About 'Solution-Sets' and 'Transformations': Kanji Setsuda
We do not deal with only a single solution for any type of magic things, but we also deal with a total set which collects every solution of that type. We want to know how many solutions it should have and how it is related with any other solution-set, for they are always important properties we should study about in any case.
We now know there are three levels of solutions in any magic thing as follows:
... (1) 'Primitive', (2) 'Standard', and (3) 'Fundamental'.
Let me show you what they look like, taking some examples of Pan-diagonal MS44.
The 'Primitive' solutions are made without any 'list-forming inequality conditions'.
** Set of 384 'Primitive Solutions' of Pan-Diagonal MS44 (Part) **
1/ 2/ 3/ 4/ 5/ 6/
1 8 10 15 1 8 10 15 1 8 11 14 1 8 11 14 1 8 13 12 1 8 13 12
12 13 3 6 14 11 5 4 12 13 2 7 15 10 5 4 14 11 2 7 15 10 3 6
7 2 16 9 7 2 16 9 6 3 16 9 6 3 16 9 4 5 16 9 4 5 16 9
14 11 5 4 12 13 3 6 15 10 5 4 12 13 2 7 15 10 3 6 14 11 2 7
7/ 8/ 9/ 10/ 11/ 12/
1 12 6 15 1 12 6 15 1 12 7 14 1 12 7 14 1 12 13 8 1 12 13 8
8 13 3 10 14 7 9 4 8 13 2 11 15 6 9 4 14 7 2 11 15 6 3 10
11 2 16 5 11 2 16 5 10 3 16 5 10 3 16 5 4 9 16 5 4 9 16 5
14 7 9 4 8 13 3 10 15 6 9 4 8 13 2 11 15 6 3 10 14 7 2 11
13/ 14/ 15/ 16/ 17/ 18/
1 14 4 15 1 14 4 15 1 14 7 12 1 14 7 12 1 14 11 8 1 14 11 8
8 11 5 10 12 7 9 6 8 11 2 13 15 4 9 6 12 7 2 13 15 4 5 10
13 2 16 3 13 2 16 3 10 5 16 3 10 5 16 3 6 9 16 3 6 9 16 3
12 7 9 6 8 11 5 10 15 4 9 6 8 11 2 13 15 4 5 10 12 7 2 13
19/ 20/ 21/ 22/ 23/ 24/
1 15 4 14 1 15 4 14 1 15 6 12 1 15 6 12 1 15 10 8 1 15 10 8
8 10 5 11 12 6 9 7 8 10 3 13 14 4 9 7 12 6 3 13 14 4 5 11
13 3 16 2 13 3 16 2 11 5 16 2 11 5 16 2 7 9 16 2 7 9 16 2
12 6 9 7 8 10 5 11 14 4 9 7 8 10 3 13 14 4 5 11 12 6 3 13
25/ 26/ 27/ 28/ 29/ 30/
2 7 9 16 2 7 9 16 2 7 12 13 2 7 12 13 2 7 14 11 2 7 14 11
11 14 4 5 13 12 6 3 11 14 1 8 16 9 6 3 13 12 1 8 16 9 4 5
8 1 15 10 8 1 15 10 5 4 15 10 5 4 15 10 3 6 15 10 3 6 15 10
13 12 6 3 11 14 4 5 16 9 6 3 11 14 1 8 16 9 4 5 13 12 1 8
31/ 32/ 33/ 34/ 35/ 36/
2 11 5 16 2 11 5 16 2 11 8 13 2 11 8 13 2 11 14 7 2 11 14 7
7 14 4 9 13 8 10 3 7 14 1 12 16 5 10 3 13 8 1 12 16 5 4 9
12 1 15 6 12 1 15 6 9 4 15 6 9 4 15 6 3 10 15 6 3 10 15 6
13 8 10 3 7 14 4 9 16 5 10 3 7 14 1 12 16 5 4 9 13 8 1 12
37/ 38/ 39/ 40/ 41/ 42/
2 13 3 16 2 13 3 16 2 13 8 11 2 13 8 11 2 13 12 7 2 13 12 7
7 12 6 9 11 8 10 5 7 12 1 14 16 3 10 5 11 8 1 14 16 3 6 9
14 1 15 4 14 1 15 4 9 6 15 4 9 6 15 4 5 10 15 4 5 10 15 4
11 8 10 5 7 12 6 9 16 3 10 5 7 12 1 14 16 3 6 9 11 8 1 14
43/ 44/ 45/ 46/ 47/ 48/
2 16 3 13 2 16 3 13 2 16 5 11 2 16 5 11 2 16 9 7 2 16 9 7
7 9 6 12 11 5 10 8 7 9 4 14 13 3 10 8 11 5 4 14 13 3 6 12
14 4 15 1 14 4 15 1 12 6 15 1 12 6 15 1 8 10 15 1 8 10 15 1
11 5 10 8 7 9 6 12 13 3 10 8 7 9 4 14 13 3 6 12 11 5 4 14
49/ 50/ 51/ 52/ 53/ 54/
3 6 9 16 3 6 9 16 3 6 12 13 3 6 12 13 3 6 15 10 3 6 15 10
10 15 4 5 13 12 7 2 10 15 1 8 16 9 7 2 13 12 1 8 16 9 4 5
8 1 14 11 8 1 14 11 5 4 14 11 5 4 14 11 2 7 14 11 2 7 14 11
13 12 7 2 10 15 4 5 16 9 7 2 10 15 1 8 16 9 4 5 13 12 1 8
55/ 56/ 57/ 58/ 59/ 60/
3 10 5 16 3 10 5 16 3 10 8 13 3 10 8 13 3 10 15 6 3 10 15 6
6 15 4 9 13 8 11 2 6 15 1 12 16 5 11 2 13 8 1 12 16 5 4 9
12 1 14 7 12 1 14 7 9 4 14 7 9 4 14 7 2 11 14 7 2 11 14 7
13 8 11 2 6 15 4 9 16 5 11 2 6 15 1 12 16 5 4 9 13 8 1 12
61/ 62/ 63/ 64/ 65/ 66/
3 13 2 16 3 13 2 16 3 13 8 10 3 13 8 10 3 13 12 6 3 13 12 6
6 12 7 9 10 8 11 5 6 12 1 15 16 2 11 5 10 8 1 15 16 2 7 9
15 1 14 4 15 1 14 4 9 7 14 4 9 7 14 4 5 11 14 4 5 11 14 4
10 8 11 5 6 12 7 9 16 2 11 5 6 12 1 15 16 2 7 9 10 8 1 15
67/ 68/ 69/ 70/ 71/ 72/
3 16 2 13 3 16 2 13 3 16 5 10 3 16 5 10 3 16 9 6 3 16 9 6
6 9 7 12 10 5 11 8 6 9 4 15 13 2 11 8 10 5 4 15 13 2 7 12
15 4 14 1 15 4 14 1 12 7 14 1 12 7 14 1 8 11 14 1 8 11 14 1
10 5 11 8 6 9 7 12 13 2 11 8 6 9 4 15 13 2 7 12 10 5 4 15
73/ 74/ 75/ 76/ 77/ 78/
4 5 10 15 4 5 10 15 4 5 11 14 4 5 11 14 4 5 16 9 4 5 16 9
9 16 3 6 14 11 8 1 9 16 2 7 15 10 8 1 14 11 2 7 15 10 3 6
7 2 13 12 7 2 13 12 6 3 13 12 6 3 13 12 1 8 13 12 1 8 13 12
14 11 8 1 9 16 3 6 15 10 8 1 9 16 2 7 15 10 3 6 14 11 2 7
79/ 80/ 81/ 82/ 83/ 84/
4 9 6 15 4 9 6 15 4 9 7 14 4 9 7 14 4 9 16 5 4 9 16 5
5 16 3 10 14 7 12 1 5 16 2 11 15 6 12 1 14 7 2 11 15 6 3 10
11 2 13 8 11 2 13 8 10 3 13 8 10 3 13 8 1 12 13 8 1 12 13 8
14 7 12 1 5 16 3 10 15 6 12 1 5 16 2 11 15 6 3 10 14 7 2 11
85/ 86/ 87/ 88/ 89/ 90/
4 14 1 15 4 14 1 15 4 14 7 9 4 14 7 9 4 14 11 5 4 14 11 5
5 11 8 10 9 7 12 6 5 11 2 16 15 1 12 6 9 7 2 16 15 1 8 10
16 2 13 3 16 2 13 3 10 8 13 3 10 8 13 3 6 12 13 3 6 12 13 3
9 7 12 6 5 11 8 10 15 1 12 6 5 11 2 16 15 1 8 10 9 7 2 16
91/ 92/ 93/ 94/ 95/ 96/
4 15 1 14 4 15 1 14 4 15 6 9 4 15 6 9 4 15 10 5 4 15 10 5
5 10 8 11 9 6 12 7 5 10 3 16 14 1 12 7 9 6 3 16 14 1 8 11
16 3 13 2 16 3 13 2 11 8 13 2 11 8 13 2 7 12 13 2 7 12 13 2
9 6 12 7 5 10 8 11 14 1 12 7 5 10 3 16 14 1 8 11 9 6 3 16
97/ 98/ 99/ 100/ 101/ 102/
5 4 9 16 5 4 9 16 5 4 14 11 5 4 14 11 5 4 15 10 5 4 15 10
10 15 6 3 11 14 7 2 10 15 1 8 16 9 7 2 11 14 1 8 16 9 6 3
8 1 12 13 8 1 12 13 3 6 12 13 3 6 12 13 2 7 12 13 2 7 12 13
11 14 7 2 10 15 6 3 16 9 7 2 10 15 1 8 16 9 6 3 11 14 1 8
. . . . . .
355/ 356/ 357/ 358/ 359/ 360/
15 10 3 6 15 10 3 6 15 10 5 4 15 10 5 4 15 10 8 1 15 10 8 1
1 8 13 12 4 5 16 9 1 8 11 14 6 3 16 9 4 5 11 14 6 3 13 12
14 11 2 7 14 11 2 7 12 13 2 7 12 13 2 7 9 16 2 7 9 16 2 7
4 5 16 9 1 8 13 12 6 3 16 9 1 8 11 14 6 3 13 12 4 5 11 14
361/ 362/ 363/ 364/ 365/ 366/
16 2 7 9 16 2 7 9 16 2 11 5 16 2 11 5 16 2 13 3 16 2 13 3
3 13 12 6 5 11 14 4 3 13 8 10 9 7 14 4 5 11 8 10 9 7 12 6
10 8 1 15 10 8 1 15 6 12 1 15 6 12 1 15 4 14 1 15 4 14 1 15
5 11 14 4 3 13 12 6 9 7 14 4 3 13 8 10 9 7 12 6 5 11 8 10
367/ 368/ 369/ 370/ 371/ 372/
16 3 6 9 16 3 6 9 16 3 10 5 16 3 10 5 16 3 13 2 16 3 13 2
2 13 12 7 5 10 15 4 2 13 8 11 9 6 15 4 5 10 8 11 9 6 12 7
11 8 1 14 11 8 1 14 7 12 1 14 7 12 1 14 4 15 1 14 4 15 1 14
5 10 15 4 2 13 12 7 9 6 15 4 2 13 8 11 9 6 12 7 5 10 8 11
373/ 374/ 375/ 376/ 377/ 378/
16 5 4 9 16 5 4 9 16 5 10 3 16 5 10 3 16 5 11 2 16 5 11 2
2 11 14 7 3 10 15 6 2 11 8 13 9 4 15 6 3 10 8 13 9 4 14 7
13 8 1 12 13 8 1 12 7 14 1 12 7 14 1 12 6 15 1 12 6 15 1 12
3 10 15 6 2 11 14 7 9 4 15 6 2 11 8 13 9 4 14 7 3 10 8 13
379/ 380/ 381/ 382/ 383/ 384/
16 9 4 5 16 9 4 5 16 9 6 3 16 9 6 3 16 9 7 2 16 9 7 2
2 7 14 11 3 6 15 10 2 7 12 13 5 4 15 10 3 6 12 13 5 4 14 11
13 12 1 8 13 12 1 8 11 14 1 8 11 14 1 8 10 15 1 8 10 15 1 8
3 6 15 10 2 7 14 11 5 4 15 10 2 7 12 13 5 4 14 11 3 6 12 13
* Counts according to the value of n1 *
[ 1: 24] [ 2: 24] [ 3: 24] [ 4: 24] [ 5: 24] [ 6: 24] [ 7: 24] [ 8: 24]
[ 9: 24] [10: 24] [11: 24] [12: 24] [13: 24] [14: 24] [15: 24] [16: 24]
* Total Count = 384 *
This list contains every group of 8 same solutions in it. Those 8 should essentially be counted as a single solution. See the following examples as follows.
P1/ P2/ P3/ P4/
1 8 10 15 14 7 12 1 4 5 11 14 15 6 9 4
12 13 3 6 11 2 13 8 9 16 2 7 10 3 16 5
7 2 16 9 5 16 3 10 6 3 13 12 8 13 2 11
14 11 5 4 4 9 6 15 15 10 8 1 1 12 7 14
P5/ P6/ P7/ P8/
1 12 7 14 15 10 8 1 4 9 6 15 14 11 5 4
8 13 2 11 6 3 13 12 5 16 3 10 7 2 16 9
10 3 16 5 9 16 2 7 11 2 13 8 12 13 3 6
15 6 9 4 4 5 11 14 14 7 12 1 1 8 10 15
Each of them is only a rotated pattern of a certain solution, or only a reflected one of it. All rows and columns of every 8 solutions have the same content pattern with them, don't you find? Even any pandiagonal has the same content, as you see. They should be recognized and counted as a single solution, don't you think?
How can you pick up a single representative solution for the 'Primitive' eight?
Why don't you take such the 'list-forming inequality conditions' as n1<n4, n1<n13, n1<n16 and n2>n5, to choose the 'Standard' solutions out of the 'Primitive' ones?
** List of the 48 Standard Solutions for Pan-diagonal MS44 **
1/ 2/ 3/ 4/ 5/ 6/
1 15 4 14 1 15 4 14 1 15 6 12 1 15 6 12 1 15 10 8 1 15 10 8
12 6 9 7 8 10 5 11 14 4 9 7 8 10 3 13 14 4 5 11 12 6 3 13
13 3 16 2 13 3 16 2 11 5 16 2 11 5 16 2 7 9 16 2 7 9 16 2
8 10 5 11 12 6 9 7 8 10 3 13 14 4 9 7 12 6 3 13 14 4 5 11
7/ 8/ 9/ 10/ 11/ 12/
1 14 4 15 1 14 4 15 1 14 7 12 1 14 11 8 1 12 6 15 1 12 7 14
12 7 9 6 8 11 5 10 8 11 2 13 12 7 2 13 8 13 3 10 8 13 2 11
13 2 16 3 13 2 16 3 10 5 16 3 6 9 16 3 11 2 16 5 10 3 16 5
8 11 5 10 12 7 9 6 15 4 9 6 15 4 5 10 14 7 9 4 15 6 9 4
13/ 14/ 15/ 16/ 17/ 18/
2 16 3 13 2 16 3 13 2 16 5 11 2 16 5 11 2 16 9 7 2 16 9 7
11 5 10 8 7 9 6 12 13 3 10 8 7 9 4 14 13 3 6 12 11 5 4 14
14 4 15 1 14 4 15 1 12 6 15 1 12 6 15 1 8 10 15 1 8 10 15 1
7 9 6 12 11 5 10 8 7 9 4 14 13 3 10 8 11 5 4 14 13 3 6 12
19/ 20/ 21/ 22/ 23/ 24/
2 13 3 16 2 13 3 16 2 13 8 11 2 13 12 7 2 11 5 16 2 11 8 13
11 8 10 5 7 12 6 9 7 12 1 14 11 8 1 14 7 14 4 9 7 14 1 12
14 1 15 4 14 1 15 4 9 6 15 4 5 10 15 4 12 1 15 6 9 4 15 6
7 12 6 9 11 8 10 5 16 3 10 5 16 3 6 9 13 8 10 3 16 5 10 3
25/ 26/ 27/ 28/ 29/ 30/
3 16 2 13 3 16 2 13 3 16 5 10 3 16 5 10 3 16 9 6 3 16 9 6
10 5 11 8 6 9 7 12 13 2 11 8 6 9 4 15 13 2 7 12 10 5 4 15
15 4 14 1 15 4 14 1 12 7 14 1 12 7 14 1 8 11 14 1 8 11 14 1
6 9 7 12 10 5 11 8 6 9 4 15 13 2 11 8 10 5 4 15 13 2 7 12
31/ 32/ 33/ 34/ 35/ 36/
3 13 2 16 3 13 2 16 3 13 8 10 3 13 12 6 4 15 1 14 4 15 1 14
10 8 11 5 6 12 7 9 6 12 1 15 10 8 1 15 9 6 12 7 5 10 8 11
15 1 14 4 15 1 14 4 9 7 14 4 5 11 14 4 16 3 13 2 16 3 13 2
6 12 7 9 10 8 11 5 16 2 11 5 16 2 7 9 5 10 8 11 9 6 12 7
37/ 38/ 39/ 40/ 41/ 42/
4 15 6 9 4 15 6 9 4 15 10 5 4 15 10 5 4 14 1 15 4 14 1 15
14 1 12 7 5 10 3 16 14 1 8 11 9 6 3 16 9 7 12 6 5 11 8 10
11 8 13 2 11 8 13 2 7 12 13 2 7 12 13 2 16 2 13 3 16 2 13 3
5 10 3 16 14 1 12 7 9 6 3 16 14 1 8 11 5 11 8 10 9 7 12 6
43/ 44/ 45/ 46/ 47/ 48/
4 14 7 9 4 14 11 5 5 16 2 11 5 16 3 10 6 15 1 12 6 15 4 9
5 11 2 16 9 7 2 16 4 9 7 14 4 9 6 15 3 10 8 13 3 10 5 16
10 8 13 3 6 12 13 3 15 6 12 1 14 7 12 1 16 5 11 2 13 8 11 2
15 1 12 6 15 1 8 10 10 3 13 8 11 2 13 8 9 4 14 7 12 1 14 7
[Count = 48] OK!
It is not difficult for you to reconstruct the original 'Primitive' solutions from the 'Standard' ones. You can use the next two types of 'Transformation' method: Mirror reflection and Rotation by 90 degrees clockwise. See the next conceptual diagrams.
If you apply them to each Standard solution, you can get every 8 Primitive ones.
[Transformation: Mirror Reflection and Rotation by 90 Degrees Clockwise]
00/ --> 01/ --> 02/ --> 03/
1 2 3 4 13 9 5 1 16 15 14 13 4 8 12 16
5 6 7 8 14 10 6 2 12 11 10 9 3 7 11 15
9 10 11 12 15 11 7 3 8 7 6 5 2 6 10 14
13 14 15 16 16 12 8 4 4 3 2 1 1 5 9 13
|
V 10/ --> 11/ --> 12/ --> 13/
1 5 9 13 4 3 2 1 16 12 8 4 13 14 15 16
2 6 10 14 8 7 6 5 15 11 7 3 9 10 11 12
3 7 11 15 12 11 10 9 14 10 6 2 5 6 7 8
4 8 12 16 16 15 14 13 13 9 5 1 1 2 3 4
You can surely remake all the complete 384 Primitive solutions, since 48 x 8 = 384
When you could have another set of 'list-forming inequality conditions' and another set of effective transformation methods, you can surely presume your 'Fundamental' solutions for all.
You can really pick up the 3 'Fundamental' solutions out of the 384 'Primitive' ones, if you take the next conditions: n1=1; n2>n5, n2>n4 and n5>n13
** The Fundamental Three of Pan-diagonal MS44 **
1/ 2/ 3/
11 8 10 5 11 8 13 8 10 3 13 8 13 12 6 3 13 12
.--.--.--.--. .--.--.--.--. .--.--.--.--.
14| 1|15| 4|14| 1 12| 1|15| 6|12| 1 8| 1|15|10| 8| 1
|--+--+--+--| |--+--+--+--| |--+--+--+--|
7|12| 6| 9| 7|12 7|14| 4| 9| 7|14 11|14| 4| 5|11|14
|--+--+--+--| |--+--+--+--| |--+--+--+--|
2|13| 3|16| 2|13 2|11| 5|16| 2|11 2| 7| 9|16| 2| 7
|--+--+--+--| |--+--+--+--| |--+--+--+--|
11| 8|10| 5|11| 8 13| 8|10| 3|13| 8 13|12| 6| 3|13|12
'--'--'--'--' '--'--'--'--' '--'--'--'--'
14 1 15 4 14 1 12 1 15 6 12 1 8 1 15 10 8 1
[Count = 3] OK!
If you want to reconstruct all the 384 Primitive solutions out of these three, take your careful look at the next conceptual diagrams below. Apply the following two transformation systems step by step to each of the three Fundamental ones.
And you can remake all the 384 solutions, since 3 x 16 x 8 = 48 x 8 = 384
* Conceptual Diagrams of Transformation Systems for PMS44 *
#1. [4x4 Transformations by the 'Row/Column Shift']
00/ 01/ 02/ 03/
1 2 3 4 13 14 15 16 9 10 11 12 5 6 7 8
5 6 7 8 1 2 3 4 13 14 15 16 9 10 11 12
9 10 11 12 5 6 7 8 1 2 3 4 13 14 15 16
13 14 15 16 9 10 11 12 5 6 7 8 1 2 3 4
10/ 11/ 12/ 13/
4 1 2 3 16 13 14 15 12 9 10 11 8 5 6 7
8 5 6 7 4 1 2 3 16 13 14 15 12 9 10 11
12 9 10 11 8 5 6 7 4 1 2 3 16 13 14 15
16 13 14 15 12 9 10 11 8 5 6 7 4 1 2 3
20/ 21/ 22/ 23/
3 4 1 2 15 16 13 14 11 12 9 10 7 8 5 6
7 8 5 6 3 4 1 2 15 16 13 14 11 12 9 10
11 12 9 10 7 8 5 6 3 4 1 2 15 16 13 14
15 16 13 14 11 12 9 10 7 8 5 6 3 4 1 2
30/ 31/ 32/ 33/
2 3 4 1 14 15 16 13 10 11 12 9 6 7 8 5
6 7 8 5 2 3 4 1 14 15 16 13 10 11 12 9
10 11 12 9 6 7 8 5 2 3 4 1 14 15 16 13
14 15 16 13 10 11 12 9 6 7 8 5 2 3 4 1
#2. [Mirror Reflection and Rotation by 90 Degrees Clockwise]
00/ 01/ 02/ 03/
1 2 3 4 13 9 5 1 16 15 14 13 4 8 12 16
5 6 7 8 14 10 6 2 12 11 10 9 3 7 11 15
9 10 11 12 15 11 7 3 8 7 6 5 2 6 10 14
13 14 15 16 16 12 8 4 4 3 2 1 1 5 9 13
10/ 11/ 12/ 13/
1 5 9 13 4 3 2 1 16 12 8 4 13 14 15 16
2 6 10 14 8 7 6 5 15 11 7 3 9 10 11 12
3 7 11 15 12 11 10 9 14 10 6 2 5 6 7 8
4 8 12 16 16 15 14 13 13 9 5 1 1 2 3 4
It seems you can directly reconstruct the complete list of the Standard solutions only by applying the transformation system #1. But you cannot really do that job. Some of the results may inevitably take a form going out of the 'list-forming inequality conditions': n1<n4, n1<n13, n1<n16 and n2>n5
You had better take the list of 384 Primitive solutions at first and then pick up the 48 Standard ones afterward by these 'inequality conditions'.
As far as any transformation system is concerned, you had better take the Primitive solution-set in the process. It is my experience that gives you such a practical advice.
... (on November 5, 2007 by Kanji Setsuda: jag12001@nifty.com)
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