Magic Triangle, 2 Numbers/Side

Date: 04/25/2003 at 20:22:19
From: Leslie
Subject: Magic Triangle/magic sum

I understand that natural numbers can be placed in an arrangement so 
that the sum of 3 numbers on each side of a triangle is always the 
same, but is it possible to find a magic triangle with 2 numbers? If 
so how?
                   2                      3
                3     5                5     7
             6     1    4           8     2     6

I don't think it's possible, since I can't get any sides to match up.

Date: 04/26/2003 at 11:33:03
From: Doctor Samus
Subject: Re: Magic Triangle/magic sum

Hi Leslie,

You're right that a 2-number/side magic triangle where no number is 
used more than once is impossible to construct. Let's see why.

Suppose we have the following magic triangle:

      a
    b   c

Where a, b, and c are all different numbers.

Since we want the numbers on each side to add to the same number, we 
therefore want the following equation to hold:

    a + b = a + c = b + c

This is actually three equations in one:

    (1) a + b = a + c
    (2) a + b = b + c
    (3) a + c = b + c

Subtracting a from both sides of (1), b from both sides of (2), and c 
from both sides of (3), we see that:

    (1) b = c
    (2) a = c
    (3) a = b

And thus

    a = b = c

However, we are assuming that no number is used more than once, so we 
have a contradiction, and thus we are unable to construct a 2-number/
side magic triangle.

We should note that we could make a 2-number/side magic triangle by 
allowing all three numbers to be the same number, but this triangle 
isn't interesting.

I hope this helps you, and please feel free to write back if you have 
any other questions.

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