Magic Squares Wheel Method-Redux Part III

New Variants of Order 7

The wheel method is a means of constructing magic squares by a random access means instead of sequentially like the Loubère and Méziriac methods which has been rewritten in a more simplified form. The method patially fills up a square to form a wheel structure using numbers chosen from a complementary table of order n then randomly fills up the rest of the square with numbers chosen from whatever is left in the complementary table. This paper is a simplification of the original paper taking a more facile approach.

Three 5^th order variants have been constructed that differ from the original at the left diagonal. These three variants are:
the Reverse: 24 → 23 → 22 → 25 → 28 → 27 → 26, the Forward II: 22 → 27 → 24 → 25 → 26 → 23 → 24 and the Reverse II: 24 → 27 → 22 → 25 → 28 → 23 → 26,
where these sequences are chosen from the values ½(n²-n+2) to ½(n²+n).

The three variants listed below were prepared according to the method employed in Part I and the squares follow the patterns shown in Part II where both Variants 2 and 4 are border squares and variant 3 is not. The variants are also accompanied by their complementary tables to show how the values are distributed. It can be seen that Variants 2 and 4 as well as Variant 1 have identical distribution.

The summation tables for these three tables are shown below:

where the parities of Variants 2 and 4 are the same. It will be shown that those of initial parity of all O will produce border squares.

The next page discusses one 9x9 variant. Go back to Part II.
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