Among the various jokes about how top mathematicians are awesome, this must be indispensable:
Once, when von Neumann was at a party, the hostess bravely posed him a riddle:
Two trains are traveling toward each other on the same track at 30 miles per hour and 1 mile apart when a fly sitting in front of one train is flying towards the other train at 60 miles per hour. When it flew to another train, it flew back quickly. It kept flying back and forth like this until the two trains inevitably collided. How many miles did the fly fly?
Almost as soon as the hostess had finished explaining the question, von Neumann replied, "1 mile."
"I'm amazed you figured it out so quickly," she said. "Most mathematicians failed to see the trick, but used infinite series to calculate, which took them a long time." "What trick? I also use infinite series to calculate." Von No Iman replied.
For a long time, I always thought it was a quack joke, but I didn't expect it to be true. Thanks to the author ZHANG Zaikun for his research: In the 1966 American documentary about von Neumann, Eugene Wigner (Eugene Wigner, theoretical physicist and mathematician, collaborator of von Neumann, Winner of the 1963 Nobel Prize in Physics) told the story in an interview. According to Wigner, the questioner to von Neumann was Max Born (Max Born, one of the founders of quantum mechanics and winner of the Nobel Prize in Physics in 1954); When Von Neumann gave the answer, "Born was astounded and he said: 'You are the first one of my scientist friends who saw the solution at once.' Johnny said: 'I can't understand that. It's a simple infinite geometrical series.'"
In the " Biography of Von Neumann " written by Norman Macrae , it is mentioned that when others mentioned this joke, Neumann would add: the figures actually put to me were not so simple. unfriendly. However, it is probably nothing to Neumann, who was able to mentally calculate 9-digit numbers and four arithmetic operations since childhood.
In terms of difficulty alone, this question is entry-level even if it is placed in the elementary school Olympiad. It takes 1/60 of a minute for two trains to meet. The fly flies back and forth during this time. It doesn’t matter how many times it flies back and forth. 60 minutes = 1 mile.
But the big guy directly uses infinite series to do mental calculations, how is this possible? It's not complicated at all, and you can do it too. For the convenience of calculation, we simplify the problem:
The distance of the train is 100 kilometers, the speed is 20 kilometers per hour, and the departure points of the two trains are named A and B respectively. The speed of the fly is 30 km/h, and it flies from point A to point B.
Every time the fly meets a train in the opposite direction, it turns around. We can calculate the distance traveled each time, and then add them all up. According to the problem conditions, the flying speed of the fly and the train relative to each other is 50 km/h, and the calculation steps are as follows:
- The train departing from point B is 100 kilometers away, meets after 2 hours, flies 60 kilometers, and the turning point is 60 kilometers away from point A. The train departing from point A has traveled 40 kilometers and is 20 kilometers away from the fly. The train departing from point B also travels 40 kilometers, and the distance from point A is 60 kilometers.
- The train departing from point A is 20 kilometers away, meets after 2/5 hours, and flies 12 kilometers. The turning point is 48 kilometers away from point A. The train departing from point B travels 8 kilometers. It was 60 kilometers away from point A before, and now it is 52 kilometers away.
- The train from point B is 4 km away, meets after 4/50 hours, and flies 12/5 km. The turning point is 48 + 5/12 kilometers from point A. The train departing from point A travels 8/5 km, the location is 48+8/5 km from point A, and the distance from the fly is 4/5 km.
- ...
After several calculations, we found that the distance traveled by the flies in each round is 1/5 of the previous round, which obviously constitutes a geometric sequence: the initial value is 60, and the proportional coefficient is 1/5, so we can apply the geometric sequence Summation formula: Sn = a * (1 - q ^ n) / (1 - q), where a is the initial value, q is the proportional coefficient, and n represents the first n items of the sequence. Because q < 1, when n is infinite, we can consider q ^ n to be 0, so the calculated result is that the fly has flown a total of 75 kilometers. It is consistent with the result obtained by our calculation with primary school mathematics method.
Some students may ask, how many times will the two trains meet, and the answer is after infinite times. So we take infinity for n when summing. This problem involves a famous paradox of the ancient Greek mathematician and philosopher Zeno : When a person goes from point A to point B, he must first complete 1/2 of the distance, and then complete the remaining 1/2 of the total distance. , and then finish the remaining 1/2... If this cycle continues, it will never reach the end. A similar question was also raised in the ancient Chinese classic "Zhuangzi · Tianxia Pian": If you take a one-foot stick, take half of it every day, and it will last forever. The first time is 60 kilometers, the second time is 12 kilometers, the third time is 12/5 kilometers, and the fourth time is 12/25 kilometers. We can write a small program to accumulate the first 20 items and try:
a = 60
q = 0.2
s = 0
for i in range(20):
s += a
a *= q
print(s)The result is 74.99999999999923, which is very, very close to 75, but always a little bit off, and the more loops the closer it gets. But why do the infinity terms add up to 75? The solution to this problem was not solved until Newton Leibniz invented calculus, but when calculus was first invented, the theory was not complete, and it was mostly used to solve engineering problems. Interested students can learn about the Berkeley paradox that detonated the second mathematical crisis . After the continuous efforts of a series of mathematicians such as Cauchy and Cantor, the theoretical system of calculus was completed and the second problem was solved. Math crisis. We can also safely use the geometric sequence summation formula. When the proportionality coefficient q < 1, the summation of infinite items is directly simplified to S = a / (1 - q).
Smart students can deduce the more general state of this problem: the speed of the train is v1, the speed of the fly is v2, and the initial distance is k. We can derive:
- a = v2 * k / (v1 + v2)
- q = (v2 - v1) / (v1 + v2)
Suppose the initial distance is 80, the speed of the train is 15, and the speed of the fly is 25. Applying the above formula:
- a = 25 * 80 / (15 + 25) = 50
- q = (25 - 15) / (25 + 15) = 1/4
Using the summation formula S = 50 / (1 - 1/4) = 50 * 4 / 3 = 200/3. Memorize this formula and practice it a few more times. When someone asks you this question, you can pretend to be B like Neumann. Not only can it report the result directly, but it can also tell how far the fly flies each time it turns back.
If there are smarter students, you can calculate a more common state: the speed of the train is v1, v2, the speed of the fly is v3, and the initial distance is k.