Magic Triangle Sums

Date: 02/04/2003 at 10:44:56
From: Keri
Subject: Consecutive Numbers

There is a triangle with spots for 6 numbers.  Each side must add up 
to the same sum using each number only once.

   Find 6 consecutive odd numbers such that their magic triangle sum 
   is 25.

   Find 6 consecutive numbers such their magic triangle sum is 4.

I'm certain this is simple but I just had a brain lock.

Date: 02/04/2003 at 12:00:35
From: Doctor Kastner
Subject: Re: Consecutive Numbers

Hi Keri -

The following answer from the Dr. Math archive will tell you some of 
what you want to know: 

   Magic Triangle Puzzle
   http://mathforum.org/library/drmath/view/57520.html 

But some intuition and trial and error will help with these problems. 

For the first one, I looked at the required sum of 25 and thought that 
the numbers 7,9,11 would probably be in there, since they are about 
1/3 of the sum. Note that you can't use 1, since if that were your 
first odd number, your set would have to be 1,3,5,7,9,11 and the 
largest sum you can make with 1 is 1+9+11 = 21, which is too small. On 
the flip side, you can't use 15, since that set would be 
5,7,9,11,13,15 and the smallest sum you can make with 15 is 
15+7+5 = 27, which is too big. Therefore, I used 3, 5, and 13, and the 
set that works is 3,5,7,9,11,13.

For the second problem, it was clear that I have to have negative 
numbers if I only want a sum of 4, so I tried starting with -2. That 
gives me the set -2,-1,0,1,2,3, but the largest sum I can get from 
using the number -2 is -2+2+3 = 3, so that can't work. Starting with 
the number -1 looked better as it would result in the set -1,0,1,2,3,4 
and it can cancel out the 5 that results from 4+1 or 3+2.

I hope this helps.  Let me know if you'd like to talk about this some 
more, or if you have any other questions.

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