PROBLEM # 1

There are two trains hurtling towards each other on a straight piece of track. They are 100 miles apart at the time of this puzzle. A dragonfly starts off from the headlamp of train A and flies towards train B at a speed of 225 miles per hour; when it reaches train B it immediately reverses course and flies towards train A. It keeps doing this (i.e. fly back and forth) till the two trains collide and it passes away to the netherworld.

The trains are each traveling at a speed of 50 miles per hour. What is the total distance traveled by the dragonfly?

Lifestyle hint: If you are summing infinite series to get the answer, you are working too hard.

PROBLEM # 2

What is the greatest number of pieces of pizza you can get if you cut a circular pizza with 5 straight cuts?

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Please send your solutions to both problems to anil@lokvani.com.

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If you need clarification on any problem, please contact rama@alum.mit.edu or anil@lokvani.com.

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SOLUTIONS TO PROBLEMS ON 3/27/03

Problem # 1: "The Itinerant Monk"

The legendary Zen monk Nue-Toan was very fond of visiting hilltop shrines.

On a lovely Fall day in 1729 BC, exactly at sunrise, he started on the path that would take him up the hill to the shrine of Lord Kor-Tezan. He stopped occasionally for a drink of pure spring water and to take deep breaths of the clean mountain air. He reached the hilltop exactly at sunset.

He spent the evening and night at the shrine, meditating and sleeping (not that an observer could easily tell the difference). The next morning, exactly at sunrise, he left the shrine and started down the hill on the same path. Again, he stopped occasionally, sometimes to drink and eat, and other times to simply take in the spectacular scenery. He reached the base of the hill at sunset.

Here's the question: prove that somewhere along the path from base to hilltop, there is a place that Nue-Toan passed by AT EXACTLY the same time on both days. You don't have to identify the place or the time - simply convince me that such a place and time exist.

Solution

Imagine that Nue-Toan has a twin brother Ol-Toan. Consider the path from hilltop to base that Nue-Toan follows on the SECOND day. Imagine that Ol-Toan traverses this path but on the FIRST day itself i.e. when Nue-Toan is making his way UP the hill.

Obviously, they will meet. The meeting point and meeting time are the ones we are looking for. Puzzle solved.

Extra credit:
There were three "puns" in the statement of the puzzle:
1. Nue-Toan is, of course, the Zen equivalent of Newton.
2. Kor-Tezan is, of course, the Zen equivalent of Cartesian.
3. The number 1729 occupies a special place in mathematics thanks to the brilliant Indian mathematician Srinivasa Ramanujan. According to math folklore, Ramanujan was in hospital in the UK being treated for tuberculosis when he was visited by his famous collaborator G.H. Hardy .....

".....in a taxi from London, Hardy noticed its number, 1729. He must have thought about it a little because he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his disappointment with it. It was, he declared, "rather a dull number," adding that he hoped that wasn't a bad omen. "No, Hardy," said Ramanujan, "it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways."" (cited in http://www.jimloy.com/number/hardy.htm)

Problem # 2

What is the largest number you can write using 3 digits. You may use a digit more than once.

Solution

9^(9^9)

About the Winner(s):

Sudheer Apte is a high-tech software professional by day who occasionally writes book reviews and commentary on news and the media.

Smita Narayan is a software engineer that develops CAD software for civil engineers. She has a Masters degrees in Civil Engineering and Computer Engineering from Syracuse University. She is from IIT Roorkee and grew up in Roorkee.

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