Million Point Word Puzzle

Date: 05/18/2005 at 11:13:04
From: Stephanie
Subject: (no subject)

If a = 1, b = 2, ... , z = 26, is there a word that when the values of
the letters are multiplied will make a product of one million?  For
example, BAD = (2)(1)(4) = 8.

It is hard for me to think logically of words and numbers together.

Date: 05/18/2005 at 11:22:27
From: Doctor Ian
Subject: Re: 

Hi Stephanie, 

This is a variation on the "dollar word" problem that we hear about
every so often:

  Words Equal to a Dollar
    http://mathforum.org/library/drmath/view/56834.html 

With a slight modification to the program there (basically, it has to
multiply 1 by the scores for the letters, rather than adding them to
zero; and the target score is a million instead of 100), I asked it to
look at every word in the dictionary, and found NO words for which the
product is a million. 

There are lots of words where the letters multiply to MORE than a
million.  But if you're looking for exactly a million, I think someone
has sent you on a wild goose chase.  

In fact, let's stop and think about this for a minute.  Suppose we DID
find some letters whose product is a million.  Since a million is a
power of 10, these could ONLY be letters whose prime factors are 2 or
5 or both:

  2, 4, 8, 16, 5, 10, 20, 25
  b  d  h   p  e   j   t   y

So those are the only letters we could possilby use.  Any OTHER letter
would introduce a prime factor that isn't 2 or 5. 

I think you can tell just by looking at those that there's very little
chance that you'd ever be able to find a "million point" word.  And
perhaps that's the point of the question--for you to figure out WHY
you shouldn't expect to find one. 

On the other hand, if we change things so that a=2, b=3, and so on, it
turns out that there _is_ a word that works:

    http://dictionary.reference.com/search?q=ixodid 

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/