Office Applications and Entertainment, Magic Squares

14.0     Special Magic Squares, Prime Numbers

14.34    Magic Squares (14 x 14), Part II

14.34.11 Bordered Magic Squares(14 x 14)
         Split Border

Alternatively a 14^th order Bordered Magic Square with Magic Sum s14 can be constructed based on:

• a Concentric Magic Center Square of order 10 with Magic Sum s10 = 10 * s14 / 14;
• 48 pairs, each summing to 2 * s14 / 14, surrounding the (Concentric) Magic Center Square;
• a split of the supplementary rows and columns into three parts:
  two summing to s4 = 4 * s14 / 14 and one to s6 = 6 * s14 / 14.

as illustrated below:

Mc14 = 16170

409	499	1823	1889	577	617	827	1609	1637	1663	1847	1747	587	439
1811	1901	487	421	1733	1693	1483	701	673	647	463	563	1871	1723
569	1741	331	337	1277	1879	1951	2017	1999	1279	281	199	1549	761
1831	479	2027	2243	229	2053	2207	97	149	2131	131	283	613	1697
787	1523	1013	173	2293	71	89	2153	2287	37	2137	1297	683	1627
859	1451	317	271	41	59	181	2113	2267	2269	2039	1993	709	1601
1009	1301	307	1997	73	2083	2297	29	211	2237	313	2003	829	1481
1399	911	379	223	107	197	43	2251	2129	2203	2087	1931	1559	751
1429	881	1327	241	2143	2281	2099	227	13	167	2069	983	1567	743
1447	863	1861	1913	2273	2239	2221	157	23	17	397	449	1583	727
1669	641	1877	2179	2081	257	103	2213	2161	179	67	433	653	1657
823	1487	2111	1973	1033	431	359	293	311	1031	2029	1979	1499	811
1619	1801	643	557	739	757	1021	1453	1471	1489	521	523	1867	1709
509	691	1667	1753	1571	1553	1289	857	839	821	1789	1787	601	443

Based on the principles described in previous sections, a fast procedure (Priem14c) can be developed:

• to read the previously generated Concentric Magic Squares (10 x 10);
• to generate the four Magic Rectangles (2 x 6);
• to generate, based on the remainder of the pairs, the four Magic Corner Segments (4 x 4);

Attachment 14.34.11 shows based on 10^th order Concentric Magic Squares as discussed in Section 14.8.7, one Prime Number Bordered Magic Square for some of the occurring Magic Sums.

Each square shown corresponds with numerous squares for the same Magic Sum.

14.34.12 Bordered Magic Squares(14 x 14)
         Composed Border

Alternatively order 14 Composed Border Magic Squares with Magic Sum s14 can be constructed based on:

• a Border composed out of:
  - 4 (Semi) Magic Squares of order 4 with Magic Sum s4 = 4 * s14 / 14;
  - 4 Associated Magic Rectangles order 6 x 4 with s6 = 6 * s14 / 14 and s4 = 4 * s14 / 14.
• a Magic Center Square of order 6 with Magic Sum s6 = 6 * s14 / 14;

as illustrated below:

Mc14 = 16170

2243	1999	199	179	2039	2027	1297	337	443	787	2207	1741	379	293
149	229	2029	2213	727	317	523	1483	1901	1979	223	449	1811	2137
97	281	2081	2161	331	409	827	1787	1993	1583	173	499	1861	2087
2131	2111	311	67	1523	1867	1973	1013	283	271	2017	1931	569	103
1627	857	709	1427	2293	71	89	2153	2287	37	1951	521	479	1669
1559	877	983	1201	41	59	181	2113	2267	2269	1877	601	691	1451
991	1019	1061	1549	73	2083	2297	29	211	2237	1093	887	839	1801
761	1249	1291	1319	107	197	43	2251	2129	2203	509	1471	1423	1217
1109	1327	1433	751	2143	2281	2099	227	13	167	859	1619	1709	433
883	1601	1453	683	2273	2239	2221	157	23	17	641	1831	1789	359
2069	1667	421	463	1747	1723	1277	653	829	701	2179	1571	431	439
397	487	1733	2003	647	673	863	1487	1567	1693	313	557	1697	2053
307	577	1823	1913	617	743	823	1447	1637	1663	257	613	1753	1997
1847	1889	643	241	1609	1481	1657	1033	587	563	1871	1879	739	131

Based on the principles described in previous sections, a fast procedure (Priem14d) can be developed:

• to read the previously generated (Concentric) Center Squares (6 x 6);
• to generate, based on the remainder of the pairs, four 4 x 4 Associated Magic Squares;
• to complete the Composed Border of order 14 with four 6 x 4 Associated Magic Rectangles.

Attachment 14.34.12a shows one Prime Number Composed Border Magic Square for some of the occurring Magic Sums.

Note:

If the applied properties are changed to:

• the opposite Semi Magic Corner Squares (4 x 4) are Anti Symmetric and Complementary;
• the opposite Rectangles (4 x 6) are Anti Symmetric and Complementary;
• the Magic Center Square (6 x 6) is Associated;

the 14^th order Composed Magic Square will be Associated.

Attachment 14.34.12b shows one Prime Number Composed Associated Magic Square for a few of the occurring Magic Sums.

Each square shown corresponds with numerous squares for the same Magic Sum.

14.34.13 Composed Magic Squares (14 x 14)
         Overlapping Sub Squares

Following 14^th order Composed Magic Square, with overlapping sub squares and Magic Sum s14, contains following sub squares:

• One 6^th order Magic Center Square C;
• Two each other overlapping 8^th order Eccentric Magic Squares A1 and A2;
• Two each other overlapping 10^th order Eccentric Magic Squares B1 and B2;
• Two 4^th order Pan Magic Squares PM1 and PM2;

as illustrated below:

Mc14 = 16170

1571	683	709	1657	241	2027	2017	307	1999	313	1993	409	1913	331
691	1567	1481	881	2069	283	293	2003	311	1997	317	1901	1979	397
1619	743	829	1429	17	2083	2251	71	379	2143	29	2267	1973	337
739	1627	1601	653	2293	227	59	2239	1931	167	43	2281	421	1889
1019	1291	2113	197	2297	1669	179	1327	521	937	2237	73	1879	431
1811	499	229	2081	617	37	1061	2053	1553	1609	223	2087	439	1871
1499	811	271	2039	1013	1279	2243	103	911	1381	2213	97	1301	1009
1433	877	211	2099	929	1399	2207	67	1031	1297	2203	107	523	1787
1663	647	1877	433	701	757	257	1249	2273	1693	2269	41	449	1861
613	1697	199	2111	1373	1789	983	2131	641	13	23	2287	673	1637
1723	587	2179	149	281	1951	2129	173	157	2221	1559	787	827	1447
577	1733	2161	131	2029	359	181	2137	2153	89	1051	1217	1093	1259
643	1741	1709	563	1753	509	1823	479	463	1867	1123	1193	1117	1187
569	1667	601	1747	557	1801	487	1831	1847	443	887	1423	1583	727

Attachment 14.34.13a shows, miscellaneous order 14 Composed Magic Squares (ref. Priem14f).

The corresponding Composed Magic Squares of order 18 contain, in addition to the sub squares mentioned above, following Corner Squares:

• Two 6^th order Eccentric Magic Squares F1 and F2 with embedded PM1 and PM2;
• Two 12^th order Eccentric Magic Squares D1 and D2 with embedded B1 and B2;

Attachment 14.34.13b shows, for the sake of completeness, miscellanous Composed Magic Squares of order 18 (ref. Priem14g).

14.34.15 Summary

The obtained results regarding the 14^th order Prime Number Magic Squares and related sub squares, as deducted and discussed in previous sections, are summarized in following table:

Comparable routines as listed above, can be used to generate miscellaneous Prime Number Composed Magic Squares, which will be described in following sections.