Thursday, December 1, 2011

waiting for cab 1729

Update: found a plate!
Korean license plates have a few small Korean characters followed by four large digits.  I have not devoted myself to the study of Korean, so the first few characters do little to distract me.  Mostly I enjoy the numbers.  A few weeks ago, it occurred to me that there may be a taxicab in this city with the plate "1729", which would be supercool.  So now I can't stop looking for it. 

The significance of 1729 as a taxi number is a legend of mathematics history.  G.H. Hardy visited Ramanujan at a hospital and noted that his taxi was number 1729, an (unfortunately) uninteresting number.  Ramanujan replied that 1729 is in fact interesting because it is the smallest number that can be written as the sum of two cubes in two different ways.  1729=13 + 123 = 93 + 103

It would be most satisfactory to see it on a taxi, but I find myself checking other license plates just in case.  I saw 1728 once, which is very close and is 123, which is in fact part of why 1729 is special.  It seems that this would be a disappointing game with occasional near misses.  If I ever found cab 1729 it would surely be anticlimactic at this point. 

Unexpectedly, it is the near misses that have proved to be the most stimulating part of the game.  I was surprised to find myself coming up with arguments in favor of each number being somehow "close" to 1729.  For example 1279, 1698, 3729, and 3458 would all be close to 1729 for different reasons.  What if a license plate has two or three of the right digits?  (2175, 9243).  It's better if they are in the correct places (1754, 1889), or if the wrong digits somehow resemble the correct missing digit (1724, 1759, 1429).  Given the variety, I started to wonder, what is the probability that a given four digit integer--at least a little bit--resembles 1729?  After all, there is about an 87% chance that a license plate contains at least one of the digits {1, 7, 2, 9}.

I can hardly walk around Seoul anymore without getting sucked into this game, which is far from a voluntary obsession.  It is really more like humming the Pororo song that plays on a loop at E-mart.  Kind of catchy.  

When I am in the US, I always try to find acronyms for the letters on license plates.  I'm just throwing that out there in case you need any traffic obsessions of your own.  Mercifully, I do not understand Korean enough for that one to take hold of me.

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