One day in A.D. 1777, French scientist D Buffon (D Buffon 1707 ~ 1788) had a full house. It turned out that they came to watch a strange experiment at the invitation of their master. The experiment began, but Mr. Buffon, who is old and rare, took out a piece of paper with great interest and drew parallel lines with equal distance in advance on the paper. Then he grabbed a small needle that had been prepared, and the length of these small needles was half of the distance between parallel lines. Then Mr. Buffon announced, "Please throw these small needles on the paper one by one! However, please be sure to tell me whether the falling needle intersects with the parallel lines on the paper. " The guests didn't know what Mr. Buffon was going to play, so they had to follow their ideas and join the experiment one by one. After throwing a small needle, I picked it up and threw it again, while Mr. Buffon himself kept counting and remembering, so he was busy for nearly an hour. Finally, Mr. Buffon announced loudly: "Gentlemen, I have recorded the result of your throwing the needle just now. * * * Throwing the needle 22 times12 times, of which 704 times crossed the parallel line. The ratio of the total number 22 12 to the number of intersections 704 is 3. 142. " Speaking of this, Mr. Buffon deliberately stopped and gave everyone a mysterious smile. Then he deliberately raised his voice and said, "gentlemen, this is the approximate value of pi!" " The crowd was in an uproar, and at that time there were many doubts. Everyone is puzzled: "Pi π? This has nothing to do with the circle! " Mr. Buffon seems to have guessed everyone's thoughts and proudly explained, "Ladies and gentlemen, the principle of probability is used here. If you are patient, you can get a more accurate approximation of π by increasing the number of injections. However, if you want to know the truth, you have to invite everyone to see my new work. " With Mr. Buffon promoting his own book "Probability Arithmetic Experiment". π appears in this chaotic situation, which is really unexpected, but it is a true fact. Because the problem of needle throwing test was first put forward by Mr. Buffon, it is called Buffon problem in the history of mathematics. Buffon's general result is: If the distance between two parallel lines on paper is d, the length of small needle is l, and the number of stitches is n, then the number of stitches crossing parallel lines is m, so when n is quite large, there is π≈(2ln)/(dm). In the above story, the length of the needle is L, and the most amazing one is the Italian mathematician Lazzerini. In 190 1, he claimed that he had carried out many injection tests, and the number of needles injected each time was 3408, with an average intersection of 2 169. Substituting Buffon's formula, π≈3. 14 15929 is obtained. This is different from the exact value of π to the seventh place after the decimal point! It is a natural creation to find such a high-precision π value in such a light way! If Zu Chongzhi is near-re-embodiment, it will be amazing! However, people have always been critical of Laszlini's results, and the reason cannot be said to be unreasonable, because in mathematics, it can be proved that it is closest to the true value of π, and the score with smaller denominator is: (1) (22)/7≈3. 14 (sparse) (2) (333)/(kloc-). (103993)/(33102) ≈ 3.141592653, and Laslini actually threw a secret rate. There can be no better result for throwing within ten thousand times. No wonder many people questioned: "Is there such a coincidence?" But most people think that Laslini is really "lucky" because of his diligence and prudence all his life. What is the fact, there is no way to investigate now! I think that a beautiful and handsome man who likes to think must also want to know the principle of teacher Buffon's injection test. There's nothing mysterious about it. Here is a simple and ingenious proof. Find a wire and bend it into a circle so that its diameter is exactly equal to the distance d between parallel lines. As you can imagine, for such a circle, no matter how it is dropped, it will have two intersections with the parallel line. Therefore, if the number of circles dropped is n, then the total number of intersection points must be 2n. Now imagine straightening the circle and turning it into a wire with a length of π d. Obviously, it is more complicated for such a wire to intersect with parallel lines when falling than a circle. There may be four intersections, three intersections, two intersections, 1 intersection, or even none. Because the length of a circle and a straight line is πd, according to the principle of equal opportunity, when they throw more and equal times, the total number of times they intersect with parallel lines is expected to be the same. That is to say, when a wire with a length of πd falls n times, the total number of times it crosses parallel lines should be about 2n. Now turning to the case that the length of the iron wire is L, when the throwing times n increase, the total number of intersections between the iron wire and the parallel lines m should be proportional to the length l, so there is: m=kl, where k is the proportional coefficient. In order to find k, it is only necessary to note that for the special case of l=πk, there is m=2n. Then k=(2n)/(πd) is obtained. If you substitute the previous formula, there are: m≈(2ln)/(πd) and π≈(2ln)/(dm). This is the famous Buffon formula! Using Buffon's formula, we can also design a needle-throwing test to find the approximate values of root 2, root 3 and root 5. Dear reader, don't you want to have a try? Just choose l/d equal to your number, but π at this time should be regarded as knowing.