# Logic — the Art of Reasoning

Mathematics  — the Art of Studying Patterns Using Logic

Infinite 3x3 Magic Square, The Puzzle

(First Draft - Work in Progress)

[This story is based on my recollections of what Arye Amitai, who I knew when I was growing up in Israel, told me over 35 years ago. While some details may be presented inaccurately, the specific mathematical details, which are the essence of this document, are accurate.]

Sometime in the late 1950's an Israeli periodical that was devoted to mathematics published a contest:

Create a 3-by-3 magic square such that, the sum of the most cell combinations equals 15. In other words, the winning solution would be the one that provides the highest count of unique cell combinations, the sum of the cells within each combination equals 15.

I do not recall the precise text and rules of the contest statement and I don't know anyone who does. So I phrased it here in the simplest form I can. However, as you will see the language of the statement has bearing on Amitai’s proposed solution. The statement above is intentionally ambiguous with respect to the issue that I will discuss below. It is possible that the original statement avoided this ambiguity by using the word number or numbers (in Hebrew).

The standard true 3 x 3 magic square is:

 6 1 8 7 5 3 2 9 4

It uses the numbers 1 through 9 and all the rows, columns and two long diagonals add up to 15. Therefore, this magic square has 8 Combinations, all triplets, that equal 15.

A trivial solution for this contest is the following:

 5 5 5 5 5 5 5 5 5

It uses only the number 5 in every cell. In this magic square any combination of three cells sums up to 15 for the grand total of 84 such sums. Since

there are 84 such all triplets.

Amitai told me his solution. When he found out that he lost the contest and that the winning solution resulted with fewer combinations than his own, he protested to the editor. When his protest was rejected on the ground his solution was not valid, he sued and during the trial a well-known mathematician testified that Amitai's solution was indeed valid.  (I know the name of the mathematician, who, I believe, is now deceased. Since I have never verified his involvement and I have no way to contact his family, I feel obliged not to disclose his identity.)