Titanium Bumble Bee - Solution

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Solution

The bee flies 200 miles before it is crushed.

Explanation 1 - the hard  way

In general, let the initial distance from the train to the station be M miles, the train travels at M mph and the bee travels at 2M mph.

Since the bee travels twice as fast as the train, by the time the bee gets to the station the train has travelled ½M miles. By the time the bee gets back to the train, the train has travelled a further 1/3 of the remaining ½M miles = 1/6M miles (since the bee travels twice as fast as the train). So, in total, the train has travelled ½M + 1/6M = 2/3M miles. The bee, in turn has travelled M miles + 2/3(½M) miles = 4/3M miles.

Now the bee is back at the train, and the train is 1/3M miles from the station. The same logic applies again, but with an initial distance of 1/3M miles rather than M miles. Thus, by the time the bee arrives back at the train again, the bee has travelled 4/3(1/3M) miles, and the train is (1/3)²M miles from the station.

The next time, the bee travels 4/3(1/3)²M miles, and the train is (1/3)³M miles from the station.

Adding this infinite series together, the bee travels the following total distance:

   4/3M + 4/3(1/3)M + 4/3(1/3)²M + 4/3(1/3)³M + ...

= 4/3M (1 + 1/3 + (1/3)² + (1/3)³ + ...)

 

	¥
Now, there is a standard theorem that, for r<1,	∑ ri = 1 / (1 - r)
	i=0

 

Using this theorem with r=1/3, we get a total distance of

 

= 4/3M (1/(1 - 1/3))

 

= 4/3M (3/2)

 

= 2M

Since M is 100 miles, the bee travels 200 miles before it is crushed.

Explanation 2 - the easy way

Since the train is 100 miles from the buffers and it is travelling at 100 mph, it takes 1 hour to hit the buffers. The bee travels at 200 mph. Therefore the bee travels 200 miles in that 1 hour.

Notes

There is an apocryphal story about this.

A student put this puzzle to his professor (John von Neumann, Dave Cleal tells me). The professor thought for a few seconds and said "200 miles".

The student, impressed at the professor's quick thinking, said that many people try to work out the puzzle by solving the infinite series of how far the bee flies between turns.

The professor looked at him quizzically and asked "Why, is there another way of doing it?"

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