Magic Square – Topographical model -
( Craig Knecht – 2007)
The above illustration represents a potential section of a magic square.
The height of the towers is specified by the value in the cell
My primary focus with the topographical model is to delineate the patterns of
Water retention within the square
ORDER 4 Magic Square WATER RETENTION PATTERNS –
WITH OCCURRENCE OF EACH – TOTAL = 880
Figure 2
a) there are 880 different order 4 magic squares
b) 137 of the 880 squares retain no water according to the topographical model
c) water can only flow on the x.,y axis according to the model
d) for the interior cells with values 9,6,12,7
1) the cell with value 9 ie a height of 9 … water could escape out on the path over
the cell with value 8 or over cell with 6, then 4 or cell 6,7,1 ….
But no water will be retained in the cell with value 9
2) the cell with value 6 .. escapes out over cell with value 4 … no water retained
3) the cell with value 12 … escapes over cell 5
4) the cell with value 7 excapes over cell with value 1
Figure 3
a) of the 880 order 4 magic squares in 396 only one cell retains water ..
b) in the example of above the cell with value 2 retains water ..
c) the amount of water this cell retains = 5 units … as it will fill to a
level of 7 before it spills off the edge
figure 4.
a) of the 880 order 4 magic squares 252 retain water in two cells
on the same row or column
b) the lowest path out for both the cells retaining water in this example
is on the path > 9 > 4 … but both cells well retain water to the level
of 9 .. so 9 units of water will be retained ( ( 9 – 7) + (9 – 2) ) = 9
figure 5
a) of the 880 order 4 magic squares – 95 retain water with two cells on the
diagonal
b) the cell with value 2… lowest path out = 7 , retains 5 units of water
the cell with value 3 …lowest path out = 6 , retains 3 units of water
total units retained by this square = 8 units
Francis Gaspalou sent me this example - 69 units retained
( Maximum retention for order 5 magic square)
ORDER 7 : largest retained value I have found so far …
Examples – retain water …but remain balanced …
13 examples for order 4 square
Example from Gaspalou … retains water … remains balanced
For order 5 … retains water … both cells on same row …
Remains balanced
figure above …. “ Big Island “
Help wanted …
1) program using a algorithm ( perhaps Greedy algorithm) that will compute the amount of retained water for any given order of square
a) apply program to the known set of associated order 5 squares
b) apply program to 20 million order 7 pandiagonal associated magic squares
Program for generating the tabular data:
(visual basic )
a1, a2, a3, a4
b1, b2, b3, b4
c1, c2, c3, c4
d1, d2, d3, d4
The above is the convention for labeling
The order 4 cells
cell b2 … b2u = units retained
cell b3 … b3u = units retained
cell c2 … c2u = units retained
cell c3 … c3u = units retained
10 pathways of escape are examined
for each interior cell. The
largest cell value of each path
is placed in the variables labeled
p1,p2,p3,p4,p5,p6,p7,p8,p9,p10
the lowest path out of the
of the p1,p2,p3,p4,p5,p6,p7,p8,p9,p10
is placed in the respective
variables b2u,b3u,c2u,c3u
negative values are removed by the equation
If b2u < 0 then b2u = 0
subtracting the lowest path value out from the
value of the cell… gives the units retained
by the equation b2u = b2u – b2
source file contains data from Harvey Heinz web site
On the 880 order 4 magic squares …
n1 = the number of the magic square 1 > 880
n2 = the number of the Dudneney Group for the square
n3 = the number of units of water retained for the square
the equation b2u + b3u + c2u + c3u adds up the
units retained by each interior cell
For y = 1 To 880
Input #1, n1, n2, n3, a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4, d1, d2, d3, d4
b2 cell retained units > variable b2u
p1 = b1
p2 = a2
p3 = b3
If a3 > b3 Then
p3 = a3
End If
p4 = b3
If b4 > b3 Then
p4 = b4
End If
p5 = b3
If c3 > b3 Then
p5 = c3
End If
If c4 > p5 Then
p5 = c4
End If
p6 = b3
If c3 > b3 Then
p6 = c3
End If
If d3 > p6 Then
p6 = d3
End If
p7 = c2
If c1 > c2 Then
p7 = c1
End If
p8 = c2
If d2 > c2 Then
p8 = d2
End If
p9 = c2
If c3 > c2 Then
p9 = c3
End If
If c4 > p9 Then
p9 = c4
End If
p10 = c2
If c3 > c2 Then
p10 = c3
End If
If d3 > p10 Then
p10 = d3
End If
b2u = p1
If p2 < b2u Then
b2u = p2
End If
If p3 < b2u Then
b2u = p3
End If
If p4 < b2u Then
b2u = p4
End If
If p5 < b2u Then
b2u = p5
End If
If p6 < b2u Then
b2u = p6
End If
If p7 < b2u Then
b2u = p7
End If
If p8 < b2u Then
b2u = p8
End If
If p9 < b2u Then
b2u = p9
End If
If p10 < b2u Then
b2u = p10
End If
b2u = b2u - b2
If b2u < 0 Then b2u = 0
B3 cell retained units > variable b3u
p1 = b4
p2 = a3
p3 = b2
If a2 > b2 Then
p3 = a2
End If
p4 = b2
If b1 > b2 Then
p4 = b1
End If
p5 = b2
If c2 > b2 Then
p5 = c2
End If
If c1 > p5 Then
p5 = c1
End If
p6 = b2
If c2 > b2 Then
p6 = c2
End If
If d2 > p6 Then
p6 = d2
End If
p7 = c3
If c4 > c3 Then
p7 = c4
End If
p8 = c3
If d3 > c3 Then
p8 = d3
End If
p9 = c3
If c2 > c3 Then
p9 = c2
End If
If c1 > p9 Then
p9 = c1
End If
p10 = c3
If c2 > c3 Then
p10 = c2
End If
If d2 > p10 Then
p10 = d2
End If
b3u = p1
If p2 < b3u Then
b3u = p2
End If
If p3 < b3u Then
b3u = p3
End If
If p4 < b3u Then
b3u = p4
End If
If p5 < b3u Then
b3u = p5
End If
If p6 < b3u Then
b3u = p6
End If
If p7 < b3u Then
b3u = p7
End If
If p8 < b3u Then
b3u = p8
End If
If p9 < b3u Then
b3u = p9
End If
If p10 < b3u Then
b3u = p10
End If
b3u = b3u - b3
If b3u < 0 Then b3u = 0
C2 cell retained units > variable c2u
p1 = c1
p2 = d2
p3 = c3
If d3 > c3 Then
p3 = d3
End If
p4 = c3
If c4 > c3 Then
p4 = c4
End If
p5 = c3
If b3 > c3 Then
p5 = c3
End If
If b4 > p5 Then
p5 = b4
End If
p6 = c3
If b3 > c3 Then
p6 = b3
End If
If a3 > p6 Then
p6 = a3
End If
p7 = b2
If b1 > b2 Then
p7 = b1
End If
p8 = b2
If a2 > b2 Then
p8 = a2
End If
p9 = b2
If b3 > b2 Then
p9 = b3
End If
If b4 > p9 Then
p9 = b4
End If
p10 = b2
If b3 > b2 Then
p10 = c3
End If
If a3 > p10 Then
p10 = a3
End If
c2u = p1
If p2 < c2u Then
c2u = p2
End If
If p3 < c2u Then
c2u = p3
End If
If p4 < c2u Then
c2u = p4
End If
If p5 < c2u Then
c2u = p5
End If
If p6 < c2u Then
c2u = p6
End If
If p7 < c2u Then
c2u = p7
End If
If p8 < c2u Then
c2u = p8
End If
If p9 < c2u Then
c2u = p9
End If
If p10 < c2u Then
c2u = p10
End If
c2u = c2u - c2
If c2u < 0 Then c2u = 0
C3 cell retained units > variable c3u
p1 = c4
p2 = d3
p3 = c2
If d2 > c2 Then
p3 = d2
End If
p4 = c2
If c1 > c2 Then
p4 = c1
End If
p5 = c2
If b2 > c2 Then
p5 = c3
End If
If b1 > p5 Then
p5 = b1
End If
p6 = c2
If b2 > c2 Then
p6 = b2
End If
If a2 > p6 Then
p6 = a2
End If
p7 = b3
If b4 > b3 Then
p7 = b4
End If
p8 = b3
If a3 > b3 Then
p8 = a3
End If
p9 = b3
If b2 > b3 Then
p9 = b2
End If
If b1 > p9 Then
p9 = b1
End If
p10 = b3
If b2 > c3 Then
p10 = b2
End If
If a2 > p10 Then
p10 = a2
End If
c3u = p1
If p2 < c3u Then
c3u = p2
End If
If p3 < c3u Then
c3u = p3
End If
If p4 < c3u Then
c3u = p4
End If
If p5 < c3u Then
c3u = p5
End If
If p6 < c3u Then
c3u = p6
End If
If p7 < c3u Then
c3u = p7
End If
If p8 < c3u Then
c3u = p8
End If
If p9 < c3u Then
c3u = p9
End If
If p10 < c3u Then
c3u = p10
End If
c3u = c3u - c3
If c3u < 0 Then c3u = 0
n3 = b2u + b3u + c2u + c3u
Rem picking up the no cell solutions .... > variable a$
If b2u = 0 And b3u = 0 And c2u = 0 And c3u = 0 Then
a$ = "+"
End If
Rem picking up the one cell only solutions > variable b$
If b2u = 0 And b3u = 0 And c2u = 0 And c3u <> 0 Then
c3p = c3p + 1
b$ = "+"
End If
If b2u = 0 And c3u = 0 And c2u = 0 And b3u <> 0 Then
b3p = b3p + 1
b$ = "+"
End If
If b3u = 0 And c3u = 0 And c2u = 0 And b2u <> 0 Then
b2p = b2p + 1
b$ = "+"
End If
If b2u = 0 And b3u = 0 And c3u = 0 And c2u <> 0 Then
b$ = "+"
End If
Rem picking up the diagonal solutions .... > variable d$
If b3u <> 0 And c2u <> 0 Then
d$ = "+"
End If
If b2u <> 0 And c3u <> 0 Then
End If
If b2u <> 0 And b3u <> 0 Then
c$ = "+"
End If
If b3u <> 0 And c3u <> 0 Then
c$ = "+"
End If
If c2u <> 0 And c3u <> 0 Then
c$ = " + "
End If
If b2u <> 0 And c2u <> 0 Then
c$ = "+"
End If
If b3u <> 0 And c2u <> 0 And b3u = c2u Then
e$ = "+"
End If
Print #2, n1, n2, n3, a$, b$, c$, d$, e$, a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4, d1, d2, d3, d4
a$ = "-"
b$ = "-"
c$ = "-"
d$ = "-"
e$ = "-"
Next y
Print "end"
End Sub
Private Sub Form_Load()
Open "I:\dudney\ugroupi.txt" For Input As #1
Open "I:\dudney2\ugroupn.txt" For Output As #2
ORDER 4 WATER RETENTION PATTERNS
Dud = Dudeney Group #
U = total units retained by square
A = pattern = no cells retain water
B = pattern = one cell only retains water
C = pattern = two cells retain water on the same row or column
D = pattern = two cells retain on the diagonal
E = pattern = water retained, remains balanced
Dud U A B C D E 16 cell values for order 4 square
|
102 1 8 - - - + - 1 8 10 15 12 13 3 6 7 2 16 9 14 11 5 4 |
|
104 1 5 - + - - - 1 8 10 15 14 11 5 4 7 2 16 9 12 13 3 6 |
|
107 1 8 - - - + - 1 8 11 14 12 13 2 7 6 3 16 9 15 10 5 4 |
|
109 1 3 - + - - - 1 8 11 14 15 10 5 4 6 3 16 9 12 13 2 7 |
|
116 1 5 - + - - - 1 8 13 12 14 11 2 7 4 5 16 9 15 10 3 6 |
|
117 1 3 - + - - - 1 8 13 12 15 10 3 6 4 5 16 9 14 11 2 7 |
|
171 1 9 - - + - - 1 12 6 15 14 7 9 4 11 2 16 5 8 13 3 10 |
|
174 1 9 - - + - - 1 12 7 14 15 6 9 4 10 3 16 5 8 13 2 11 |
|
177 1 9 - - + - - 1 12 13 8 14 7 2 11 4 9 16 5 15 6 3 10 |
|
178 1 9 - - + - - 1 12 13 8 15 6 3 10 4 9 16 5 14 7 2 11 |
|
201 1 9 - - + - - 1 14 7 12 15 4 9 6 10 5 16 3 8 11 2 13 |
|
204 1 9 - - + - - 1 14 11 8 15 4 5 10 6 9 16 3 12 7 2 13 |
|
279 1 8 - - - + - 2 7 9 16 11 14 4 5 8 1 15 10 13 12 6 3 |
|
281 1 7 - + - - - 2 7 9 16 13 12 6 3 8 1 15 10 11 14 4 5 |
|
292 1 8 - - - + - 2 7 12 13 11 14 1 8 5 4 15 10 16 9 6 3 |
|
294 1 1 - + - - - 2 7 12 13 16 9 6 3 5 4 15 10 11 14 1 8 |
|
304 1 7 - + - - - 2 7 14 11 13 12 1 8 3 6 15 10 16 9 4 5 |
|
305 1 1 - + - - - 2 7 14 11 16 9 4 5 3 6 15 10 13 12 1 8 |
|
355 1 11 - - + - - 2 11 5 16 13 8 10 3 12 1 15 6 7 14 4 9 |
|
365 1 9 - - + - - 2 11 8 13 16 5 10 3 9 4 15 6 7 14 1 12 |
|
372 1 11 - - + - - 2 11 14 7 13 8 1 12 3 10 15 6 16 5 4 9 |
|
375 1 9 - - + - - 2 11 14 7 16 5 4 9 3 10 15 6 13 8 1 12 |
|
393 1 9 - - + - - 2 13 8 11 16 3 10 5 9 6 15 4 7 12 1 14 |
|
396 1 9 - - + - - 2 13 12 7 16 3 6 9 5 10 15 4 11 8 1 14 |
|
469 1 7 - + - - - 3 6 9 16 13 12 7 2 8 1 14 11 10 15 4 5 |
|
473 1 1 - + - - - 3 6 12 13 16 9 7 2 5 4 14 11 10 15 1 8 |
|
483 1 7 - + - - - 3 6 15 10 13 12 1 8 2 7 14 11 16 9 4 5 |
|
485 1 1 - + - - - 3 6 15 10 16 9 4 5 2 7 14 11 13 12 1 8 |
|
530 1 11 - - + - - 3 10 5 16 13 8 11 2 12 1 14 7 6 15 4 9 |
|
532 1 9 - - + - - 3 10 8 13 16 5 11 2 9 4 14 7 6 15 1 12 |
|
536 1 11 - - + - - 3 10 15 6 13 8 1 12 2 11 14 7 16 5 4 9 |
|
537 1 9 - - + - - 3 10 15 6 16 5 4 9 2 11 14 7 13 8 1 12 |
|
560 1 9 - - + - - 3 13 8 10 16 2 11 5 9 7 14 4 6 12 1 15 |
|
565 1 9 - - + - - 3 13 12 6 16 2 7 9 5 11 14 4 10 8 1 15 |
|
621 1 5 - + - - - 4 5 10 15 14 11 8 1 7 2 13 12 9 16 3 6 |
|
623 1 3 - + - - - 4 5 11 14 15 10 8 1 6 3 13 12 9 16 2 7 |
|
646 1 5 - + - - - 4 5 16 9 14 11 2 7 1 8 13 12 15 10 3 6 |
|
647 1 3 - + - - - 4 5 16 9 15 10 3 6 1 8 13 12 14 11 2 7 |
|
690 1 9 - - + - - 4 9 6 15 14 7 12 1 11 2 13 8 5 16 3 10 |
|
691 1 9 - - + - - 4 9 7 14 15 6 12 1 10 3 13 8 5 16 2 11 |
|
702 1 9 - - + - - 4 9 16 5 14 7 2 11 1 12 13 8 15 6 3 10 |
|
704 1 9 - - + - - 4 9 16 5 15 6 3 10 1 12 13 8 14 7 2 11 |
|
744 1 11 - - + - - 4 14 7 9 15 1 12 6 10 8 13 3 5 11 2 16 |
|
748 1 11 - - + - - 4 14 11 5 15 1 8 10 6 12 13 3 9 7 2 16 |
|
785 1 0 + - - - - 5 4 14 11 16 9 7 2 3 6 12 13 10 15 1 8 |
|
788 1 0 + - - - - 5 4 15 10 16 9 6 3 2 7 12 13 11 14 1 8 |
|
828 1 0 + - - - - 6 3 13 12 15 10 8 1 4 5 11 14 9 16 2 7 |
|
839 1 0 + - - - - 6 3 16 9 15 10 5 4 1 8 11 14 12 13 2 7 |
|
21 2 4 - - - + - 1 4 14 15 13 16 2 3 8 5 11 10 12 9 7 6 |
|
22 2 1 - + - - - 1 4 14 15 13 16 2 3 12 9 7 6 8 5 11 10 |
|
27 2 3 - + - - - 1 4 15 14 13 16 3 2 8 5 10 11 12 9 6 7 |
|
28 2 0 + - - - - 1 4 15 14 13 16 3 2 12 9 6 7 8 5 10 11 |
|
56 2 8 - - - + - 1 6 12 15 11 16 2 5 8 3 13 10 14 9 7 4 |
|
57 2 3 - + - - - 1 6 12 15 11 16 2 5 14 9 7 4 8 3 13 10 |
|
62 2 5 - + - - - 1 6 15 12 11 16 5 2 8 3 10 13 14 9 4 7 |
|
63 2 1 - + - - - 1 6 15 12 11 16 5 2 14 9 4 7 8 3 10 13 |
|
82 2 8 - - - + - 1 7 12 14 10 16 3 5 8 2 13 11 15 9 6 4 |
|
83 2 2 - + - - - 1 7 12 14 10 16 3 5 15 9 6 4 8 2 13 11 |
|
89 2 6 - + - - - 1 7 14 12 10 16 5 3 8 2 11 13 15 9 4 6 |
|
90 2 1 - + - - - 1 7 14 12 10 16 5 3 15 9 4 6 8 2 11 13 |
|
213 2 4 - - - + - 2 3 13 16 14 15 1 4 7 6 12 9 11 10 8 5 |
|
214 2 3 - + - - - 2 3 13 16 14 15 1 4 11 10 8 5 7 6 12 9 |
|
233 2 1 - + - - - 2 3 16 13 14 15 4 1 7 6 9 12 11 10 5 8 |
|
234 2 0 + - - - - 2 3 16 13 14 15 4 1 11 10 5 8 7 6 9 12 |
|
246 2 8 - - - + - 2 5 11 16 12 15 1 6 7 4 14 9 13 10 8 3 |
|
247 2 5 - + - - - 2 5 11 16 12 15 1 6 13 10 8 3 7 4 14 9 |
|
269 2 3 - + - - - 2 5 16 11 12 15 6 1 7 4 9 14 13 10 3 8 |
|
270 2 3 - + - - - 2 5 16 11 12 15 6 1 13 10 3 8 7 4 9 14 |
|
316 2 8 - - - + - 2 8 11 13 9 15 4 6 7 1 14 12 16 10 5 3 |
|
317 2 1 - + - - - 2 8 11 13 9 15 4 6 16 10 5 3 7 1 14 12 |
|
323 2 6 - + - - - 2 8 13 11 9 15 6 4 7 1 12 14 16 10 3 5 |
|
324 2 2 - + - - - 2 8 13 11 9 15 6 4 16 10 3 5 7 1 12 14 |
|
421 2 3 - + - - - 3 2 13 16 15 14 1 4 6 7 12 9 10 11 8 5 |
|
422 2 3 - + - - - 3 2 13 16 15 14 1 4 10 11 8 5 6 7 12 9 |
|
445 2 0 + - - - - 3 2 16 13 15 14 4 1 6 7 9 12 10 11 5 8 |
|
446 2 0 + - - - - 3 2 16 13 15 14 4 1 10 11 5 8 6 7 9 12 |
|
450 2 6 - + - - - 3 5 10 16 12 14 1 7 13 11 8 2 6 4 15 9 |
|
464 2 2 - + - - - 3 5 16 10 12 14 7 1 6 4 9 15 13 11 2 8 |
|
465 2 5 - + - - - 3 5 16 10 12 14 7 1 13 11 2 8 6 4 9 15 |
|
503 2 1 - + - - - 3 8 10 13 9 14 4 7 16 11 5 2 6 1 15 12 |
|
505 2 5 - + - - - 3 8 13 10 9 14 7 4 6 1 12 15 16 11 2 5 |
|
506 2 3 - + - - - 3 8 13 10 9 14 7 4 16 11 2 5 6 1 12 15 |
|
583 2 1 - + - - - 4 1 14 15 16 13 2 3 5 8 11 10 9 12 7 6 |
|
584 2 1 - + - - - 4 1 14 15 16 13 2 3 9 12 7 6 5 8 11 10 |
|
591 2 0 + - - - - 4 1 15 14 16 13 3 2 5 8 10 11 9 12 6 7 |
|
592 2 0 + - - - - 4 1 15 14 16 13 3 2 9 12 6 7 5 8 10 11 |
|
648 2 5 - + - - - 4 6 9 15 11 13 2 8 14 12 7 1 5 3 16 10 |
|
661 2 2 - + - - - 4 6 15 9 11 13 8 2 5 3 10 16 14 12 1 7 |
|
662 2 6 - + - - - 4 6 15 9 11 13 8 2 14 12 1 7 5 3 10 16 |
|
668 2 3 - + - - - 4 7 9 14 10 13 3 8 15 12 6 1 5 2 16 11 |
|
678 2 3 - + - - - 4 7 14 9 10 13 8 3 5 2 11 16 15 12 1 6 |
|
679 2 5 - + - - - 4 7 14 9 10 13 8 3 15 12 1 6 5 2 11 16 |
|
768 2 0 + - - - - 5 2 16 11 15 12 6 1 4 7 9 14 10 13 3 8 |
|
779 2 0 + - - - - 5 3 16 10 14 12 7 1 4 6 9 15 11 13 2 8 |
|
818 2 0 + - - - - 6 1 15 12 16 11 5 2 3 8 10 13 9 14 4 7 |
|
844 2 0 + - - - - 6 4 15 9 13 11 8 2 3 5 10 16 12 14 1 7 |
|
112 3 0 + - - - - 1 8 12 13 14 11 7 2 15 10 6 3 4 5 9 16 |
|
113 3 0 + - - - - 1 8 12 13 15 10 6 3 14 11 7 2 4 5 9 16 |
|
120 3 1 - + - - - 1 8 14 11 12 13 7 2 15 10 4 5 6 3 9 16 |
|
122 3 1 - + - - - 1 8 14 11 15 10 4 5 12 13 7 2 6 3 9 16 |
|
124 3 1 - + - - - 1 8 15 10 12 13 6 3 14 11 4 5 7 2 9 16 |
|
126 3 1 - + - - - 1 8 15 10 14 11 4 5 12 13 6 3 7 2 9 16 |
|
175 3 5 - - + - - 1 12 8 13 14 7 11 2 15 6 10 3 4 9 5 16 |
|
176 3 5 - - + - - 1 12 8 13 15 6 10 3 14 7 11 2 4 9 5 16 |
|
183 3 8 - - + - - 1 12 14 7 15 6 4 9 8 13 11 2 10 3 5 16 |
|
185 3 7 - - + - - 1 12 15 6 14 7 4 9 8 13 10 3 11 2 5 16 |
|
203 3 7 - - + - - 1 14 8 11 15 4 10 5 12 7 13 2 6 9 3 16 |
|
206 3 8 - - + - - 1 14 12 7 15 4 6 9 8 11 13 2 10 5 3 16 |
|
289 3 0 + - - - - 2 7 11 14 13 12 8 1 16 9 5 4 3 6 10 15 |
|
290 3 0 + - - - - 2 7 11 14 16 9 5 4 13 12 8 1 3 6 10 15 |
|
297 3 3 - + - - - 2 7 13 12 11 14 8 1 16 9 3 6 5 4 10 15 |
|
299 3 3 - + - - - 2 7 13 12 16 9 3 6 11 14 8 1 5 4 10 15 |
|
306 3 2 - + - - - 2 7 16 9 11 14 5 4 13 12 3 6 8 1 10 15 |
|
308 3 2 - + - - - 2 7 16 9 13 12 3 6 11 14 5 4 8 1 10 15 |
|
360 3 5 - - + - - 2 11 7 14 13 8 12 1 16 5 9 4 3 10 6 15 |
|
361 3 5 - - + - - 2 11 7 14 16 5 9 4 13 8 12 1 3 10 6 15 |
|
368 3 12 - - + - - 2 11 13 8 16 5 3 10 7 14 12 1 9 4 6 15 |
|
377 3 7 - - + - - 2 11 16 5 13 8 3 10 7 14 9 4 12 1 6 15 |
|
392 3 7 - - + - - 2 13 7 12 16 3 9 6 11 8 14 1 5 10 4 15 |
|
395 3 12 - - + - - 2 13 11 8 16 3 5 10 7 12 14 1 9 6 4 15 |
|
476 3 5 - + - - - 3 6 13 12 10 15 8 1 16 9 2 7 5 4 11 14 |
|
478 3 5 - + - - - 3 6 13 12 16 9 2 7 10 15 8 1 5 4 11 14 |
|
487 3 3 - + - - - 3 6 16 9 10 15 5 4 13 12 2 7 8 1 11 14 |
|
489 3 3 - + - - - 3 6 16 9 13 12 2 7 10 15 5 4 8 1 11 14 |
|
535 3 13 - - + - - 3 10 13 8 16 5 2 11 6 15 12 1 9 4 7 14 |
|
539 3 8 - - + - - 3 10 16 5 13 8 2 11 6 15 9 4 12 1 7 14 |
|
558 3 8 - - + - - 3 13 6 12 16 2 9 7 10 8 15 1 5 11 4 14 |
|
562 3 13 - - + - - 3 13 10 8 16 2 5 11 6 12 15 1 9 7 4 14 |
|
628 3 6 - + - - - 4 5 14 11 9 16 7 2 15 10 1 8 6 3 12 13 |
|
632 3 6 - + - - - 4 5 14 11 15 10 1 8 9 16 7 2 6 3 12 13 |
|
635 3 5 - + - - - 4 5 15 10 9 16 6 3 14 11 1 8 7 2 12 13 |
|
637 3 5 - + - - - 4 5 15 10 14 11 1 8 9 16 6 3 7 2 12 13 |
|
695 3 11 - - + - - 4 9 14 7 15 6 1 12 5 16 11 2 10 3 8 13 |
|
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