Sums in a Triangle

Date: 9/2/95 at 18:30:30
From: Anonymous
Subject: Algebra

Six numbered bottle caps are arranged in a triangle. The sum of the three
corner numbers, 1+6+5, is three more than the sum of the remaining numbers.
  
			        1
			       4 3
			      6 2 5

How can you rearrange the bottle caps so that the sum of the corner 
number is: 

 a. Twice the sum of the remaining numbers?

 b. The same as the sum of the remaining numbers?

Date: 9/6/95 at 11:32:48
From: Doctor Ken
Subject: Re: Algebra

Hello!

To do these problems, we can write an equation.  If I were given the first
problem, the one that's already solved, I would use the variable X to represent
the sum of the numbers on the corners, and write down the following equation:

X + (X-3) = 1+2+3+4+5+6
          = 21

The X is the numbers on the corners, and the X-3 is the other numbers 
(which add up to 3 less than X, hence the x-3).
Then I could solve for X:

2X - 3 = 21
    2X = 24
     X = 12

So then all you'd have to do is fiddle around and try to find three numbers
that add up to 12: 1+6+5 does fine, and so does 3+4+5 and a bunch of other
solutions.
How would you write an equation for the next problem?

If the sum of the corner numbers is twice the sum of the remaining numbers,
then the corner numbers are X and the other numbers are X/2, right?

-Doctor Ken,  The Geometry Forum