General Method for Large Order Staircase Squares (Part II)

A stairs

Loubère Square Background

The Siamese method which includes both the Loubère and Méziriac magic squares have the property that the center cell must always contain the middle number of the series of numbers used, i.e. a number which is equal to one half the sum of the first and last numbers of the series, or ½(n2 + 1). In addition, the sum of the horizontal rows, vertical columns and corner diagonals are equal to the magic sum S. Both squares also require an upward stepwise addition of consecutive numbers, i.e., 1,2,3... It's also a fact that only one Loubère square per order n has been handed down thru the centuries. In addition, construction of the square requires a one down shift after filling of a diagonal to move to the next diagonal until the square is filled.

This page is a continuation of Part I which contains the general rules for construction of the staircase squares. Also previous methods have shown how to construct these squares but in a specific manner for each of the methods as was depicted for first row, middle column and left diagonal generated squares. Two types of squares will be covered, the prime 11 and the composite 15 orders.

Four Staircase Squares of Order 11

Four 11th order squares were constructed according to the new general method where the initial starting 1's were chosen at random as depicted in H1, H2, H3 and H4. All are magic which is typical for a prime order square, except where the 1 is on the main diagonal, the first column or the last row. In addition, a knight move may be shown to proceed in several manners. For example, one may also consider going via an alternate knight route in order to lower the amount of cell counting during knight breaks, especially when tackling large squares. Thus, H2 (3→ 9↓) may also proceed via (3→ 2↑) by subtracting 9 from 11 and H4 (8→ 4↓) via (3← 4↓) by subtracting 8 by 11 where the direction of one of the arrows in the ordered pair is reversed.

H1 (3→ 6↓)
135790 246810k/j
112 48 1054198 23 8016 739 661
47 104 409733 79 1572 865 1113
103 39 963278 14 717 64121 465
38 95 318813 70 663 12045 1027
94 30 871269 5 62119 55101 379
29 86 22684 61 11854 10036 930
852167360117 531103592 282
20 77 259116 52 10934 9127 844
76 1 5811551 108 4490 2683 196
11 57 11450107 43 8925 8218 758
56 113 4910642 99 2481 1774 1010
H2 (3→ 9↓)
135790 246810k/j
76 86 96106116 5 1525 3545 661
85 95 1051154 14 2434 5565 753
94 104 114313 23 4454 6474 845
103 113 21233 73 5363 7383 937
112 1 223242 52 6272 8292 1029
112131415161 718191101 1110
20 30 405060 70 8090 100121 102
29 39 495969 79 89110 1209 194
38 48 586878 99 109119 818 286
47 57 678898 108 1187 1727 378
56 77 8797107 117 616 2636 4610
H3 (4→ 2↓)
135790 246810k/j
83 100 72452 69 97114 2138 661
110 6 235168 96 11320 3765 823
5 33 506795 112 1936 6481 1095
32 49 7794111 18 3563 80108 47
48 76 9312117 34 6279 1073 319
75 92 1201644 61 78106 230 470
9111915436088 10512946 742
118 14 425987 104 1128 4573 904
13 41 5886103 10 2755 7289 1176
40 57 851029 26 5471 99116 128
56 84 101825 53 7498 11522 3910
H4 (8→ 4↓)
135790 246810k/j
71 87 92108113 8 1329 3450 661
86 91 1071127 12 2844 4965 703
90 106 111622 27 4348 6469 855
105 121 52126 42 4763 6884 897
120 4 202541 46 6267 8399 1049
31924404561 778298103 1190
18 23 395560 76 8197 102118 22
33 38 545975 80 96101 1171 174
37 53 587479 95 100116 1116 326
52 57 737894 110 11510 1531 368
56 72 8893109 114 914 3035 5110

Two Staircase Squares of Order 15

The following two squares of order 15 are L1 and L2 both magic.

L1 (0→ 7↓)
1357911130 2468101214k/j
89582722119015912897 663541981671361201
57 26 220189158127966534 3 197166150119883
252191881571269564332 19618014911887 565
2181871561259463321210 179148117 8655 247
186155 12493623115 209178147 11685 5423 2179
154 123 92614514208177146 115 8453 22216 18511
122 91 75441320717614511483 5221215184 15313
105 74 431220617514411382 51 20214 183152 1210
73 42 11205174143 11281 50 19 2131821511351042
41 10 204173142111 8049 18212 181165134103724
9 203 172141110794817211 195 164133 10271 406
202171140109784716225 194163132101703988
170 139 108774630224193162 131 10069 387 20110
138 107 766029223 19216113099 6837 6200 16912
106 90 592822219116012998 67 365 199168 13714


L2 (3→ 10↓) or (3→ 5↑)
1357911130 2468101214k/j
21294201831907217946 1533514224131131201
93 200 82189711786015234 141 23130121192113
19981188701775915133140 2212911118225 925
8018769176581653213921 12810117 22491 1987
18668 1755716431138 201279 116223 105197 799
67 174 5616345137191268 115 222104 19678 18511
1735516244136181257114221 10321077184 6613
54 161 43150171246113220 102 20976 18365 1720
160 422 149161235 112 219 101208 9018264171532
41 148 301224111 218100 20789 18163170521594
147 29 12131102179920688 195 62169 51158 406
2813521092169820587 1946116850157391468
134 1 108215972048619375 167 49156 38145 2710
15 107 214962038519274166 48 15537 14426 13312
106 213 95202841917318047 154 36143 25132 1414

As stated previously all the composite squares generate a lower number of magic squares. Two 15x15 squares that are not magic (not shown) have their initial 1 at k/j position (7,1) and (0,4). The former has an M=31 in column 1, and the latter an M=16 on row 15.

This completes this section (Part II). To return to Part I or to return to homepage.


Copyright © 2020 by Eddie N Gutierrez. E-Mail: enaguti1949@gmail.com