Written Numbers

Date: 05/12/97 at 15:22:58
From: Anonymous
Subject: A big number?

What is the biggest number that can be made using letters only once to 
make the number? For example, EIGHTY-FOUR. Can anyone find a bigger 
number?

What is the smallest number that can be made?

Grant Whitaker

Date: 06/25/97 at 15:19:37
From: Doctor Sydney
Subject: Re: A big number?

Dear Grant,

This is a hard question!  We are not sure what the biggest and 
smallest numbers are that are of the type you describe.  That would be 
a hard problem to solve.  However, we do have some ideas for you that 
might help you think about the solution to the problem.

First, think about which numbers from 0 to 9 cannot be used at all 
because the word representing them contains duplicate letters. For 
instance, we know that none of our numbers will have a 3 in it when 
the 3 is positioned such that the 3 has to be said when the number is 
read. This is because the word three has two e's in it. That means 
that any number with a 3 in the ones place will not work since when we 
say all numbers with a three in the ones place we end by saying the 
word "three" (e.g. 33 = thirty-three).  

The one exception is numbers that end in 13 - in these cases we say 
"thirteen," but that too violates the rules for valid numbers because 
we can't have two e's in our word. So, any number that meets the 
qualifications specified above will not have a 3 in the ones place.  
Can a valid number have a 3 in the tens place? Hundreds place?  Well, 
when it is in the tens place, we say thirty-something, right?  So, it 
is possible that we could have a 3 in the tens place -- "thirty-one" 
works, for example.  I'll let you think on your own about whether or 
not 3 could be in the hundreds place, thousands place, and so on. Be 
careful! This can be tricky.  

Continue with bigger numbers. Are there other single-digit numbers 
that have problems similar to 3's problems?

Now, continue by systematically going through words that are used when 
we say numbers and eliminate ones that are bad on their own, like 
three, million, trillion, and so on. Then work with the words that are 
still okay, and try to construct the biggest and smallest numbers with 
what you have remaining. This could turn into a pretty big job, but 
you might enjoy it. 

Just so you know, we did find a number bigger than eighty-four that 
works:

FIVE THOUSAND

A small number we found is: MINUS TWO.

However, we do not know whether or not these two numbers are the 
biggest and smallest possible. That is something you should work on!  

Good luck with this, and thanks for writing!  It was sure fun to think 
about. If have any questions about what we said above or if you have 
questions about something else, please do write back to us! 

-Doctors Barney and Sydney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/