VARIATIONS BASED ON ROTATION OF A PRIME BORDER JOHNSON SQUARE

Picture of a 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:

000 ≡ 111 100 ≡ 011010 ≡ 101 001 ≡ 110

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 25th 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.
1031 1049 1061 1097 1181 1217 1229 1259 1301 1367 1409 1427 1481 1499 1511 1559 1601 1607 1619 1721 1871 1877 1889 1931
1979
2927 2909 2897 2861 2777 2741 2729 2699 2657 2591 2549 2531 2477 2459 2447 2399 2357 2351 2339 2237 2087 2081 2069 2027
000 - Johnson Square
2777 1409 2339 14811061 26992087
253118892237 2459122920811427
136723572399 1511 20271601 2591
290910311607 1979 235129271049
130127411931 2447 1559 12172657
109718771721 149927292069 2861
187125491619 2477289712591181
100 Square
1181 1259 2897 24771619 25491871
286118892237 2459122920811097
265723572399 1511 20271601 1301
104910311607 1979 235129272909
259127411931 2447 1559 12171367
142718771721 149927292069 2531
208726991061 1481233914092777
+
010 Square
2777 1409 2339 14811061 26992087
253120692729 1499172118771427
136712172399 1511 20272741 2591
290929271607 1979 235110311049
130116011931 2447 1559 23572657
109720811229 245922371889 2861
187125491619 2477289712591181
+
001 Square
2777 1409 2339 14811061 26992087
253118892237 2459122920811427
136723571559 2447 19311601 2591
290910312351 1979 160729271049
130127412027 1511 2399 12172657
109718771721 149927292069 2861
187125491619 2477289712591181

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


Copyright © 2010 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com