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In this article we will use a computer model to solve the question "Is there a finite number of solutions to the M3 problem, and if so, what are they?"

Past experience solving other problems demonstrated value in creating a programming model of the thinking process since it allowed a study of perspectives not likely found using just paper and pencil. In such a case a computer program model used as a tool for problem solving becomes "A Tool of Thought". There are many programming languages that have been used in this manner. Depending upon your background and experience and the problem itself, some languages are more appropriate than others. Following is the output of such a program written in "A Programming Language" (APL).

PERMUTATIONS OF 3LETS					=	9*3	=	729
PROGRAMMING MODEL GENERATED							729	3LETS
OF WHICH			61	3LETS SUM TO 15
AFTER REMOVING DUP NO.S IN 3LETS
THERE ARE			48	3LETS LEFT
FOLLOWING ARE ALL THE 3LETS GROUPED BY CENTER NO.

519	429	438	249	159	168	276	186	195
618	528	537	348	258	267	375	285	294
816	627	735	546	357	465	573	384	492
915	726	834	645	456	564	672	483	591
	825		843	654	762		582
	924		942	753	861		681
				852
				951

PROCESS - USING GROUP WITH CENTER NO. = 2 SET
WE MUST SELECT ALL SUBSETS OF 3 3LETS THAT
MEET THE FOLLOWING CRITERIA:

1. THE SUM OF THEIR FIRST VALUES = 15
2. THE SUM OF THEIR LAST VALUES = 15
3. THERE MUST BE A REMAINING 3LET
   THAT FILLS THE 2 EMPTY CELLS AND STILL MEETS THE
   SUM TO 15 CRITERIA

4		7
5	2	8
6		9

THE CENTER NO.=2 GROUP FAILS CRITERIA (2) AND (3).


NOW LET US EXAM THE GROUP WITH CENTER NO.= 5
159
258
357
456
654
753
852
951

THE ONLY SUBSET OF THE '5' GROUP THAT MEET CRITERIA IS:
456 357 852 159

492
357
816

THE ROTATIONS OF THIS MAGIC SQUARE ARE:

816	672	294	438	618	834	492	276
357	159	753	951	753	159	357	951
492	834	618	276	294	672	816	438

IN POLITICS THEY WOULD CALL IT 'SPIN', BUT 'SPIN' SOMETIME HIDES A MORE USEFUL VIEW.
HERE IT IS JUST A DIFFERENT WAY OF LOOKING AT THE SAME THING.

This approach to solving the M3 problem can easily be programmed in DOS basic or Visual Basic and still qualify as a "A Tool of Thought" process.

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