Write a Sum That Totals 100 Using Digits 1 - 9

Date: 05/30/2004 at 00:31:40
From: Claire
Subject: Digits 1-9 totaling 100

Is it possible to arrange the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 so 
that when added they total 100?  Only adding is permitted and the 
numbers can be rearranged.  Each number can only be used once.

I've been trying for ages to figure this one out and am having a great 
deal of trouble--I keep getting 99 and 108.  Can it be done?

Date: 05/30/2004 at 21:02:58
From: Doctor Vogler
Subject: Re: Digits 1-9 totaling 100

Hi Claire,

Thanks for writing to Dr Math.  I gather that you have to use every
digit.  If you don't have to use all of them, then it's pretty easy. 
So I will assume that you have to use them all.  In that case, the
answer to your question is no, it is not possible.  I will try to
explain why.

First of all, you clearly can't use numbers with three digits or more,
because even one such number is already too much.  So every number in
the sum has to be either a one-digit number or a two-digit number.

Second, if you take a digit used on the tens place, and you move it to
the tens place of a different number, then that doesn't change 
anything.  For example,

  12+3+4+57+6+8+9=99
  2+13+4+57+6+8+9=99
  2+13+4+7+6+58+9=99
  52+3+4+17+6+8+9=99
  etc.

So really the only question we have to answer is:  Which digits go in
the tens place, and which digits go in the ones place?

Now let's look at the sum like this:

  1*(sum of digits in ones place) + 10*(sum of digits in tens place)

Now if we split up the second part into two pieces, according to

  10 = 9 + 1,

then we can look at the sum as

  1*(sum of ALL digits) + 9*(sum of digits in tens place).

Well, I expect you can figure out that the sum of all of the digits is

  1+2+3+4+5+6+7+8+9 = 45,

which means that you want

  100 = 1*45 + 9*(sum of digits in tens place)

but that requires

  55/9 = sum of digits in tens place,

and 55/9 is not an integer.

Does all of that make sense?  If you have any questions about this or
need more help, please write back and show me what you have been able
to do, and I will try to offer further suggestions.

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

Date: 06/10/2004 at 11:03:30
From: Doctor Douglas
Subject: Re: Digits 1-9 totalling 100

Hi Claire.

Dr. Vogler's answer explains why it is impossible to obtain 100 if you 
are restricted to using only whole numbers.  However, there is a trick 
that allows you to get a solution; you need to use a decimal point and 
a vinculum to get a repeating decimal.  Check out these pages: 

  Recurring Decimal Notation
    http://mathforum.org/library/drmath/view/62850.html 

  0.9999... = 1
    http://mathforum.org/dr.math/faq/faq.0.9999.html 

Once you understand the notation, I don't think you'll have much 
difficulty finding a solution.

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