76 of 100: Daredevil Fly

The two train will collide after 300/(100+100)=3/2 hours.

Therefore the fly move at this time with constant speed 150 km / hr .

Therefore total distance = 150×3/2=225 km

Instead of counting the number of times the bee flies in the other direction, we take the information that the fly flies at a constant speed. Because there are two trains, they will cover the 300 km in 1.5 hours. The fly may fly back and forth a lot of times, but it flies for 1.5 hours overall. Because the fly flies at 150 km per hour for 1.5 hours, he flies a distance of $150\text{ km} \cdot 1.5 = \boxed{225 \text{ km}}$ .

$d=vt\\ d=v(\frac { 300km }{ 100km/h+100km/h } )\\ \\ d=v(1.5h)\\ \\ d=(150km/h)(1.5h)\\ \\ d=225km$

Relevant wiki: Geometric Progression Sum

Consider the fly's first trip, when it reaches the second train:

$\begin{aligned} 150t & = 300 - 100t \tag{1} \\ t_1 & = \frac 65 \\ \implies d_1 & = 150t_1 = 180 \end{aligned}$

It then turns around and reaches the first train again when

$\begin{aligned} 180 - 150t & = 120 + 100t \tag{2} \\ t_2 & = \frac{6}{25} \\ \implies d_2 & = 150t_2 = 36 \end{aligned}$

Continuing, the total distance traveled by the fly is given by summing the series

$S_d = 150\sum_{n \ = \ 1}^{\infty} \frac{6}{5^n} = \boxed{225}$

By using brute force, we gained an intuition that the fly's trip can be modeled by a geometric series - the same intuition that tells us that an elastic bouncy ball with a constant height ratio will also follow a geometric series.

Fermi supposedly heard this problem at a cocktail party. He thought for a couple of seconds and replied with the right answer. The puzzle poser replied, "Very nice Professor! Usually men of your intellect try to solve this using infinite series." Fermi, surprised, "You mean there is another way?"

The fly is constantly flying until the two trains collide. To find the distance the fly travels, one would just multiply its speed by the time until the trains collide. The two trains both travel at the constant speed of 100 kmph, and are 300 km apart, so each train would have to travel 150 km to collide. That takes 150/100 = 3/2 hours. Multiplying that by the fly's speed, 3/2 * 150 = 225 kilometers.

The relative speed of the trains is 200 km/h, so the ratio of the distance travelled by the trains (total) to the distance travelled by the fly is 200 : 150 = 4:3 = 300 : 225. Hence the fly travelled 225 km.

Considering each lap traversed by the fly, with each respective time interval decreasing the remaining inter-train distance by 200km/hr * time interval elapsed by the fly:

$\Delta x_n = 300 - 200(\sum_{k=0}^{n-1} \Delta t_k)$ $= 300 - \frac{200}{150}(\sum_{k=0}^{n-1} \Delta x_k)$

The distance between the trains is monotone function of time and achieves its low bound value of 0 after some finite time (composed of the union of many time intervals). $\Delta x_n$ represents the remaining distance between the trains after $n$ time intervals. The fly will need to travel an infinite number of reversals because at each reversal there will always be an vanishingly small residue of distance to cover.

Therefore:

$\lim_{n \to \infty} \Delta x_n = 0$

$\lim_{n \to \infty} \sum_{k=0}^{n-1} \Delta x_k \triangleq S$

Taking limits of both sides:

$0 = 300 - \frac{200}{150} S_n$ $\implies S = \frac{300\cdot 150}{200} = 225$

The problem can also be done as the sum of an infinite series of distances.

The first distance the fly travel before turning around is 180 km. The second is 36 km. The third is 36/5 km. This continues with a common ratio of 1/5. We can sum the series with first term 180 km and r=1/5.

Total distance = 180 km /(1-1/5)

Total distance = 180 km/ (4/5)

Total distance = 180 km (5/4)

Total distance = 225 km

More complicated than the other way, but interesting.

This is a classic - first time I remember seeing it I was a grad student reading about Von Neumann. As the story goes, when the question was posed to him he immediately gave the right answer but when someone commented that it was interesting that he saw the shortcut rather than attempting to sum the infinite series, Von Neumann replied that he did sum the series.

The trains will crash in the km 150.

It'll take 1,5h at 100km/h for each train to get there.

The fly will fly for 1,5h travelling 1,5*150= 225km

Look at the informations given. At first if you wish to start from bee, stop, look from a different view, notice that bee's speed is constant.
So, at first, count the time needed for collition- as both the trains are at same speed of 100km/hr, and the distant is 300km, so they will collide aftergoing 300/2 or 150 km. At speed of 100 km/hr, it will take150/100 or 3/2 hour.

At speed of 150 km/hr, in 3/2 hour the bee will fly- (150*3/2) km = 225 km

CAN'T THE RAIL DRIVERS NOTICE THE BEE AND STOP THE ACCIDENT???

let the two train be A and B vA=100km/hr vB=-100km/hr.. relative velocity,vAB=vA-vB =100-(-100) =200km/h :d=300Km time taken to collide=200/300 =2/3h :distance travel by robo fly=150/2/3 =150*3/2 =225Km

relative velocity of the two train is 200 Km so the trains will collide after duration 90 min (1 and half hour). In this duration the distance covered by the bee is 150 km /hr *1.5 hr which comes out 225 Km.

What Saswata said! :)

The distance between the two trains is shrinking at a rate of 200km/h, so the initial distance (300km) will be zero after one hour and a half. So basically the fly will keep on flying for the same period of time until it crushes. Which is to say: d = 1.5 hours x 150 km/h = 225 km.

an object traveling at a constant rate of speed will,in an hour and a half, travel a distance equal to one and half times the speed at which it is traveling.

The trains will crash in an hour and a half, in one hour the fly goes 150 km per hour, half of 150 km = 75 km. So 150 km + 75 km = 225 km. the answer is 225 km.

The two trains reach other after 1/2(300/100) = 1.5 hours. The fly travels the distance between the two trains in 300/150 = 2 hours. Since 2 > 1.5, the fly can not travel the total distance between the trains but it will travel only 150(1.5) = 225 km before it gets crushed by the trains.

The trains collide when the sum of their distances equals the 300 miles. So, add up their distances and solve for t (time). Then calculate the distance the fly flew in this amount of time. t=1.5 hours. the fly's rate (150) x time (1.5)=225.

The trains will collide after each has travelled... 300 miles / 2 = 150 miles

This will take the trains... 150miles / 100 mph = 1.5 hours

During this time, the super-fast fly will travel back and forth a distance of 150mph x 1.5 hours = 225 miles

The trains will collide in 1.5 hours, during which the fly traveled at 150 x1.5 hr. = 225. Ed Gray

The two trains will meet each other in 3 hours (300 km, and each is traveling at 100 km/hour),

so the fly is flying for:

$distance= rate \times time \implies d= \frac{3}{2}\times 150 = 225.$

for a distance of 225 km.

The two trains will collide in the middle, because they have the same speed, so each train will travel for 150km.

At 100km/h will take 1.5 h.

At 150km/h in 1.5h the fly will travel for 225km.

The puzzle is SO OLD

That I knew the answer immediately

Even John von Neumann knows that puzzle

Relative speed of two trains $= 100 +100 = 200$ , as they are moving in opposite directions. Since the distance between the trains is 300 km, so collision will occur after $\dfrac{300}{200} = 1.5$ hrs. And during this entire time this super-fast fly keep on flying at the speed of 150 km/hr. So total distance = $150 \times 1.5 = \boxed{225}$

From the Wikipedia excerpt on John von Neumann :

Halmos recounts a story told by Nicholas Metropolis, concerning the speed of von Neumann's calculations, when somebody asked von Neumann to solve the famous fly puzzle:[169]

Two bicyclists start 20 miles apart and head toward each other, each going at a steady rate of 10 mph. At the same time a fly that travels at a steady 15 mph starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner till he is crushed between the two front wheels. Question: what total distance did the fly cover? The slow way to find the answer is to calculate what distance the fly covers on the first, northbound, leg of the trip, then on the second, southbound, leg, then on the third, etc., etc., and, finally, to sum the infinite series so obtained. The quick way is to observe that the bicycles meet exactly one hour after their start, so that the fly had just an hour for his travels; the answer must therefore be 15 miles. When the question was put to von Neumann, he solved it in an instant, and thereby disappointed the questioner: "Oh, you must have heard the trick before!" "What trick?" asked von Neumann, "All I did was sum the geometric series."

Since the mosquito is traveling for the same amount of time as the trains, we just multiply $150$ km/h by $\frac{300}{100-(-100)}$ . We therefore get $150 \cdot 1.5 =225$ km.