Logic — the Art of Reasoning

Mathematics  — the Art of Studying Patterns Using Logic

Infinite 3x3 Magic Square, Amitai's Solution

(First Draft - Work in Progress)

What Amitai entered in each and every cell was

+ 15 +

or
+ 15 + .

With his entries, the magic square looks like this:

 
+ 15
+ 15
+ 15
+ 15
+ 15
+ 15
+ 15
+ 15
+ 15
or
 
+ 15
+ 15
+ 15
+ 15
+ 15
+ 15
+ 15
+ 15
+ 15

Each cell entry can be interpreted as:

(± 7.5) + 15 + (± 7.5)

Now you can add up any number of cells, selectively taking the positive or the negative value of the square root, such that the total equals 15.

For a single cell:

(+7.5) + 15 + (–7.5) = 15 For two cells: [(+7.5) + 15 + (–7.5)] + [(–7.5) + 15 + (–7.5)] = 15 For any number of cells n, n > 1: [(+7.5) + 15 + (–7.5)] + [(–7.5) + 15 + (–7.5)]+ …+ [(–7.5) + 15 + (–7.5)] = where the expression [(–7.5) + 15 + (–7.5)] is added n–1 times. [(+7.5) + 15 + (–7.5)] + (n–1) * [(–7.5) + 15 + (–7.5)] =

[(+7.5) + 15 + (–7.5)] + (n–1) * 0 = 15

Interpreting the contest rules strictly, Amitai's entry yields that the separate sums of all single cells, all pairs, all triplets, …, all quintuplets and the single nine-cell combinations, each and every one of them equals 15.

And further, interpreting the rules liberally, since there is no prohibition against using a cell more than once, there are infinitely many cell combinations that sum up to 15.

The editor's objection was that Amitai's entries in the magic squares are not numbers. Amitai argued that they are numbers and so testified a distinguished mathematician.
 

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It turns out that Amitai's solution can be generalized:

  • Any magic shape, such as rectangle, triangle, pentagon, hexagon, etc., of any dimensions can be used.
  • The target sum can be any number, even real one, e.g.,(pi) or   (square root of pi). Posing the original problem serves to restrict the audience's imagination from thinking of the most general solution.
  • Questioning Amitai's Solution

    There is no doubt that, if we use Amitai's scheme, we can generate a magic square of any size having infinitely many combinations the sum of which can be any real number of our choice. The question is:

    Are the entries that Amitai proposed themselves numbers (as required by the original challenge)?
    The point of contention arises from the fact that Amitai inserted an expression that includes radicals, radicals have roots and roots are real numbers. The issue at hand can be boiled down to the question
    Is a radical a number?
    Searching for an answer, I was unable to find an affirmative answer, either positive or negative, to this question. On the other hand, considering various definitions of number, they implicitly imply that a number as a unique value. I reach this conclusion because, although numbers can be conceived independently of counting or measuring, all numbers, at least all real numbers, can be used for counting or measuring. And a count or a measurement cannot be ambiguous.
     
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    Obviously a number can be expressed in many different ways. For example, the number twelve can be expressed as 10 + 2 and as 20 - 8. When we write it as 12 simply use the common and convenient notation just as the Romans Wrote XII. So the presence of an operation does not render Amitai's entries in each magic square cell from being numbers.

    What about using a radical in each entry? The affirmative answer goes back to the time of the ancient Greeks. When the Pythagoreans discovered that  is not a rational number, they definitely  recognized it is a number. But here we assume that the cause of Pythagoreans' frustration was the positive root of .

    What happens when we consider both the positive and negative roots of the square root? Is it necessary to precede the radical symbol with the plus-minus sign in order to indicate that both positive and negative roots are acceptable?

    Finally, what about exploiting the ambiguity of the value of the square root? When we evaluate each magic-sqaure cell entry and some times we opt to take the positive value of the square root and at other times we select its negative root, it seems that we are using each entry as an expression, not as having constant. And a number, after all, is a constant. So here lies the reason to my question.

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    Last Update: Nov. 26, 2005