# Two trains and a bumblebee problem

I was intrigued by this math problem a while ago. It took me a couple of pages of calculations to get the solution. Then I kicked myself because the solution was actually staring in my face.

I can’t find the original problem now, although there seem to be other variations of it. So I’m going to give you my (highly embellished) version. Then see if you can get the answer. Here goes the story of Blitz the bumblebee and the fateful encounter of two trains…

Blitz the bumblebee and the meeting of two caterpillars
Blitz the bumblebee was an adventurous kind of guy. He’d wander further out in the fields than his fellow bumblebees. Now he’d seen these huge black caterpillars (trains) spewing out lots of grey and black clouds passing his fields before. Now Blitz, despite his big size, could fly very fast. But these caterpillars zoomed past him, leaving him in a whirlwind spiral as he tried to get back into control.

So one sunny morning, he decided to test his limits. He waited for one of these caterpillars, and when he heard their telltale “choo choo” from a distance, he started flying as fast as he could in the direction where the caterpillar would move. When the caterpillar moved close to his level, Blitz simply flew to the head of the caterpillar and hung on.

What a thrill! The wind was howling around him and the scenery was speeding past him. Then Blitz wondered where the caterpillar was going. So, using the caterpillar’s speed as initial momentum, Blitz took off.

To his surprise, he flew even faster than the caterpillar! So Blitz shot straight ahead. After some time, Blitz was starting to worry where he’s going, when he heard the “choo choo” coming from in front of him. Before he realised it, he bounced off another caterpillar. Somehow Blitz survived the bounce and flew, out of instinct, back in the direction where he came from.

This is fun!” Blitz thought. And wondered if he’d meet the first caterpillar again. Sure enough, the first caterpillar appeared, and Blitz bounced off it, and flew towards the second caterpillar. Through this bouncing and flying, it never occurred to Blitz what would happen when the two caterpillars meet.

When the two caterpillars were within sight of each other, Blitz realised that his bounces were getting more frequent. By the time this revelation came, he found he couldn’t stop nor get out of the bouncing cycle anymore.

They’re gonna crash! I’m too young to die!

And Blitz promptly fainted, while the two caterpillars amazingly brushed past each other and continued on their merry way…

The end.

The real math problem
Alright, maybe the original problem wasn’t phrased like that, so I’ll give you the short version.

There are two trains running at 45km/h in opposite directions towards each other on two separate (straight) tracks. Assume for the discussion that the difference caused by the separate tracks is negligible. A bumblebee is positioned at the head of the first train. When the trains are 90km apart, the bumblebee flies at 60km/h towards the second train. When the bumblebee meets the second train, it returns and flies back towards the first train. It continues to fly back and forth in this manner until the two trains meet. What is the distance covered by the bumblebee?

I wrote down tons of calculations and numbers. I scrutinised my work and finally found a relationship between the numbers. Then I used mathematical induction to convince myself that the relationship can be written in the form of a geometric progression. The ratio of the geometric progression turned out to be less than 1, so that simplified the equation further. And the answer was … 60km.

Immediately after I got this answer, I found that since the time taken for the two trains to meet is 1 hour (time = distance / speed = 90km / (45km/h + 45km/h)), it also means the bumblebee only flew for 1 hour. With the speed of 60km/h, flying for 1 hour means a distance of 60km covered. I spent about an hour or so and a couple of pieces of paper to figure this out…

Moral of the story? Sometimes, it’s better to think through a problem instead of jumping on the first solution that pops up in your head.

1. Jack Bremer - 3B Web Design - Mac OSX and Windows tech support as well as leading edge web development

From reading the initial “story” I had assumed he’d actually got stuck between one carriage and the next, and was just bouncing between them!

Once I read the mathematics behind it at the bottom, I understood the problem!

The mathematical explanation is sound, but doesn’t seem in keeping with the description in the story – the bee using momentum to fly off faster, and “bounce” off the other train etc. If he is just flying for the amount of time until the trains meet, with bouncing time considered negligible, then fine but that’s far less exciting!

2. Vincent Tan

You know, I forgot about the momentum and bouncing part… ðŸ™‚ I was intent on writing an exciting story instead…