VARIATIONS BASED ON ROTATION OF A PRIME BORDER JOHNSON SQUARE

A 7x7 Prime Magic Square

A 7x7 magic square, discovered by A. Johnson, contains only prime numbers and its internal 5x5 and 3x3 squares are also magic at their shared borders. (See square 000 below referenced from Ian Stuart's book¹ and A. Johnson's original paper²). Because of the structure of the square, this square can be converted into other variants by different degrees of rotation of the individual borders. In this page only rotation by 180° is being employed.

If we depict the Johnson square as 000 (no rotation), then its counterpart is 111 (rotation of all three borders). Similarly 100 (rotation of external border) is equivalent to 011 (rotation of the two internal borders), 010 (rotation of the internal border) equivalent to 101 (rotation of the internal and external border), and 001 (rotation of only the internal border) equivalent to 110 (rotation of the two external borders).

Accordingly, the four equivalencies are summarized as:

Construction of the Johnson 7x7 Magic Square and Rotational Variants

1. Construct the complementary table of prime pairs where each pair of complements, as well as the 25^th number doubled, sums to 3958.
2. The internal 3x3 square is generated using the nine khaki colored numbers whose difference (Δ) between numbers (as shown in the figure above) is 48.
3. Generate the Johnson square (000).
4. Generate the three other rotational squares.

This concludes the rotational variants of a Johnson prime square.
A new prime square based on the Johnson square is shown in the next webpage.
Go back to homepage.

References

1. Ian Stuart: Professors Stuart's Hoard of Mathematical Treasures(2009) Page 192
2. A. W. Johnson, Jr., Journal of Recreational Mathematics 15:2, 1982-83, p. 84