General Method for Large Order Staircase Squares (Part II)
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↓)
1 | 3 | 5 | 7 | 9 | 0 |
2 | 4 | 6 | 8 | 10 | k/j |
112 | 48 | 105 | 41 | 98 |
23 | 80 | 16 | 73 | 9 | 66 | 1 |
47 | 104 | 40 | 97 | 33 |
79 | 15 | 72 | 8 | 65 | 111 | 3 |
103 | 39 | 96 | 32 | 78 |
14 | 71 | 7 | 64 | 121 | 46 | 5 |
38 | 95 | 31 | 88 | 13 |
70 | 6 | 63 | 120 | 45 | 102 | 7 |
94 | 30 | 87 | 12 | 69 |
5 | 62 | 119 | 55 | 101 | 37 | 9 |
29 | 86 | 22 | 68 | 4 |
61 | 118 | 54 | 100 | 36 | 93 | 0 |
85 | 21 | 67 | 3 | 60 | 117 |
53 | 110 | 35 | 92 | 28 | 2 |
20 | 77 | 2 | 59 | 116 |
52 | 109 | 34 | 91 | 27 | 84 | 4 |
76 | 1 | 58 | 115 | 51 |
108 | 44 | 90 | 26 | 83 | 19 | 6 |
11 | 57 | 114 | 50 | 107 |
43 | 89 | 25 | 82 | 18 | 75 | 8 |
56 | 113 | 49 | 106 | 42 |
99 | 24 | 81 | 17 | 74 | 10 | 10 |
|
|
H2 (3→ 9↓)
1 | 3 | 5 | 7 | 9 | 0 |
2 | 4 | 6 | 8 | 10 | k/j |
76 | 86 | 96 | 106 | 116 |
5 | 15 | 25 | 35 | 45 | 66 | 1 |
85 | 95 | 105 | 115 | 4 |
14 | 24 | 34 | 55 | 65 | 75 | 3 |
94 | 104 | 114 | 3 | 13 |
23 | 44 | 54 | 64 | 74 | 84 | 5 |
103 | 113 | 2 | 12 | 33 |
73 | 53 | 63 | 73 | 83 | 93 | 7 |
112 | 1 | 22 | 32 | 42 |
52 | 62 | 72 | 82 | 92 | 102 | 9 |
11 | 21 | 31 | 41 | 51 | 61 |
71 | 81 | 91 | 101 | 111 | 0 |
20 | 30 | 40 | 50 | 60 |
70 | 80 | 90 | 100 | 121 | 10 | 2 |
29 | 39 | 49 | 59 | 69 |
79 | 89 | 110 | 120 | 9 | 19 | 4 |
38 | 48 | 58 | 68 | 78 |
99 | 109 | 119 | 8 | 18 | 28 | 6 |
47 | 57 | 67 | 88 | 98 |
108 | 118 | 7 | 17 | 27 | 37 | 8 |
56 | 77 | 87 | 97 | 107 |
117 | 6 | 16 | 26 | 36 | 46 | 10 |
|
H3 (4→ 2↓)
1 | 3 | 5 | 7 | 9 | 0 |
2 | 4 | 6 | 8 | 10 | k/j |
83 | 100 | 7 | 24 | 52 |
69 | 97 | 114 | 21 | 38 | 66 | 1 |
110 | 6 | 23 | 51 | 68 |
96 | 113 | 20 | 37 | 65 | 82 | 3 |
5 | 33 | 50 | 67 | 95 |
112 | 19 | 36 | 64 | 81 | 109 | 5 |
32 | 49 | 77 | 94 | 111 |
18 | 35 | 63 | 80 | 108 | 4 | 7 |
48 | 76 | 93 | 121 | 17 |
34 | 62 | 79 | 107 | 3 | 31 | 9 |
75 | 92 | 120 | 16 | 44 |
61 | 78 | 106 | 2 | 30 | 47 | 0 |
91 | 119 | 15 | 43 | 60 | 88 |
105 | 1 | 29 | 46 | 74 | 2 |
118 | 14 | 42 | 59 | 87 |
104 | 11 | 28 | 45 | 73 | 90 | 4 |
13 | 41 | 58 | 86 | 103 |
10 | 27 | 55 | 72 | 89 | 117 | 6 |
40 | 57 | 85 | 102 | 9 |
26 | 54 | 71 | 99 | 116 | 12 | 8 |
56 | 84 | 101 | 8 | 25 |
53 | 74 | 98 | 115 | 22 | 39 | 10 |
|
|
H4 (8→ 4↓)
1 | 3 | 5 | 7 | 9 | 0 |
2 | 4 | 6 | 8 | 10 | k/j |
71 | 87 | 92 | 108 | 113 |
8 | 13 | 29 | 34 | 50 | 66 | 1 |
86 | 91 | 107 | 112 | 7 |
12 | 28 | 44 | 49 | 65 | 70 | 3 |
90 | 106 | 111 | 6 | 22 |
27 | 43 | 48 | 64 | 69 | 85 | 5 |
105 | 121 | 5 | 21 | 26 |
42 | 47 | 63 | 68 | 84 | 89 | 7 |
120 | 4 | 20 | 25 | 41 |
46 | 62 | 67 | 83 | 99 | 104 | 9 |
3 | 19 | 24 | 40 | 45 | 61 |
77 | 82 | 98 | 103 | 119 | 0 |
18 | 23 | 39 | 55 | 60 |
76 | 81 | 97 | 102 | 118 | 2 | 2 |
33 | 38 | 54 | 59 | 75 |
80 | 96 | 101 | 117 | 1 | 17 | 4 |
37 | 53 | 58 | 74 | 79 |
95 | 100 | 116 | 11 | 16 | 32 | 6 |
52 | 57 | 73 | 78 | 94 |
110 | 115 | 10 | 15 | 31 | 36 | 8 |
56 | 72 | 88 | 93 | 109 |
114 | 9 | 14 | 30 | 35 | 51 | 10 |
|
Two Staircase Squares of Order 15
The following two squares of order 15 are L1 and L2 both magic.
L1 (0→ 7↓)
1 | 3 | 5 | 7 | 9 | 11 | 13 | 0 |
2 | 4 | 6 | 8 | 10 | 12 | 14 | k/j |
89 | 58 | 27 | 221 | 190 | 159 | 128 | 97 |
66 | 35 | 4 | 198 | 167 | 136 | 120 | 1 |
57 | 26 | 220 | 189 | 158 | 127 | 96 | 65 | 34 |
3 | 197 | 166 | 150 | 119 | 88 | 3 |
25 | 219 | 188 | 157 | 126 | 95 | 64 | 33 | 2 |
196 | 180 | 149 | 118 | 87 | 56 | 5 |
218 | 187 | 156 | 125 | 94 | 63 | 32 | 1 | 210 |
179 | 148 | 117 | 86 | 55 | 24 | 7 |
186 | 155 | 124 | 93 | 62 | 31 | 15 |
209 | 178 | 147 | 116 | 85 | 54 | 23 | 217 | 9 |
154 | 123 | 92 | 61 | 45 | 14 | 208 | 177 | 146 |
115 | 84 | 53 | 22 | 216 | 185 | 11 |
122 | 91 | 75 | 44 | 13 | 207 | 176 | 145 | 114 | 83 |
52 | 21 | 215 | 184 | 153 | 13 |
105 | 74 | 43 | 12 | 206 | 175 | 144 | 113 | 82 |
51 | 20 | 214 | 183 | 152 | 121 | 0 |
73 | 42 | 11 | 205 | 174 | 143 | 112 | 81 | 50 |
19 | 213 | 182 | 151 | 135 | 104 | 2 |
41 | 10 | 204 | 173 | 142 | 111 | 80 | 49 |
18 | 212 | 181 | 165 | 134 | 103 | 72 | 4 |
9 | 203 | 172 | 141 | 110 | 79 | 48 | 17 | 211 |
195 | 164 | 133 | 102 | 71 | 40 | 6 |
202 | 171 | 140 | 109 | 78 | 47 | 16 | 225 |
194 | 163 | 132 | 101 | 70 | 39 | 8 | 8 |
170 | 139 | 108 | 77 | 46 | 30 | 224 | 193 | 162 |
131 | 100 | 69 | 38 | 7 | 201 | 10 |
138 | 107 | 76 | 60 | 29 | 223 |
192 | 161 | 130 | 99 | 68 | 37 | 6 | 200 | 169 | 12 |
106 | 90 | 59 | 28 | 222 | 191 | 160 | 129 | 98 |
67 | 36 | 5 | 199 | 168 | 137 | 14 |
L2 (3→ 10↓) or (3→ 5↑)
1 | 3 | 5 | 7 | 9 | 11 | 13 | 0 |
2 | 4 | 6 | 8 | 10 | 12 | 14 | k/j |
212 | 94 | 201 | 83 | 190 | 72 | 179 | 46 |
153 | 35 | 142 | 24 | 131 | 13 | 120 | 1 |
93 | 200 | 82 | 189 | 71 | 178 | 60 | 152 | 34 |
141 | 23 | 130 | 12 | 119 | 211 | 3 |
199 | 81 | 188 | 70 | 177 | 59 | 151 | 33 | 140 |
22 | 129 | 11 | 118 | 225 | 92 | 5 |
80 | 187 | 69 | 176 | 58 | 165 | 32 | 139 | 21 |
128 | 10 | 117 | 224 | 91 | 198 | 7 |
186 | 68 | 175 | 57 | 164 | 31 | 138 |
20 | 127 | 9 | 116 | 223 | 105 | 197 | 79 | 9 |
67 | 174 | 56 | 163 | 45 | 137 | 19 | 126 | 8 |
115 | 222 | 104 | 196 | 78 | 185 | 11 |
173 | 55 | 162 | 44 | 136 | 18 | 125 | 7 | 114 | 221 |
103 | 210 | 77 | 184 | 66 | 13 |
54 | 161 | 43 | 150 | 17 | 124 | 6 | 113 | 220 |
102 | 209 | 76 | 183 | 65 | 172 | 0 |
160 | 422 | 149 | 16 | 123 | 5 | 112 |
219 | 101 | 208 | 90 | 182 | 64 | 171 | 53 | 2 |
41 | 148 | 30 | 122 | 4 | 111 | 218 | 100 |
207 | 89 | 181 | 63 | 170 | 52 | 159 | 4 |
147 | 29 | 121 | 3 | 110 | 217 | 99 | 206 | 88 |
195 | 62 | 169 | 51 | 158 | 40 | 6 |
28 | 135 | 2 | 109 | 216 | 98 | 205 | 87 |
194 | 61 | 168 | 50 | 157 | 39 | 146 | 8 |
134 | 1 | 108 | 215 | 97 | 204 | 86 | 193 | 75 |
167 | 49 | 156 | 38 | 145 | 27 | 10 |
15 | 107 | 214 | 96 | 203 | 85 | 192 | 74 | 166 |
48 | 155 | 37 | 144 | 26 | 133 | 12 |
106 | 213 | 95 | 202 | 84 | 191 | 73 | 180 | 47 |
154 | 36 | 143 | 25 | 132 | 14 | 14 |
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