873:
5870:
5865:
5861:
5854:
5851:
5849:
5839:
5822:
5818:
5806:
5801:
5797:
5784:
5780:
5774:
5769:
5766:
5763:
5759:
5755:
5753:
5743:
5739:
5732:
5729:
5726:
5718:
5714:
5707:
5704:
5693:
5692:
5691:
5674:
5666:
5662:
5658:
5653:
5649:
5642:
5639:
5636:
5633:
5625:
5621:
5614:
5611:
5608:
5600:
5596:
5589:
5586:
5583:
5575:
5571:
5567:
5562:
5558:
5551:
5548:
5541:
5540:
5539:
5535:
5528:
5497:
5491:
5487:
5483:
5478:
5474:
5465:
5460:
5457:
5454:
5450:
5444:
5441:
5436:
5427:
5417:
5416:
5415:
5400:
5385:
5381:
5377:
5373:
5369:
5346:
5343:
5339:
5334:
5330:
5323:
5320:
5316:
5311:
5308:
5304:
5300:
5296:
5289:
5286:
5280:
5277:
5272:
5269:
5263:
5259:
5256:
5253:
5250:
5244:
5241:
5235:
5228:
5227:
5226:
5222:
5218:
5211:
5207:
5200:
5196:
5192:
5168:
5162:
5158:
5155:
5148:
5145:
5139:
5136:
5130:
5127:
5122:
5119:
5114:
5112:
5104:
5098:
5087:
5084:
5080:
5074:
5071:
5065:
5062:
5056:
5053:
5048:
5045:
5040:
5038:
5030:
5024:
5013:
5012:
5011:
5007:
5003:
4996:
4992:
4988:
4968:
4962:
4959:
4955:
4950:
4947:
4944:
4938:
4933:
4929:
4925:
4918:
4914:
4910:
4904:
4901:
4895:
4892:
4886:
4883:
4880:
4874:
4871:
4868:
4861:
4857:
4853:
4847:
4844:
4841:
4835:
4832:
4826:
4823:
4820:
4813:
4810:
4805:
4802:
4795:
4788:
4787:
4786:
4782:
4778:
4774:
4754:
4747:
4744:
4739:
4736:
4729:
4726:
4720:
4716:
4713:
4706:
4703:
4696:
4693:
4689:
4683:
4680:
4677:
4674:
4668:
4665:
4659:
4652:
4651:
4650:
4646:
4642:
4636:
4632:
4622:
4608:
4604:
4600:
4596:
4590:
4586:
4582:
4578:
4546:
4541:
4529:
4524:
4518:
4513:
4510:
4503:
4502:
4501:
4464:
4459:
4447:
4442:
4436:
4431:
4428:
4421:
4420:
4419:
4408:
4405:
4356:
4353:
4350:
4343:
4339:
4335:
4329:
4326:
4323:
4317:
4314:
4308:
4305:
4298:
4297:
4296:
4275:
4269:
4266:
4263:
4256:
4252:
4248:
4242:
4239:
4236:
4230:
4227:
4221:
4218:
4215:
4209:
4206:
4203:
4196:
4192:
4188:
4185:
4182:
4179:
4176:
4173:
4170:
4167:
4164:
4161:
4158:
4155:
4149:
4144:
4139:
4135:
4125:
4124:
4123:
4106:
4101:
4097:
4093:
4090:
4087:
4084:
4081:
4078:
4075:
4072:
4069:
4066:
4063:
4060:
4057:
4054:
4051:
4044:
4038:
4032:
4029:
4021:
4018:
4013:
4009:
4005:
4000:
3996:
3988:
3987:
3986:
3969:
3961:
3958:
3955:
3952:
3942:
3938:
3931:
3928:
3922:
3914:
3911:
3908:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3879:
3876:
3873:
3870:
3867:
3861:
3858:
3855:
3852:
3849:
3846:
3837:
3834:
3831:
3828:
3825:
3822:
3816:
3810:
3804:
3797:
3796:
3795:
3788:
3784:
3780:
3770:
3766:
3762:
3755:
3741:
3733:
3727:
3713:
3698:
3694:
3683:
3679:
3659:
3656:
3649:
3643:
3637:
3634:
3626:
3623:
3618:
3614:
3610:
3605:
3601:
3593:
3592:
3591:
3589:
3588:J.V. Uspensky
3583:
3577:
3574:
3570:
3559:
3546:
3538:
3524:
3513:
3509:
3502:
3498:
3470:
3466:
3459:
3455:
3449:
3445:
3421:
3417:
3413:
3392:
3383:
3363:
3334:
3329:
3325:
3322:
3316:
3313:
3306:
3305:
3304:
3287:
3282:
3279:
3274:
3269:
3266:
3261:
3256:
3253:
3244:
3243:
3242:
3199:
3196:
3191:
3186:
3183:
3178:
3175:
3168:
3167:
3166:
3147:
3114:
3108:
3088:
3065:
3064:approximation
3046:
3040:
3037:
3032:
3029:
3026:
3023:
3017:
3014:
3007:
3006:
3005:
2998:
2990:
2968:
2943:
2937:
2934:
2929:
2926:
2920:
2917:
2910:
2909:
2908:
2884:
2877:
2872:
2868:
2863:
2831:
2823:
2819:
2803:
2791:
2789:
2785:
2745:
2739:
2736:
2731:
2728:
2722:
2720:
2706:
2703:
2696:
2691:
2686:
2683:
2678:
2676:
2665:
2662:
2656:
2653:
2647:
2643:
2637:
2634:
2630:
2627:
2624:
2618:
2615:
2609:
2605:
2598:
2593:
2590:
2585:
2583:
2572:
2569:
2565:
2559:
2556:
2550:
2546:
2540:
2537:
2533:
2530:
2527:
2524:
2518:
2515:
2509:
2505:
2498:
2496:
2491:
2480:
2479:
2478:
2465:
2457:
2446:
2442:
2428:
2424:
2417:
2410:
2402:
2389:
2371:
2363:
2352:
2345:
2341:
2317:
2312:
2309:
2304:
2301:
2298:
2293:
2290:
2285:
2283:
2275:
2272:
2265:
2259:
2254:
2251:
2247:
2241:
2236:
2232:
2226:
2223:
2218:
2216:
2208:
2205:
2199:
2192:
2186:
2183:
2175:
2170:
2166:
2162:
2160:
2153:
2149:
2137:
2136:
2135:
2127:
2123:
2119:
2093:
2074:
2060:
2057:
2054:
2042:
2028:
2017:
2013:
2006:
1998:
1987:
1967:
1949:
1929:
1919:
1913:
1904:
1895:
1888:
1884:
1874:
1848:
1845:
1841:
1835:
1832:
1827:
1824:
1821:
1818:
1811:
1807:
1803:
1798:
1794:
1787:
1780:
1777:
1773:
1768:
1762:
1759:
1754:
1751:
1745:
1743:
1734:
1729:
1725:
1720:
1717:
1709:
1706:
1698:
1695:
1690:
1687:
1683:
1678:
1674:
1670:
1665:
1661:
1658:
1655:
1649:
1646:
1642:
1635:
1632:
1629:
1624:
1621:
1614:
1611:
1608:
1604:
1596:
1593:
1588:
1585:
1580:
1577:
1574:
1570:
1565:
1561:
1559:
1554:
1543:
1542:
1541:
1528:
1520:
1512:
1504:
1493:
1489:
1485:
1475:
1467:
1459:
1452:
1448:
1444:
1434:
1416:
1412:
1404:
1399:
1394:
1391:
1386:
1381:
1378:
1373:
1370:
1366:
1360:
1357:
1352:
1347:
1344:
1339:
1334:
1331:
1326:
1323:
1318:
1315:
1310:
1307:
1300:
1299:
1298:
1281:
1278:
1274:
1268:
1265:
1260:
1257:
1254:
1251:
1244:
1240:
1236:
1231:
1227:
1220:
1213:
1210:
1206:
1201:
1195:
1192:
1187:
1184:
1178:
1175:
1168:
1167:
1166:
1163:
1159:
1153:
1140:
1130:
1116:
1105:
1101:
1081:
1078:
1075:
1071:
1068:
1061:
1058:
1054:
1044:
1038:
1033:
1030:
1027:
1023:
1016:
1013:
1007:
1004:
1001:
997:
989:
988:
987:
984:
980:
970:
966:
945:
939:
936:
931:
928:
922:
919:
916:
912:
909:
902:
899:
895:
888:
885:
882:
877:
874:
867:
864:
861:
857:
850:
847:
841:
838:
835:
831:
827:
824:
817:
816:
815:
808:
804:
797:
780:
777:
774:
771:
766:
763:
758:
755:
748:
747:
746:
717:
712:
702:
699:
693:
690:
687:
684:
678:
672:
669:
663:
660:
657:
654:
648:
639:
636:
632:
623:
618:
612:
609:
606:
595:
592:
588:
580:
579:
578:
576:
564:
559:
557:
543:
535:
528:
496:
491:
481:
478:
472:
469:
466:
463:
457:
449:
446:
437:
432:
426:
414:
406:
405:
404:
391:
377:
364:
356:
349:
317:
312:
302:
299:
293:
290:
287:
284:
278:
270:
267:
258:
253:
247:
239:
235:
227:
226:
225:
212:
199:
194:
183:
167:
165:
159:
153:
134:
129:
126:
121:
116:
113:
108:
105:
98:
97:
96:
82:
78:
70:
66:
62:
58:
54:
50:
49:
48:
46:
42:
38:
21:
7678:. Retrieved
7674:the original
7663:
7610:cut-the-knot
7601:cut-the-knot
7580:
7548:
7544:
7524:
7498:
7494:
7469:
7465:
7454:Bibliography
7438:
7433:, 1937, 255.
7425:
7412:
7404:
7395:Lee Badger,
7379:
7375:
7357:
7348:
7342:
7329:
7317:. Retrieved
7304:
7263:
7259:
7246:
7236:
7230:
7226:
7176:
7169:
7165:
7105:Solving for
7104:
6941:
6850:
6843:
6712:
6496:
6355:
6265:
5689:
5530:
5523:
5515:
5398:
5383:
5379:
5375:
5371:
5367:
5364:
5220:
5216:
5209:
5205:
5198:
5194:
5190:
5188:Solving for
5187:
5005:
5001:
4994:
4990:
4986:
4983:
4780:
4776:
4772:
4769:
4644:
4640:
4637:
4630:
4620:
4606:
4602:
4598:
4594:
4588:
4584:
4580:
4576:
4573:
4491:
4409:
4403:
4383:
4290:
4121:
3984:
3786:
3782:
3778:
3768:
3764:
3760:
3753:
3739:
3731:
3725:
3711:
3696:
3692:
3681:
3677:
3674:
3581:
3578:
3572:
3568:
3560:
3544:
3536:
3522:
3511:
3507:
3500:
3496:
3468:
3464:
3457:
3453:
3447:
3443:
3427:
3418:
3414:
3390:
3381:
3361:
3349:
3302:
3233:, he needed
3216:
3145:
3115:
3061:
2996:
2988:
2969:
2958:
2902:
2882:
2875:
2829:
2821:
2817:
2769:
2477:is given by
2463:
2455:
2444:
2440:
2426:
2422:
2418:
2408:
2400:
2387:
2369:
2361:
2350:
2343:
2339:
2336:
2125:
2121:
2117:
2091:
2075:
2040:
2037:
2026:
2004:
1996:
1985:
1983:, we obtain
1965:
1947:
1927:
1920:
1911:
1902:
1893:
1886:
1882:
1875:
1872:
1526:
1518:
1510:
1502:
1491:
1487:
1483:
1473:
1465:
1457:
1450:
1446:
1442:
1435:
1431:
1296:
1161:
1157:
1154:
1138:
1128:
1114:
1103:
1099:
1096:
982:
978:
975:
968:
964:
813:
806:
802:
795:
744:
560:
541:
533:
526:
523:
389:
378:
362:
354:
347:
344:
210:
196:The uniform
195:
184:
173:
149:
74:
40:
34:
7670:Brady Haran
7665:Numberphile
3420:described.
2794:Estimating
2124:) = arccos(
1495:, that is,
69:probability
7692:Categories
7680:2013-04-09
7634:slideshare
7551:(9): 010.
7382:: 140–143.
7218:References
7184:. Because
3776:, we have
3241:such that
2871:Matplotlib
2841:2 × 9 × 17
2416:as above.
722:elsewhere.
501:elsewhere.
322:elsewhere.
59:strips of
7647:animation
7296:124429237
7280:0003-9519
7148:≈
7141:⟹
7080:≈
7070:π
7062:−
7049:π
7041:⋅
7009:
6968:^
6956:
6900:∑
6880:^
6822:≈
6806:π
6794:−
6791:π
6743:^
6731:
6689:π
6677:−
6674:π
6649:π
6637:−
6634:π
6611:π
6603:−
6590:π
6572:π
6564:−
6551:π
6511:
6467:π
6455:−
6452:π
6441:π
6433:⋅
6428:π
6420:−
6414:π
6370:
6356:yielding
6338:π
6205:−
6135:
6115:π
6107:−
6094:π
6070:π
6062:−
6046:π
6038:−
6011:π
6003:−
5989:π
5958:π
5950:−
5897:π
5889:−
5807:−
5760:∑
5733:
5708:
5643:
5615:
5590:
5552:
5451:∑
5431:^
5347:π
5324:π
5312:−
5301:−
5278:−
5273:π
5257:−
5123:π
5049:π
4963:π
4951:−
4939:π
4911:−
4887:−
4869:π
4854:−
4827:−
4727:−
4704:−
4681:−
4351:π
4336:−
4264:π
4249:−
4222:−
4204:π
4177:−
4165:−
4156:π
4140:∗
4082:−
4070:−
4061:π
4055:φ
4045:φ
4030:π
4019:π
4014:−
4010:∫
4001:∗
3962:φ
3956:
3915:φ
3912:
3895:−
3892:φ
3889:
3877:−
3862:φ
3859:
3850:−
3838:φ
3835:
3826:−
3811:φ
3660:φ
3650:φ
3635:π
3624:π
3619:−
3615:∫
3606:∗
3275:⋅
3192:⋅
3179:≈
3176:π
3030:⋅
3018:≈
3015:π
2918:π
2740:π
2704:π
2692:⋅
2666:θ
2654:π
2644:∫
2638:θ
2631:θ
2628:
2616:π
2606:∫
2599:⋅
2573:θ
2557:π
2547:∫
2541:θ
2534:θ
2531:
2516:π
2506:∫
2313:π
2299:⋅
2294:π
2260:
2252:−
2233:∫
2227:π
2200:π
2187:θ
2167:∫
1828:
1804:−
1781:π
1769:−
1763:π
1730:θ
1721:π
1707:π
1691:
1684:∫
1666:θ
1650:π
1636:θ
1633:
1605:∫
1589:
1575:θ
1571:∫
1460:≤ arcsin
1382:−
1374:−
1353:⋅
1348:π
1327:
1319:π
1261:
1237:−
1214:π
1202:−
1196:π
1079:θ
1062:π
1045:θ
1024:∫
1014:π
1002:θ
998:∫
940:π
920:θ
903:π
889:θ
886:
858:∫
848:π
836:θ
832:∫
778:θ
775:
759:≤
700:π
694:≤
691:θ
688:≤
664:≤
658:≤
640:π
613:θ
599:Θ
479:π
473:≤
470:θ
467:≤
450:π
427:θ
419:Θ
294:≤
288:≤
200:(PDF) of
122:⋅
117:π
7645:package
7573:32470555
7319:14 March
7288:41134124
7206:See also
7202:method.
7182:) < 0
6853:< 200
5159:′
5088:′
4814:′
4806:′
4748:′
4740:′
4717:′
4697:′
4492:and let
3579:Now let
2867:Python 3
2852:≈ 3.1 ≈
1900:, where
976:Suppose
561:The two
170:Solution
57:parallel
55:made of
7553:Bibcode
7515:2317945
7486:2690682
7374:].
5538:where
5403:
5388:
4635:drops.
4625:
4611:
4500:-axis)
4418:-axis)
3750:
3736:
3722:
3708:
3555:
3541:
3533:
3519:
3401:
3386:
3372:
3357:
3231:
3219:
3150:
3135:
3130:
3118:
3103:
3091:
3079:
3067:
3001:
2985:
2850:
2838:
2834:
2813:
2474:
2460:
2430:(where
2413:
2396:
2392:
2378:
2374:
2358:
2009:
1993:
1976:
1962:
1958:
1944:
1938:
1924:
1537:
1523:
1515:
1499:
1497:arcsin
1478:
1462:
1149:
1135:
1125:
1111:
556:radians
552:
538:
400:
386:
373:
359:
221:
207:
7571:
7532:
7513:
7484:
7294:
7286:
7278:
6497:Thus,
3675:where
3461:. Let
2847:9 × 11
1825:arcsin
1688:arcsin
1586:arcsin
1324:arccos
1258:arcsin
1097:where
682:
652:
524:Here,
461:
345:Here,
282:
65:needle
7569:S2CID
7511:JSTOR
7482:JSTOR
7314:(PDF)
7292:S2CID
7284:JSTOR
7133:0.000
7122:0.217
7085:0.217
6825:0.000
5386:′) =
4633:= 100
3456:<
3446:<
2885:= 2.6
2878:= 5.0
1517:<
1490:>
1160:>
981:>
967:>
53:floor
7549:2009
7530:ISBN
7321:2015
7276:ISSN
7191:and
7168:Cov(
7151:222.
6942:and
6829:976.
6717:is
6479:<
5522:Var(
5378:) =
5203:and
4999:and
4609:) =
4601:) =
4410:Let
3785:sin
3767:cos
3752:cos
3734:− 2(
3724:cos
3506:0 ≤
3495:0 ≤
3433:and
3237:and
3107:Milü
3081:for
2454:0 ≤
2443:cos
2425:cos
2080:and
2076:Let
1486:sin
1456:0 ≤
1445:sin
1132:and
1127:sin
569:and
185:Let
61:wood
23:The
7632:at
7608:at
7599:at
7561:doi
7503:doi
7474:doi
7464:".
7268:doi
7137:976
7109:,
7006:Var
6953:Var
6775:100
6761:200
6728:Var
6508:Var
6367:Cov
6132:Cov
5730:Var
5705:Var
5640:Cov
5612:Var
5587:Var
5549:Var
5466:100
5445:100
4587:= 2
3953:sin
3909:sin
3886:cos
3856:sin
3832:cos
3573:πab
3398:213
3389:113
3369:213
3360:113
3330:213
3323:113
3257:113
3254:355
3228:113
3222:355
3165:as
3100:113
3094:355
3076:113
3070:355
2790:".
2625:cos
2528:cos
2248:cos
1630:sin
1297:or
883:sin
772:sin
529:= 0
403:is
350:= 0
224:is
35:In
7694::
7668:.
7662:.
7567:.
7559:.
7547:.
7509:.
7499:76
7497:.
7480:.
7470:67
7468:.
7403:,
7388:^
7290:.
7282:.
7274:.
7264:55
7262:.
7258:.
6861:q̂
6715:p̂
5529:+
5518:p̂
5412:p̂
5384:AB
5221:AB
5212:′)
5210:AB
5008:′)
5006:AB
4783:′)
4645:AB
4583:=
4579:=
3781:±
3763:−
3590:,
3576:.
3571:=
3558:.
3539:≤
3535:≤
3515:,
3510:≤
3504:,
3499:≤
3489:,
3485:,
3451:,
3384:=
2967:.
2880:,
2865:A
2836:=
2830:th
2820:·
2458:≤
2409:tπ
2394:=
2376:·
2356:=
2349:·
2342:=
2115::
2073:.
2012:.
1991:=
1960:+
1892:·
1885:=
1521:≤
1449:≤
1152:.
805:≤
565:,
536:=
357:=
47::
39:,
7683:.
7643:R
7575:.
7563::
7555::
7538:.
7517:.
7505::
7488:.
7476::
7462:π
7401:'
7399:π
7380:4
7372:π
7368:"
7366:π
7337:.
7323:.
7298:.
7270::
7195:n
7193:y
7188:n
7186:x
7179:n
7177:y
7175:,
7172:n
7170:x
7145:M
7130:=
7125:M
7107:M
7088:M
7076:)
7067:1
7059:1
7055:(
7046:1
7036:M
7033:1
7028:=
7025:)
7020:m
7016:x
7012:(
7003:)
7000:M
6997:(
6990:2
6986:M
6982:1
6977:=
6974:)
6965:q
6959:(
6925:m
6921:x
6915:M
6910:1
6907:=
6904:m
6894:M
6891:1
6886:=
6877:q
6857:M
6851:M
6846:M
6818:)
6810:2
6802:2
6797:8
6788:5
6782:(
6778:)
6772:(
6765:2
6757:1
6752:=
6749:)
6740:p
6734:(
6693:2
6685:2
6680:8
6671:5
6665:=
6661:)
6653:2
6645:4
6640:4
6628:(
6624:2
6621:+
6617:)
6608:1
6600:1
6596:(
6587:1
6582:+
6578:)
6569:1
6561:1
6557:(
6548:1
6543:=
6540:)
6535:n
6531:y
6527:+
6522:n
6518:x
6514:(
6482:0
6471:2
6463:4
6458:4
6446:=
6438:1
6425:1
6411:4
6407:1
6402:=
6399:)
6394:n
6390:y
6386:,
6381:n
6377:x
6373:(
6335:4
6331:1
6326:=
6323:)
6320:B
6317:A
6314:(
6311:P
6308:=
6305:)
6300:n
6296:y
6290:n
6286:x
6282:(
6278:E
6247:)
6242:n
6238:y
6234:(
6230:E
6226:)
6221:n
6217:x
6213:(
6209:E
6202:)
6197:n
6193:y
6187:n
6183:x
6179:(
6175:E
6171:=
6164:)
6159:n
6155:y
6151:,
6146:n
6142:x
6138:(
6125:.
6121:)
6112:1
6104:1
6100:(
6091:1
6086:=
6081:2
6076:)
6067:1
6058:(
6052:)
6043:1
6035:1
6031:(
6027:+
6022:2
6017:)
6008:1
6000:1
5996:(
5986:1
5981:=
5969:2
5964:)
5955:1
5947:0
5943:(
5938:)
5935:0
5932:=
5927:i
5923:x
5919:(
5916:P
5913:+
5908:2
5903:)
5894:1
5886:1
5882:(
5877:)
5874:1
5871:=
5866:i
5862:x
5858:(
5855:P
5852:=
5840:2
5834:)
5828:)
5823:i
5819:x
5815:(
5811:E
5802:i
5798:x
5792:(
5785:i
5781:p
5775:2
5770:1
5767:=
5764:i
5756:=
5749:)
5744:n
5740:y
5736:(
5727:=
5724:)
5719:n
5715:x
5711:(
5675:.
5672:)
5667:n
5663:y
5659:,
5654:n
5650:x
5646:(
5637:2
5634:+
5631:)
5626:n
5622:y
5618:(
5609:+
5606:)
5601:n
5597:x
5593:(
5584:=
5581:)
5576:n
5572:y
5568:+
5563:n
5559:x
5555:(
5536:)
5533:n
5531:y
5526:n
5524:x
5512:.
5498:2
5492:n
5488:y
5484:+
5479:n
5475:x
5461:1
5458:=
5455:n
5442:1
5437:=
5428:p
5408:π
5399:π
5397:4
5394:/
5391:3
5382:(
5380:P
5376:B
5374:′
5372:A
5370:(
5368:P
5344:4
5340:1
5335:=
5331:)
5321:4
5317:7
5309:1
5305:(
5297:)
5293:)
5290:B
5287:A
5284:(
5281:P
5270:1
5264:(
5260:2
5254:1
5251:=
5248:)
5245:B
5242:A
5239:(
5236:P
5223:)
5219:(
5217:P
5208:(
5206:P
5201:)
5199:B
5197:′
5195:A
5193:(
5191:P
5169:.
5166:)
5163:B
5156:A
5152:(
5149:P
5146:+
5143:)
5140:B
5137:A
5134:(
5131:P
5128:=
5120:1
5115:=
5108:)
5105:B
5102:(
5099:P
5092:)
5085:B
5081:A
5078:(
5075:P
5072:+
5069:)
5066:B
5063:A
5060:(
5057:P
5054:=
5046:1
5041:=
5034:)
5031:A
5028:(
5025:P
5004:(
5002:P
4997:)
4995:B
4993:′
4991:A
4989:(
4987:P
4969:.
4960:4
4956:7
4948:1
4945:=
4934:2
4930:l
4926:4
4919:2
4915:l
4908:)
4905:l
4902:4
4899:(
4896:l
4893:2
4884:1
4881:=
4875:b
4872:a
4862:2
4858:l
4851:)
4848:b
4845:+
4842:a
4839:(
4836:l
4833:2
4824:1
4821:=
4818:)
4811:B
4803:A
4799:(
4796:P
4781:B
4779:′
4777:A
4775:(
4773:P
4755:.
4752:)
4745:B
4737:A
4733:(
4730:P
4724:)
4721:B
4714:A
4710:(
4707:P
4701:)
4694:B
4690:A
4687:(
4684:P
4678:1
4675:=
4672:)
4669:B
4666:A
4663:(
4660:P
4647:)
4643:(
4641:P
4631:N
4621:π
4617:/
4614:1
4607:B
4605:(
4603:P
4599:A
4597:(
4595:P
4589:l
4585:t
4581:b
4577:a
4547::
4542:0
4530::
4525:1
4519:{
4514:=
4511:y
4498:y
4494:B
4465::
4460:0
4448::
4443:1
4437:{
4432:=
4429:x
4416:x
4412:A
4404:N
4402:2
4398:N
4394:π
4390:π
4386:π
4379:π
4372:.
4357:b
4354:a
4344:2
4340:l
4333:)
4330:b
4327:+
4324:a
4321:(
4318:l
4315:2
4309:=
4306:P
4293:P
4276:.
4270:b
4267:a
4257:2
4253:l
4246:)
4243:b
4240:+
4237:a
4234:(
4231:l
4228:2
4219:1
4216:=
4210:b
4207:a
4197:2
4193:l
4189:+
4186:l
4183:a
4180:2
4174:l
4171:b
4168:2
4162:b
4159:a
4150:=
4145:V
4136:V
4107:.
4102:2
4098:l
4094:+
4091:l
4088:a
4085:2
4079:l
4076:b
4073:2
4067:b
4064:a
4058:=
4052:d
4048:)
4042:(
4039:F
4033:2
4022:2
4006:=
3997:V
3970:.
3966:|
3959:2
3949:|
3943:2
3939:l
3932:2
3929:1
3923:+
3919:|
3905:|
3901:l
3898:a
3883:l
3880:b
3874:b
3871:a
3868:=
3865:)
3853:l
3847:b
3844:(
3841:)
3829:l
3823:a
3820:(
3817:=
3814:)
3808:(
3805:F
3792:φ
3787:φ
3783:l
3779:b
3774:b
3769:φ
3765:l
3761:a
3756:)
3754:φ
3747:2
3744:/
3740:l
3732:a
3726:φ
3719:2
3716:/
3712:l
3703:a
3699:)
3697:φ
3695:(
3693:F
3688:φ
3684:)
3682:φ
3680:(
3678:F
3657:d
3653:)
3647:(
3644:F
3638:2
3627:2
3611:=
3602:V
3584:*
3582:V
3569:V
3564:V
3552:2
3549:/
3545:π
3537:φ
3530:2
3527:/
3523:π
3517:−
3512:b
3508:y
3501:a
3497:x
3491:φ
3487:y
3483:x
3479:x
3475:φ
3471:)
3469:y
3467:,
3465:x
3463:(
3458:b
3454:l
3448:a
3444:l
3439:l
3435:b
3431:a
3406:π
3395:/
3391:n
3382:x
3377:n
3366:/
3362:n
3352:n
3335:.
3326:n
3317:=
3314:x
3288:,
3283:x
3280:n
3270:3
3267:5
3262:=
3239:x
3235:n
3225:/
3200:x
3197:n
3187:3
3184:5
3163:π
3159:x
3155:n
3146:π
3144:3
3141:/
3138:5
3127:6
3124:/
3121:5
3111:π
3097:/
3083:π
3073:/
3047:.
3041:h
3038:t
3033:n
3027:l
3024:2
2997:n
2993:/
2989:h
2980:P
2976:h
2972:n
2965:π
2961:P
2944:.
2938:P
2935:t
2930:l
2927:2
2921:=
2905:P
2897:x
2893:y
2891:(
2889:π
2883:l
2876:t
2857:.
2854:π
2844:/
2826:/
2822:n
2818:l
2816:2
2807:π
2796:π
2780:t
2776:t
2772:p
2746:.
2737:t
2732:l
2729:2
2723:=
2707:2
2697:1
2687:t
2684:l
2679:=
2663:d
2657:2
2648:0
2635:d
2619:2
2610:0
2594:t
2591:l
2586:=
2570:d
2566:t
2560:2
2551:0
2538:d
2525:l
2519:2
2510:0
2499:=
2492:P
2471:2
2468:/
2464:π
2456:θ
2450:t
2445:θ
2441:l
2436:t
2432:θ
2427:θ
2423:l
2405:/
2401:l
2399:2
2388:π
2384:/
2381:2
2370:t
2366:/
2362:l
2354:2
2351:P
2347:1
2344:P
2340:P
2318:.
2310:2
2305:=
2302:1
2291:2
2286:=
2276:x
2273:d
2269:)
2266:x
2263:(
2255:1
2242:1
2237:0
2224:2
2219:=
2209:x
2206:d
2196:)
2193:x
2190:(
2184:2
2176:1
2171:0
2163:=
2154:2
2150:P
2132:x
2128:)
2126:x
2122:x
2120:(
2118:θ
2113:x
2109:θ
2105:x
2101:x
2097:π
2092:θ
2090:2
2086:x
2082:θ
2078:x
2061:2
2058:=
2055:l
2044:2
2041:P
2032:x
2027:θ
2025:2
2021:x
2005:t
2001:/
1997:l
1989:1
1986:P
1981:t
1973:2
1970:/
1966:l
1955:2
1952:/
1948:l
1935:2
1932:/
1928:l
1915:2
1912:P
1906:1
1903:P
1897:2
1894:P
1890:1
1887:P
1883:P
1878:P
1849:1
1846:+
1842:)
1836:l
1833:t
1822:t
1819:+
1812:2
1808:t
1799:2
1795:l
1788:(
1778:t
1774:2
1760:t
1755:l
1752:2
1746:=
1735:)
1726:d
1718:2
1710:2
1699:l
1696:t
1679:(
1675:+
1671:)
1662:d
1659:x
1656:d
1647:t
1643:4
1625:2
1622:l
1615:0
1612:=
1609:x
1597:l
1594:t
1581:0
1578:=
1566:(
1562:=
1555:P
1534:2
1531:/
1527:π
1519:θ
1511:l
1507:/
1503:t
1492:t
1488:θ
1484:l
1474:l
1470:/
1466:t
1458:θ
1451:t
1447:θ
1443:l
1438:θ
1417:.
1413:)
1405:2
1400:)
1395:l
1392:t
1387:(
1379:1
1371:1
1367:(
1361:t
1358:l
1345:2
1340:+
1335:l
1332:t
1316:2
1311:=
1308:P
1282:1
1279:+
1275:)
1269:l
1266:t
1255:t
1252:+
1245:2
1241:t
1232:2
1228:l
1221:(
1211:t
1207:2
1193:t
1188:l
1185:2
1179:=
1176:P
1162:t
1158:l
1146:2
1143:/
1139:t
1129:θ
1122:2
1119:/
1115:l
1106:)
1104:θ
1102:(
1100:m
1082:,
1076:d
1072:x
1069:d
1059:t
1055:4
1048:)
1042:(
1039:m
1034:0
1031:=
1028:x
1017:2
1008:0
1005:=
983:t
979:l
972:)
969:t
965:l
946:.
937:t
932:l
929:2
923:=
917:d
913:x
910:d
900:t
896:4
878:2
875:l
868:0
865:=
862:x
851:2
842:0
839:=
828:=
825:P
810:)
807:t
803:l
781:.
767:2
764:l
756:x
718::
713:0
703:2
685:0
679:,
673:2
670:t
661:x
655:0
649::
637:t
633:4
624:{
619:=
616:)
610:,
607:x
604:(
596:,
593:X
589:f
571:θ
567:x
549:2
546:/
542:π
534:θ
527:θ
497::
492:0
482:2
464:0
458::
447:2
438:{
433:=
430:)
424:(
415:f
397:2
394:/
390:π
381:θ
370:2
367:/
363:t
355:x
348:x
318::
313:0
303:2
300:t
291:x
285:0
279::
271:t
268:2
259:{
254:=
251:)
248:x
245:(
240:X
236:f
218:2
215:/
211:t
202:x
191:θ
187:x
180:t
176:l
163:π
157:π
135:.
130:t
127:l
114:2
109:=
106:p
93:t
89:l
85:p
29:b
25:a
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