March 18, 2002
Problem of the Month

The Train and Bee Problem

Two trains, 150 miles part, are approaching each other on the same track, each traveling 75 mph.

A bee, perched on the front of train A, begins to fly at a speed of 137.5 mph toward train B; on reaching train B, it reverses direction, always flying at the same speed of 137.5 mph, until it once more reaches train A, whereupon it again reverses direction and flies toward train B, and so on.

How far does the bee fly before it and the two trains collide?

Solution to the Problem:

Because the trains are 150 miles apart and are approaching each other at a relative velocity of 150 mph, they will collide at the end of one hour.
Since the bee is travelling at 137.5 mph for one hour, it must travel 137.5 miles.

According to mathematical folklore, mathematician John von Neumann was enjoying himself at a cocktail party, when another guest proposed a similar problem to him. Von Neumann solved the problem instantaneously by summing an infinite series in his head!

John von Neumann would have solved this week's problem in the following manner:

1. The bee and the opposite train covered a distance of 150 miles at a combined rate of 212.5 mph, so the time that it took for the bee to meet that train was 150 / 212.5 hours or .70588 hours.
   (Time = Distance / Rate)
2. So the bee travels (137.5) x (150/212.5) miles or approximately 97.058823 miles when it reaches the train and turns around.
   (Distance = Rate x Time)
3. Meanwhile, the first train has traveled a distance of 75 x (150/212.5) miles or about 52.94 miles.
   (Distance = Rate x Time)
4. Therefore, the distance between the bee and the first train is
   (137.5)x(150/212.5) - 75x(150/212.5) or approximately 97.0588 - 52.94 = 44.11 miles.
   In factored form, the distance would be
   (150/212.5) x (137.5 - 75) miles.
5. When the bee is traveling back toward the first train, that train and the bee are travelling at a combined rate of 212.5 mph, so the time it will take for them to meet will be
   (150x(137.5 - 75)) / (212.5 x 212.5) hours or
   approximately .20761 hours.
   (Time = Distance / Rate)
6. So the bee travels a distance of
   137.5 x (150x(137.5 - 75)) / (212.5 x 212.5)
   miles or approximately 28.5467 miles.
   (Distance = Rate x Time)
7. Continuing in the same manner, you would get the following series:
   97.05882352941177
   28.546712802768166
   8.396092000814166
   2.469438823768872
   0.7263055364026094
   0.21361927541253217
   0.06282919865074475
   0.01847917607374846
   0.005435051786396606
   0.001598544643057825
   0.0004701601891346544
   0.000138282408569016
   0.00004067129663794589
   ...
   --------------------------------
   137.5 miles

   I don't know how John von Neumann was able to keep all that in his head!

8. Another way to solve this problem is to use the formula for the sum of an infinite geometric series:

   Sum = a1 / (1-r)

   where a1 is the first term of an infinite geometric series, and r is the ratio between any two terms of the series (as long as r is less than 1).

9. Use the information above to determine what the infinte series of successive distances travelled by the bee would look like.
   From step (2), the first term, or a1, equals
   (137.5) x (150/212.5) miles or approximately 97.05882355.
   From step (6), the second term is
   (137.5) x (150/212.5) x (137.5 - 75)/212.5 or
   approximately 28.5467128
10. Therefore, the ratio must be (137.5 - 75)/212.5
    or approximately .2941176471
11. Substituting the values of a1 and r into the formula for the sum of an infinite series, we get:
    Sum = (137.5 x (150/212.5)) /
    (1 - (137.5 - 75)/212.5)
    which equals 137.5 miles!!

    Using the approximations, we get the same answer:
    Sum = 97.05882355 / (1 - .2941176471) = 137.5!

    An interesting twist to this problem would have been asking the same question but have the bee travel at 37.5 mph instead of 137.5 mph! Then it would travel 0 miles since the train would squash it at the beginning since the train is travelling twice as fast as the bee!!

    Rich Murray sent in the following correction to the paragraph above:
    The bee started "perched" on Train A. Even if it was squashed, it would still travel 75 miles. To travel 0 miles requires Train A to travel faster than the speed of light, such that we mere humans didn't see it coming. (The train, that is!).
    
    

Correctly solved by:

1. Richard K. Johnson	La Jolla, California
2. Walt Arrison	Philadelphia, Pennsylvania
3. Rick Jones	Kennett Square, Pennsylvania
4. Renata Sommerville	Austin, Texas
5. Keith Mealy	Cincinnati, Ohio
6. David & Judy Dixon	Bennettsville, South Carolina
7. Rich Murray	Ridgetown, Ontario, Canada
8. John Beasley	Winchester, Virginia
9. Laurence O'Neill	Winchester, Virginia
10. James Alarie	University of Michigan -- Flint, Michigan
11. John Funk	Ventura, California
12. David Powell	Winchester, Virginia
13. George Gaither	Winchester, Virginia
14. Tori Eads	Winchester, Virginia
15. Justin Collins	Winchester, Virginia

Send any comments or questions to: David Pleacher