Digits that Average

Date: 7/19/96 at 8:13:7
From: Anonymous
Subject: Counting No. with Digits that Average...

How many three-digit numbers from 100 to 999 inclusive have one digit 
that is the average of the other two?

(A) 121 (B) 117 (112) (D) 115 (E) 105

Date: 7/22/96 at 22:10:35
From: Doctor Erich
Subject: Re: Counting No. with Digits that Average...

There are lots of questions similar to this one, counting how many 
times a certain type of number occurs... in this case, how many times 
between 100 and 999 does one of the digits equal the average of the 
other two?  

In many types of math problems, including ones like this, one of the 
easier ways to deal with it is to simplify the problem into the less 
complex cases.  In your question, we could figure out the number of 
three digit numbers from 100 to 999 that have a 1 and two other digits 
whose average is 1.  (An example of a number like this would be 201). 

To start this simpler problem, we need to think about what pairs of 
digits have an average of 1.  Well, we know the average of 1 and 1 is 
1 and the average of 0 and 2 is 1 and that's it!  So now we have to 
figure out how many different three digit numbers we can get from 
1,1,1 and 0,1,2.  Let's start with the first one.  Since all the 
digits are the same, we can only make one three digit number 111. 
With the numbers 0,1,2, though we can make several numbers... 012, 
021, 102, 120, 201, 210 are all of them.  Two of these (012 and 021) 
aren't between 100 and 999, so we have to throw them out.  

So what did we find out?  At least for 1, we know the only three digit 
numbers that fit the requirements of the question are 102, 111, 120, 
201 and 210.  So there are five of these numbers for 1. 

There's a start; now if you figure out how many three digit numbers 
there are that fit for 2 to 9 you'll have the answer.  

Just so you can check your work, I figured out how many numbers there
are that have a 2 and two other digits whose average is 2, and how many 
numbers have a 3 and two other digits whose average is 3.  (For 2 
there are 11 and for 3 there are 17.)  

If you want a real challenge, try to figure out a formula for this 
problem, so if anyone walked up to you and asked you how many three 
digit numbers had a 7 and two other digits that averaged to a 7, you 
could put 7 into some formula and get the number of three digit 
numbers that worked. 

Good luck, solving! If you have any other questions, write us again!

-Doctor Erich,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/