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George C. Reese,

Office for Mathematics, Science and Technology Education,

University of Illinois Champaign-Urbana,

505 East Armory Avenue,

#341

Champaign, Illinois 61820.

__E-mail__: g-reese@uiuc.edu or mste@uiuc.edu

Added to MAP: October 1999.

Buffon's Needle is one of the oldest problems in the field of geometrical probability. It involves dropping a needle on a lined sheet of paper and determining the probability of the needle crossing one of the lines.

Language: | N/A |

Product form: | Executable file for use on an Apple Macintosh computer (binhex format). |

Complete program.

Buffon's Needle is one of the oldest problems in the field of geometrical probability. It was first stated in 1777. It involves dropping a needle on a lined sheet of paper and determining the probability of the needle crossing one of the lines. The remarkable result is that the probability is directly related to the value of pi.

The program simulates the needle drop in the simplest case scenario in which the length of the needle is the same as the distance between the lines (1 unit). There are two variables, the angle at which the needle falls, theta, and the distance from the center of the needle to the closest line, D. Theta can vary from 0 to 180 degrees and is measured against a line parallel to the lines on the paper. The distance from the centre to the closest line can never be more that half the distance between the lines. The diagram below depicts this situation.

In the graph below, we plot D along the ordinate and (1/2)sine(theta) along the abscissa. The values on or below the curve represent a hit (D <= (1/2)sin(theta)). Thus, the probability of a success is the ratio of the shaded area to the entire rectangle. What is this value equal to?

The area if the shaded portion is found by evaluating the definite integral of (1/2)sin(theta) from zero to pi. The result is that the shaded portion has a value of 1. The value of the entire rectangle is (1/2)(pi) or pi/2. So, the probability of a hit is 1/(pi/2) or 2/pi. That's approximately 0.6366197. To calculate pi from the needle drops, simply take the number of drops and multiply it by two, then divide by the number of hits, or (approximately)

__Buffon's Needle and metallography__

The problem of Buffon's Needles has relevance in metallography where it is used to relate the length of grain boundary line in the surface of a polished specimen to the number of intersections made by boundaries with a test line on the specimen surface [3]:

where L_{A} is the length of boundary per unit area and N_{L} is the average number of intersections with a random test line, per unit length of test line.

__Downloading and running the program__

The executable file has been compressed using STUFFIT software on the Macintosh and can be downloaded in binhex format. The compressed file, `buffon.sea.sit`, can be uncompressed simply by double clicking on the file icon; it is self extracting. To run the program double-click on the executable file, `Buffon's Needles`.

- W. Cheney and D. Kincaid, 1985,
*Numerical Mathematics and Computing*, 2nd Ed., (Pace Grove, California: Brooks/Cole Publishing Company), pp. 354-354. - L. Schroeder, 1974,
*Mathematics Teacher*,**67**, 183-186. -
*Quantitative Microscopy*, 1968, Ed. R.T. DeHoff and F.N. Rhines, (New York: McGraw-Hill), p 78.

**Drop 1/10/100**- Number of needles to be dropped onto the page. Groups of 1, 10 or 100 needles can be dropped at a time.

**Number of needle drops**- The total number of needles dropped onto the page.
**Number of hits**- Number of needles crossing one of the lines.
**Ratio hits:drops**- Ratio of the number of needles crossing one of the line to the number of needles dropped.
**Estimate of pi**- The estimate for pi obtained using the formula above.

None.

No information supplied.

There are two other possibilities for the relationship between the length of the needles and the distance between the lines. A good discussion of these can be found in Schroeder's paper [2]. The situation in which the distance between the lines is greater than the length of the needle is an extension of the above explanation and the probability of a hit is 2L/Kpi where L is the length of the needle and K is the distance between the lines. The situation in which the needle is longer than the distance between the lines leads to a more complicated result.

Further information and an online program are available at http://www.mste.uiuc.edu/reese/buffon/buffon.html.

Complete program.

Total number of needle drops = 19100

Number of needle drops: 19100 Number of Hits 12099 Ratio hits to drops .6334554974 Estimate of PI is 3.157285726

None.

Buffon, needle, Buffon's needle, probability, grain boundary, intersection

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MAP originated from a joint project of the National Physical Laboratory and the University of Cambridge.
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