What is the equation for finding the magic constant of a magic square?

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Discussion Overview

The discussion centers on deriving an equation for the magic constant of a magic square, exploring different approaches and formulas related to this concept. Participants engage in clarifying definitions and attempting to prove relationships between series and the magic constant.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant seeks to convert a series into an equation for the magic constant of a magic square.
  • Another participant clarifies that the magic constant refers to the sum of numbers in a magic square, specifically noting the example of a 3x3 square summing to 15.
  • A participant expresses the intention to derive the magic constant formula from the series provided, indicating a need for proof of correctness.
  • Concerns are raised about the validity of the initial formula proposed, with a participant noting that it yields incorrect results for certain values of n.
  • One participant proposes a different formula for the magic constant, suggesting it is M_{2}(n) = n(n² + 1)/2.
  • Another participant connects the discussion to the sum of integers and references Gauss's method for summing series, indicating a relationship to the magic constant derivation.

Areas of Agreement / Disagreement

Participants express differing views on the validity of the initial formula for the magic constant, with some proposing alternative formulas. The discussion remains unresolved regarding the correct derivation and formula for the magic constant.

Contextual Notes

There are unresolved mathematical steps in deriving the magic constant from the series, and participants have differing assumptions about the definitions and applications of the formulas discussed.

BobFijiwinkle
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Hey all. I'm trying to convert a series which gives me the magic constant for a magic square into an equation. How would I go about doing this?

The series is:
[tex]\mbox{S}=\left[ \frac{n+1}{2}+\left( n-1 \right)n \right][/tex]
 
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What do you mean by the magic constant? The sum - so for a 3 x 3 from 1-9 you want 15?

Simply sum 1 to n2, then divide by n?
 
Yes, that's what I mean by the magic constant.

Yes, I know that, but I'm trying to derive the formula from the series to prove that that is correct.

BF
 
I'm not sure that it is. Plugging 3 into your formula gives 10, to say nothing of what happens if n is even.

What is the sum of 1 + 2 + 3 + ... a, where a is an integer?
 
Let me get something clear.

I'm trying to prove that the magic constant is

[tex]M_{2}\left( n \right)\; =\; \frac{n\left( n^{2}+1 \right)}{2}[/tex]
 
Ahh, that looks a lot more like the formula I've got.

To explain where I'm coming from with my request for the sum of 1 + 2 + 3 + ... a (hint Gauss), if you sum these, then divide by the number of rows / colums you ought to get somewhere. I didn't want to say the sum of 1 to n2, in case it got confused as 1 + 4 + 9 etc.
 

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