eleven wrote:udosuk wrote:I have a conjecture that CS+\M implies the factor of 8 (CS+RS+\M+/M+HT or CS+RS+QT) and am trying to prove it rigidly.
No more proof needed, you can directly read it from the list (you cant find 134 and 37 without 79 and 86), same for QT+ \M. [Added:] Please dont misunderstand me - i would appreciate a more intuitive proof very much
Here comes a very intuitive proof.
Turns out it's not very difficult if you label the cells appropriately, however the verifying process is a little bit tedious.
Proof: CS+\M implies CS+RS+\M+/M+HT and CS+RS+QTSuppose r456c258 form the axis of the Column Sticks, and d\ (leading diagonal) forms the axis of the Diagonal Mirror.
Label the cell values of the central block as [ABCDEFGHJ].
Note we now have these 2 symmetries:
CS: (B)(E)(H)(AC)(DF)(GJ)
\M: (A)(E)(J)(BD)(CG)(FH)
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The * cells must be from {AEJ}
The # cells must be from {BEH}
* . . | . . . | . . .
. * . | . . . | . . .
. . * | . . . | . . .
----------+----------+----------
. # . | A B C | . # .
. # . | D E F | . # .
. # . | G H J | . # .
----------+----------+----------
. . . | . . . | * . .
. . . | . . . | . * .
. . . | . . . | . . *
Without loss of generality, we can always morph r456c2 to be [EHB] and r456c8 to be [HBE].
Also we can always morph r1c1 and r9c9 to be E, leaving r2c2+r3c3 and r7c7+r8c8 to be from {AJ}.
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The * cells must be from {AJ}
E . . | . . . | . . .
. * . | . . . | . . .
. . * | . . . | . . .
----------+----------+----------
. E . | A B C | . H .
. H . | D E F | . B .
. B . | G H J | . E .
----------+----------+----------
. . . | . . . | * . .
. . . | . . . | . * .
. . . | . . . | . . E
Applying CS & \M symmetry, we can fill in the following cells: r28c456, r3c7, r7c3.
Also, we can label the remaining diagonals cells as Wi,Xi,Yi,Zi with i from 2 to 3.
(r2c2+r3c3 as W23, r1c9+r2c8 as X32, r8c2+r9c1 as Y23, r7c7+r8c8 as Z32.)
Note now with CS Symmetry: [Wi,Yi]=[AC|JG], [Xi,Zi]=[CA|GJ].
Thus with \M symmetry: [Xi,Yi]=[CG|GC]={CG}.
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Wi,Zi must be from {AJ}
Xi,Yi must be from {CG}
E . . | . . . | . . X3
. W2 . | E F D | . X2 .
. . W3 | . . . | E . .
----------+----------+----------
. E . | A B C | . H .
. H . | D E F | . B .
. B . | G H J | . E .
----------+----------+----------
. . E | . . . | Z3 . .
. Y2 . | F D E | . Z2 .
Y3 . . | . . . | . . E
Now we label r1c234568 as K234568 respectively.
Similarly, label r234568c1 as L234568, r245678c9 as M865432, r9c245678 as N865432.
Note now with \M symmetry: [Ki,Li] and [Mi,Ni] must be {AA|BD|CG|FH|JJ}.
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E K2 K3 | K4 K5 K6 | . K8 X3
L2 W2 . | E F D | . X2 M8
L3 . W3 | . . . | E . .
----------+----------+----------
L4 E . | A B C | . H M6
L5 H . | D E F | . B M5
L6 B . | G H J | . E M4
----------+----------+----------
. . E | . . . | Z3 . M3
L8 Y2 . | F D E | . Z2 M2
Y3 N8 . | N6 N5 N4 | N3 N2 E
Then we label r3c245689 as P845623 respectively.
Similarly, label r245689c3 as Q845623, r124568 as R326548, r7c124568 as S326548.
Note now with \M symmetry: [Pi,Qi] and [Ri,Si] must be {AA|BD|CG|FH|JJ}.
Also with CS symmetry: [Ki,Si], [Li,Qi], [Mi,Ri], [Ni,Pi] must be {AC|BB|DF|GJ|HH}.
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E K2 K3 | K4 K5 K6 | R3 K8 X3
L2 W2 Q8 | E F D | R2 X2 M8
L3 P8 W3 | P4 P5 P6 | E P2 P3
----------+----------+----------
L4 E Q4 | A B C | R6 H M6
L5 H Q5 | D E F | R5 B M5
L6 B Q6 | G H J | R4 E M4
----------+----------+----------
S3 S2 E | S6 S5 S4 | Z3 S8 M3
L8 Y2 Q2 | F D E | R8 Z2 M2
Y3 N8 Q3 | N6 N5 N4 | N3 N2 E
Now with the whole grid set up we can start "linking up" the cell relationships.
With [Wi,Yi]=[AC|JG], [Xi,Zi]=[CA|GJ], [Xi,Yi]=[CG|GC]
=> [Wi,Xi,Yi,Zi]=[AGCJ|JCGA]
With [Ki,Li], [Mi,Ni], [Pi,Qi], [Ri,Si] being {AA|BD|CG|FH|JJ}
Also [Ki,Si], [Li,Qi], [Mi,Ri], [Ni,Pi] being {AC|BB|DF|GJ|HH}
=> [Ki,Li,Mi,Ni,Pi,Qi,Ri,Si]=[AAJJGCGC|BDFHHFDB|CGCGJJAA|DBHFDBHF|FHBDFHBD|GCGCAAJJ|HFDBBDFH|JJAACGCG]
And finally we can set out to prove each extra symmetries.
/M:
[Wi,Zi]=[AJ|JA]={AJ}
[Ki,Mi]=[AJ|BF|CC|DH|FB|GG|HD|JA]={AJ|BF|CC|DH|GG}
[Li,Ni]=[AJ|DH|GG|BF|HD|CC|FB|JA]={AJ|BF|CC|DH|GG}
[Pi,Ri]=[GG|HD|JA|DH|FB|AJ|BF|CC]={AJ|BF|CC|DH|GG}
[Qi,Si]=[CC|FB|JA|BF|HD|AJ|DH|GG]={AJ|BF|CC|DH|GG}
Therefore all pairs of cells symmetrical across d/ (non-leading diagonal) must be {AJ|BF|CC|DH|EE|GG}
And all cells on d/ are from {CEG}
Hence we have /M symmetry: (C)(E)(G)(AJ)(BF)(DH)
RS:
[Wi,Xi]=[AG|JC]
[Yi,Zi]=[CJ|GA]
[Ki,Pi]=[AG|BH|CJ|DD|FF|GA|HB|JC]={AG|BH|CJ|DD|FF}
[Li,Ri]=[AG|DD|GA|BH|HB|CJ|FF|JC]={AG|BH|CJ|DD|FF}
[Mi,Qi]=[JC|FF|CJ|HB|BH|GA|DD|AG]={AG|BH|CJ|DD|FF}
[Ni,Si]=[JC|HB|GA|FF|DD|CJ|BH|AG]={AG|BH|CJ|DD|FF}
Therefore within each band, all corresponding cells on pairs of minirows symmetrical across
the central mini-row (r258c456) must be {AG|BH|CJ|DD|EE|FF}
And r258c456 are from {DEF}
Hence we have RS symmetry: (D)(E)(F)(AG)(BH)(CJ)
HT:
[Wi,Zi]=[AJ|JA]={AJ}
[Xi,Yi]=[GC|CG]={CG}
[Ki,Ni]=[AJ|BH|CG|DF|FD|GC|HB|JA]={AJ|BH|CG|DF}
[Li,Mi]=[AJ|DF|GC|BH|HB|CG|FD|JA]={AJ|BH|CG|DF}
[Pi,Si]=[GC|HB|JA|DF|FD|AJ|BH|CG]={AJ|BH|CG|DF}
[Qi,Ri]=[CG|FD|JA|BH|HB|AJ|DF|GC]={AJ|BH|CG|DF}
Therefore all pairs of cells symmetrical across r5c5 must be {AJ|BH|CG|DF|EE}
And r5c5=E
Hence we have HT symmetry: (E)(AJ)(BH)(CG)(DF)
To prove the grid has QT symmetry, we can do the following transformation:
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r789456123
(i.e. swapping top and bottom band)
S3 S2 E | S6 S5 S4 | Z3 S8 M3
L8 Y2 Q2 | F D E | R8 Z2 M2
Y3 N8 Q3 | N6 N5 N4 | N3 N2 E
----------+----------+----------
L4 E Q4 | A B C | R6 H M6
L5 H Q5 | D E F | R5 B M5
L6 B Q6 | G H J | R4 E M4
----------+----------+----------
E K2 K3 | K4 K5 K6 | R3 K8 X3
L2 W2 Q8 | E F D | R2 X2 M8
L3 P8 W3 | P4 P5 P6 | E P2 P3
[Wi,Yi,Zi,Xi]=[ACJG|JGAC]
[Ki,Qi,Ni,Ri]=[ACJG|BFHD|CJGA|DBFH|FHDB|GACJ|HDBF|JGAC]
[Li,Si,Mi,Pi]=[ACJG|DBFH|GACJ|BFHD|HDBF|CJGA|FHDB|JGAC]
Therefore all quadruplets of cells clockwisely symmetrical around r5c5
must be [ACJG|BFHD|CJGA|DBFH|EEEE|FHDB|GACJ|HDBF|JGAC]
And r5c5=E
Hence we have QT symmetry: (E)(ACJG)(BFHD)
(And this isomorph must also have the same CS and RS symmetries as the one above.)
Q.E.D.