Magic Triangle, 2 Numbers/SideDate: 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/ |
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