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The 6th International Young Naturalists’ Tournament
Team «12FM» Diana Ignatovich
Problem № 1 «Buffon’s needle»
The taskDraw a series of parallel equally spaced lines on a horizontal
surface. Pick a bunch of sticks (e.g. matches or needles)
slightly shorter or longer than the separation between the
lines, and randomly drop them on the surface. It is claimed
that the number of times the sticks cross the lines allows
estimating the constant π to a high precision. What accuracy
can you achieve?
"2
Hypothesis
Aim of the study
If you draw parallel lines on paper and drop matches on it, accuracy π will depend on the distances between the lines.
Get the most accurate possible value of π using the Buffon method.
"3
Objectives
1. Study the literature; 2. Study possible cases of intersection of the needle and
the straight line; 3. Use the Buffon method to calculate the number π; 4. Compare the experimental data with the theoretical
data.
"4
Theory
π — irrational number: that is, it cannot be expressed as a fraction a/b, where a is an integer and b is a non-zero integer.
π — transcendental number: that is, it cannot be the root of any p o l y n o m i a l w i t h i n t e g e r coefficients.
d
Circumference
Diameter
"5
TheoryMN, M1N1 – parallel lines; а – distances between lines; C – the center of the needle; CD (y) – the perpendicular from the center of the needle to the line; A – the end of the needle; х – the angle between the perpendicular and the needle;
"6
Theory
"7
S OMTS ORST
2
2
"8
Experiment 1Distances 45 mm
2,5
2,675
2,85
3,025
3,2
1 18 35 52 69 86 103
120
137
154
171
188
205
222
239
256
273
290
307
324
341
358
375
392
409
426
443
460
477
494
Экспериментальное значение Число пи
Distances 50 mm
2,5
2,75
3
3,25
3,5
1 17 33 49 65 81 97 113
129
145
161
177
193
209
225
241
257
273
289
305
321
337
353
369
385
401
417
433
449
465
481
497
Экспериментальное значение Число пи
Distances 35 mm
2,8
2,95
3,1
3,25
3,4
1 18 35 52 69 86 103
120
137
154
171
188
205
222
239
256
273
290
307
324
341
358
375
392
409
426
443
460
477
494
Экспериментальное значение Число пи
Distances 28 mm
2
2,35
2,7
3,05
3,4
1 16 31 46 61 76 91 106
121
136
151
166
181
196
211
226
241
256
271
286
301
316
331
346
361
376
391
406
421
436
451
466
481
496
Экспериментальное значение Число πThe experimental value of π The value of π
The experimental value of π The value of πThe experimental value of π The value of π
The experimental value of π The value of π"9
Distances 35 mm
2,8
2,95
3,1
3,25
3,4
1 17 33 49 65 81 97 113
129
145
161
177
193
209
225
241
257
273
289
305
321
337
353
369
385
401
417
433
449
465
481
497
Экспериментальное значение Число пиThe experimental value of π The value of π
Experiment 1
m – intersections n – number of the thrown matches
intersections"10
Experiment 2Purpose: to determine the most accurate value of π. Equipment: matches (length - 42 mm); a sheet of paper lined with the parallel straight lines at a distance of 84 mm.
Conclusion: the experimental value of the number π: 3,14159292.
Distances 84 mm
2,5
2,775
3,05
3,325
3,6
1 14 27 40 53 66 79 92 105
118
131
144
157
170
183
196
209
222
235
248
261
274
287
300
313
326
339
352
365
378
391
404
417
430
443
456
469
482
495
Экспериментальное значение Число пиThe experimental value of π The value of π
"11
Experiment 3Purpose: to determine the most accurate value of π. Equipment: 250 matches (length - 42 mm); a sheet of paper lined with the parallel straight lines at a distance of 84 mm.
Conclusion: the possibility to get exact value of the π is slight. The error of the closest result is 0.03.
The
valu
e of
π
2,5
2,775
3,05
3,325
3,6
Number of experience0 3 5 8 10
Экспериментальное значение Число πThe experimental value of π The value of π
"12
Experiment 4Purpose: to calculate the value of π by using the ratio of the circumference to diameter. Equipment: tennis ball, glass, mug, glass jar, box of tennis balls.
Conclusion: To calculate the accuracy of the number π by using this method is small: in only one case out of 5, the found constant contains the correct number in the hundredth digit.
Object Circumference, cm Diameter, cm
The ratio of the circumference to
its diameterA tennis ball 20 6,4 3,125
A glass 17,5 5,5 3,18181818A mug 26,7 8,5 3,14117647
A glass jar 19,0 6,0 3,16666667
A box of tennis balls 23,7 7,5 3,16
"13
• Even if we have a slight difference between the length of the "needle" and the distance between the lines, we do not get even a close value of the number π. But if the distance between the lines is twice larger than the length of the needle, then the possibility of obtaining the most accurate value of the number π increases several times.
• The experimental value of the number π: 3,14159292.
Conclusion
"14
References
1. A. V. Zhukov, the Ubiquitous number π. - M.: editorial URSS, 2004;
2. G. I. Glazer, history of mathematics at school: 9-10 CL. Manual for teachers. - M.: Education, 1983;M. Balk, Math after school. - M.: Education, 1971;
3. O. V. Manturov, Mathematics in concepts, definitions and terms. Part 2. – M: Education, 1982;S. Shumikhin, A. Shumikhina, the Number π. The history of 4000 years. - Moscow: Eksmo, 2011;
4. Universal school encyclopedia. Vol.2. M – I/ Chapters.ed. E. Khlebalin. - M.: AVANTA+, 2004.
5. http://storyof.ru/chisla/chto-oznachaet-chislo-pi/ 6. http://nsportal.ru/ap/library/drugoe/2014/10/15/istoriya-
vozniknoveniya-chisla-pi 7. http://chtooznachaet.ru/chislo_pi.html 8. http://festival.1september.ru/articles/522274/
"15
The 6th International Young Naturalists’ Tournament
Team «12FM» Diana Ignatovich
Problem № 1 «Buffon’s needle»
