The New Sudoku Players' Forum

by **gsf** » Mon Nov 24, 2008 7:59 pm

Mauricio wrote:Have fun with the following puzzles:
Each one has 6 distinct automorphisms (one of them being the trivial one). See if you can find them all.

my solver -f%#aC gets these automorphism counts, respectively

Code: Select all: 6 18 6 54 54 18 54 6 6 6 108

by **eleven** » Tue Nov 25, 2008 5:19 am

Mauricio wrote:Have fun with the following puzzles:

Thanks for the puzzles.

As to the first one (only boxes in band 1 might map on one another)

Code: Select all: +-------+-------+-------+ | . . . | . . 1 | . 2 . | | . 1 . | . . . | . . 3 | | . . 2 | . 3 . | . . . | +-------+-------+-------+ | . . 4 | . 5 . | 6 . . | | 5 . . | . . 7 | . 8 . | | . 6 . | 8 . . | . . 9 | +-------+-------+-------+ | . . 7 | . 8 . | 9 . . | | 4 . . | . . 9 | . 6 . | | . 5 . | 7 . . | . . 4 | +-------+-------+-------+

2 cyclic column changes (123,456) and one for rows 123 lead to

Code: Select all: +-------+-------+-------+ | . . 1 | . . . | . . 3 | | 2 . . | 3 . . | . . . | | . . . | . 1 . | . 2 . | +-------+-------+-------+ | 4 . . | 5 . . | 6 . . | | . 5 . | . 7 . | . 8 . | | . . 6 | . . 8 | . . 9 | +-------+-------+-------+ | 7 . . | 8 . . | 9 . . | | . 4 . | . 9 . | . 6 . | | . . 5 | . . 7 | . . 4 | +-------+-------+-------+

which is a well known symmetry with number cycles (132)(479)(586), the mapping is given by cyclic stack changes and row changes in all bands.

I dont know now, where the other 3 automorphisms are, but its already easy to solve.
[Added: ah, band 2 maps to band 3 with (457896)]

by **Mauricio** » Tue Nov 25, 2008 9:53 am

gsf wrote:
Mauricio wrote:Have fun with the following puzzles:
Each one has 6 distinct automorphisms (one of them being the trivial one). See if you can find them all.

my solver -f%#aC gets these automorphism counts, respectively
Code: Select all
6 18 6 54 54 18 54 6 6 6 108

I guess that counts the number of automorphisms of the solutions, not of the puzzles.

by **eleven** » Tue Nov 25, 2008 10:12 pm

Btw, i found [edit:] 2 automorphisms of the puzzle above as described (starting from the equivalent puzzle shown, A1 is " bands right, rows down in all bands, number cycling", A2 is change bands 23, number cycling [edit 2: when i retried it now , i needed band/stack/row/column switches and number cycles (13)(2)(46)(59)(78)]).
You can apply A1 2 times and you can mix them, e.g. A1A2A1 to get 6 different automorphic mappings. But should we count that ?
We also dont count the identity more times, though we can get it with A2A2 or A1A1A1.

[Added:]
The 3rd puzzle was quite interesting.

Code: Select all: +-------+-------+-------+ | . . . | . . 1 | . . 2 | | . . 3 | . . . | . 4 . | | 5 . . | . 6 . | 7 . . | +-------+-------+-------+ | . . 6 | . . . | . . 5 | | . 1 . | . . 7 | . . . | | 8 . . | 4 . . | . 9 . | +-------+-------+-------+ | . . 4 | . . 9 | . . . | | . . . | . 5 . | . . 8 | | . 2 . | 3 . . | 1 . . | +-------+-------+-------+

With bands down (cyclically), stacks right and number cycles (154)(269)(378) you have this familiar symmetry of cyclic boxes mapping 159|267|348
But this does not easily solve it.

If you move the bands up however you have a diagonal symmetry, and its solved with singles.

Code: Select all: +-------+-------+-------+ | . . 6 | . . . | . . 5 | | . 1 . | . . 7 | . . . | | 8 . . | 4 . . | . 9 . | +-------+-------+-------+ | . . 4 | . . 9 | . . . | | . . . | . 5 . | . . 8 | | . 2 . | 3 . . | 1 . . | +-------+-------+-------+ | . . . | . . 1 | . . 2 | | . . 3 | . . . | . 4 . | | 5 . . | . 6 . | 7 . . | +-------+-------+-------+

by **udosuk** » Wed Nov 26, 2008 4:52 am

I think these are the 6 automorphisms of the 1st puzzle:

Code: Select all: .....1.2. .1......3 ..2.3.... ..4.5.6.. 5....7.8. .6.8....9 ..7.8.9.. 4....9.6. .5.7....4 Automorphism 1 (trivial) r123456789 c123456789 d123456789 -> d123456789 Automorphism 2 r231564897 c456789123 d123456789 -> d231968457 Automorphism 3 r312645978 c789123456 d123456789 -> d312785964 Automorphism 4 r132987654 c132798465 d123456789 -> d213547698 Automorphism 5 r321879546 c798465132 d123456789 -> d321694875 Automorphism 6 r213798465 c465132798 d123456789 -> d132879546

by **eleven** » Wed Nov 26, 2008 6:00 am

Thanks, i see, what is meant.

I cannot check that in detail now, but i suppose, your automorphisms 2 and 4 are equivalent to the two i found.
And you get number 3 by applying 2 two times, and number 5 by applying 2 and 4 and number 6 is 3+4.

What i wanted to say is, that there are only 2 relevant automorphisms, one with cycle 3 (to get the identity again) and the other with cycle 2. The rest are combinations of them. From a solvers point of view, those will not give you any more useful information.

So i am thinking in symmetries, not automorphisms.

by **udosuk** » Wed Nov 26, 2008 7:06 am

Yep, there are only 2 simultaneous symmetry classes for each puzzle, one with cycle-2 & one with cycle-3. Together they form 2x3=6 automorphisms.

I take it you haven't noticed what the cycle-2 symmetry really was?

Try reordering the rows/columns as:

Code: Select all: r456213789 c564213789

And you can do a similar thing for puzzle 4... :idea:

by **eleven** » Thu Nov 27, 2008 2:12 am

Not sure, what you mean. I tried nr 4 now. One symmetry is obvious in the given puzzle:

Code: Select all: +-------+-------+-------+ | . . . | . . 1 | . . 2 | | . 1 . | 3 . . | 4 . . | | 2 . . | . 4 . | . 5 . | +-------+-------+-------+ | . . 6 | . . . | . . 5 | | 7 . . | . 5 . | 2 . . | | . 8 . | 6 . . | . 7 . | +-------+-------+-------+ | . . 8 | . . 3 | . . . | | 6 . . | 9 . . | . 8 . | | . 9 . | . 1 . | 3 . . | +-------+-------+-------+

A1: bands down, stacks right, numbers (158)(263)(479)
This also can be used for solving relatively easy, boxes 149|267|348 map cyclically.

Since we have 3 boxes with 2 givens, they either can map cyclically or by mirroring in the diagonal or by exchanging two of them.
Diagonal symmetry is not possible here, because there are no candidates for the diagonal: All numbers appear at least twice in a band or stack.

So at last you can try to swap two of the 2 number boxes. It does not matter which ones here, because if one mapping works, you can get the others with the first symmetry.
Exchanging 2 boxes in the diagonal can be done by swapping the 2 bands and stacks. (Maybe you meant this, but i dont know, why this mapping also swaps rc12 and cycles c456). Then (if the numbers cant match) with row/col changes you can try to get other pattern matches. In this case with
A2: switch bands 12, stacks 12, rows 23,56,89, cols 12,45,78
you get

Code: Select all: +-------+-------+-------+ | . . . | . . 6 | . . 5 | | . 6 . | 8 . . | 7 . . | | 5 . . | . 7 . | . 2 . | +-------+-------+-------+ | . . 1 | . . . | . . 2 | | 4 . . | . 2 . | 5 . . | | . 3 . | 1 . . | . 4 . | +-------+-------+-------+ | . . 3 | . . 8 | . . . | | 1 . . | 9 . . | . 3 . | | . 9 . | . 6 . | 8 . . | +-------+-------+-------+

and the number cycles (16)(25)(38)(47)(9) map the original puzzle here.

This symmetry is hard to use for solving, because there is no easy way (i know) to see, where each cell maps to.

[Added:] I just noticed, that this second symmetry leaves r7c9 unchanged (so it must be 9). But it cant be equivalent to the 180 degree symmetry.
[Added 2:] No wait, think it is. (I really need a better tool than an editor.)

by **udosuk** » Thu Nov 27, 2008 8:15 pm

Mauricio wrote:Have fun with the following puzzles:
Code: Select all
.....1.2..1......3..2.3......4.5.6..5....7.8..6.8....9..7.8.9..4....9.6..5.7....4 .....1.2..1......3..2.3......45...6..5...78..6...8...9..97...8..6...49..4...5...7 .....1..2..3....4.5...6.7....6.....5.1...7...8..4...9...4..9.......5...8.2.3..1.. .....1..2.1.3..4..2...4..5...6.....57...5.2...8.6...7...8..3...6..9...8..9..1.3.. .....1..2.1.3..4..2...5..6...7.....65...6.2...8.7...9...8..3...7..9...8..4..1.3.. .....1..2.3.2..4..5...6..1...7.....48...5.7...4.9...2...8..6...6..1...9..7..8.3.. .....1.2...32....4.5..6.1...3......7..6..83..7...1..9...5.8....8....9..2.4.5...7. ..1..2..3.2..3..4.5..1..6....5..7..6.1..6..5.8..9..7....8..9..4.9..7..8.4..2..3.. ..1..2.3..3.4....25...6.1...4...7..8..8.4.9..7..2...1...5.9...63....5.9..7.6..8.. .....1..2..3.2..4..5.6..7....7.....1.4...5.7.8...9.6....1..8....8..4...96..2...3. .....1..2.1.3..4..2...5..6...7.....64...6.2...8.7...5...8..3...7..4...8..5..1.3..

The following row/column rearrangement will expose all symmetry classes associated with the corresponding puzzle:

Code: Select all: r978123456 c213546879 r897123456 c213546879 r456789123 c123456789 r213546879 c132465798 r213546879 c132465798 r213546879 c132465798 r213546879 c213546879 r231564897 c123456789 r123456789 c123456789 r456123789 c123456789 r213546879 c132465798

Some of them are dead easy and some a little bit more tricky.

For #4 I still can't find any easy path to crack it, even applying symmetrical tricks on both classes.

by **udosuk** » Sun Nov 30, 2008 2:15 am

Okay, let's get things into perspective regarding the 11 doubly-symmetrical puzzles posted by Mauricio.

Two of them (1,2) have band/stack-permutation symmetry plus 180-degree-rotation symmetry
Four of them (3,8,9,10) have block-(diagonal-)permutation symmetry plus diagonal-reflection symmetry
Five of them (4,5,6,7,11) have block-(diagonal-)permutation symmetry plus 180-degree-rotation symmetry

The first 2 groups are all very easy to solve, as in the following summary:

Puzzle 1 - solvable by gludo+naked pairs
Puzzle 2 - solvable by gludo+naked pairs
Puzzle 3 - solvable by diagonal placements+singles
Puzzle 8 - solvable by triples or diagonal placements+naked singles
Puzzle 9 - solvable by diagonal placements+naked pairs
Puzzle 10 - solvable by naked pairs or diagonal placements+singles

However, the 3rd group (of 5 puzzles) are all quite tricky, as we have not (yet) developed any powerful technique for "block symmetry" that's comparable to gludo. Here I'll list out my solution paths for each of them, in ascending order of their difficulties (as perceived by me):

Code: Select all: Puzzle 5: r213546879 c132465798 +----------------------+----------------------+----------------------+ | 689 569 1 | 3 26789 2789 | 4 5789 57 | | 34689 34569 35679 | 468 1 4789 | 5789 2 357 | | 2 349 379 | 48 4789 5 | 1789 13789 6 | +----------------------+----------------------+----------------------+ | 5 1349 39 | 148 489 6 | 2 13478 1347 | | 1349 7 239 | 12458 24589 23489 | 158 6 1345 | | 1346 12346 8 | 7 245 234 | 15 1345 9 | +----------------------+----------------------+----------------------+ | 7 12356 2356 | 9 2456 24 | 156 145 8 | | 169 8 2569 | 2456 3 247 | 15679 14579 12457 | | 69 2569 4 | 2568 25678 1 | 3 579 257 | +----------------------+----------------------+----------------------+

GSP (Gurth's Symmetrical Placement): r5c5=4
BkS (Block Symmetry): r8c8=5, r2c2=9
Now r1c2 from {56} & r1c9 from {57} must have 6 or 7
=> r1c1+r9c9 can't be [67], must be [82]
BkS: r3c3+r4c4+r6c6+r7c7=[7136]
Naked Pair @ c2: r34c2={34}
All singles from here.

Code: Select all: Puzzle 6: r213546879 c132465798 +----------------------+----------------------+----------------------+ | 179 169 3 | 2 5789 79 | 4 56789 5678 | | 479 469 689 | 34578 1 3479 | 5689 2 35678 | | 5 249 289 | 3478 34789 6 | 89 3789 1 | +----------------------+----------------------+----------------------+ | 8 12369 1269 | 346 234 5 | 7 1369 36 | | 1239 7 12569 | 368 238 123 | 15689 4 3568 | | 13 1356 4 | 9 378 137 | 1568 13568 2 | +----------------------+----------------------+----------------------+ | 6 2345 25 | 1 23457 2347 | 258 578 9 | | 12349 8 1259 | 3457 6 23479 | 125 157 457 | | 1249 12459 7 | 45 2459 8 | 3 156 456 | +----------------------+----------------------+----------------------+

GSP: r5c5=3
BkS: r8c8=5, r2c2=9
SPP (Symmetrical Pointing Pair): r3c3+r7c7={28} => r3c7=9
BkS: r6c1=3, r9c4=5
Hidden Single @ b2: r1c5=5
SPP: r2c1+r8c9={47} => r8c1<>4
BkS: r2c4<>8, r5c7<>1
Intersection: 8 @ r3,b2 locked @ r3c45
All singles from here.

Code: Select all: Puzzle 11: r213546879 c132465798 +----------------------+----------------------+----------------------+ | 5689 569 1 | 3 26789 2789 | 4 5789 79 | | 35689 34569 34679 | 689 1 4789 | 5789 2 379 | | 2 349 3479 | 89 4789 5 | 1789 13789 6 | +----------------------+----------------------+----------------------+ | 4 1359 39 | 1589 589 6 | 2 13789 1379 | | 1359 7 239 | 12589 24589 23489 | 189 6 1349 | | 1369 12369 8 | 7 249 2349 | 19 1349 5 | +----------------------+----------------------+----------------------+ | 7 12369 2369 | 4 2569 29 | 1569 159 8 | | 169 8 2469 | 2569 3 279 | 15679 14579 12479 | | 69 2469 5 | 2689 26789 1 | 3 479 2479 | +----------------------+----------------------+----------------------+

GSP: r5c5=9
BkS: r8c8=9, r2c2=9
BkS: r2c4+r5c7+r8c1={168} => 1 locked @ r5c7+r8c1
Also 1 @ c9 locked @ r458c9
Grouped Turbot Fish: r8c19 can't be both 1
=> At least one of r45c9+r5c7 must be 1
=> r4c8+r7c78, seeing r45c9+r5c7, can't have 1
Intersection: 1 @ r6,b4 locked @ r6c12
Naked Pair @ b4: r4c2+r5c1={35} => r5c3=2
BkS: r8c6=7, r2c9=3
All singles from here.

Code: Select all: Puzzle 7: r213546879 c213546879 +----------------------+----------------------+----------------------+ | 16789 169 3 | 579 2 57 | 568 56789 4 | | 6789 469 4789 | 34579 34789 1 | 2 56789 35689 | | 5 249 24789 | 6 34789 347 | 38 1 389 | +----------------------+----------------------+----------------------+ | 129 12459 6 | 24579 479 8 | 145 3 15 | | 3 12459 12489 | 2459 469 2456 | 14568 24568 7 | | 28 7 248 | 1 346 23456 | 9 24568 568 | +----------------------+----------------------+----------------------+ | 167 8 17 | 347 13467 9 | 13456 456 2 | | 12679 12369 5 | 8 13467 23467 | 1346 469 1369 | | 4 12369 129 | 23 5 236 | 7 689 13689 | +----------------------+----------------------+----------------------+

GSP: r5c5=4
BkS: r8c8=6, r2c2=9
BkS: r3c7+r6c1+r9c4={238} => 2 locked @ r6c1+r9c4
Also 2 @ c8,b6 locked @ r56c8
Turbot Fish: r6c18 can't be both 2
=> At least one of r5c8+r9c4 must be 2
=> r5c4, seeing r5c8+r9c4, can't be 2
XYZ-wing: r1c4 from {579}, r1c6 from {57}, r5c4 from {59}
=> r1c6+r25c4 can't be [759]
=> r2c4, seeing r1c46+r5c4, can't be 5
Intersection: 5 @ r1,b2 locked @ r1c46
Now 5 @ r2,b3 locked @ r2c89
Also 5 @ c2,b4 locked @ r45c2
Turbot Fish: r2c8+r5c2 can't be both 5 (BkS)
=> At least one of r2c9+r4c2 must be 5
=> r4c9, seeing r2c9+r4c2, can't be 5, must be 1
Intersection: 1 @ r9,b7 locked @ r9c23
All singles from here.

Code: Select all: Puzzle 4: r213546879 c132465798 +-------------------------+-------------------------+-------------------------+ | 589 579 1 | 3 256789 26789 | 4 6789 69 | | 34589 34579 34567 | 578 1 6789 | 6789 2 369 | | 2 379 367 | 78 6789 4 | 16789 136789 5 | +-------------------------+-------------------------+-------------------------+ | 7 1349 34 | 148 489 5 | 2 134689 13469 | | 1349 6 234 | 12478 24789 23789 | 189 5 1349 | | 13459 123459 8 | 6 249 239 | 19 1349 7 | +-------------------------+-------------------------+-------------------------+ | 6 123457 23457 | 9 2457 27 | 157 147 8 | | 145 8 2457 | 2457 3 267 | 15679 14679 12469 | | 45 2457 9 | 24578 245678 1 | 3 467 246 | +-------------------------+-------------------------+-------------------------+

(Quick and dirty path)
Grouped Turbot Chain: 1 @ c9 locked @ r458c9
=> r6c78 & r8c1 can't both have 1
Now 1 @ r6 locked @ r6c1278 & 1 @ b7 locked @ r7c2+r8c1
=> At least one of r6c12+r7c2 must have 1
=> r4c2, seeing r6c12+r7c2, can't be 1
180 Symmetry: r6c8 can't be 3
BkS: r9c2 can't be 2
Hidden Pair @ c2: r67c2={12}
All singles from here.

For #4 there ought to be a more elegant way to crack it (I hate anything associated with the word "chain"), but I don't have time to dig deep now. Good luck to whoever trying it (eleven? Glyn? Mauricio?).

PS: Wholehearted regards for Glyn's mum who has encountered a little mishap recently. Fingers crossed for a great recovery!

Added later:

Here is an alternative path for #4:

GSP: r5c5=7
BkS: r8c8=9, r2c2=4
Now 2 @ c4 locked @ r589c4
=> r5c3+r7c6 can't both be 2
BkS: r5c3+r78c6 can't be [226]
=> r78c6 can't be [26], must have 7
=> 7 @ c6,b8 locked @ r78c6
X-wing: 7 @ c28 locked @ r19c28
Again, 2 @ c4 locked @ r589c4
=> r5c3+r7c5 can't both be 2
BkS: r4c23+r5c3+r7c5 can't be [3422]
=> r4c2 can't be 3
Now r3c2 from {39}, r4c2 from {19}
HEUR (Half Emerald Unlocking Rectangle): r37c2 can't be [31]
=> r347c2 can't be [391]
=> r7c2 can't be 1
Hidden Single @ b7: r8c1=1
Intersection: 1 @ c9,b6 locked @ r45c9
All singles from here.

It is not necessarily more elegant than the "quick and dirty path", but for me the critical moves are somehow easier to spot.

:idea:

by **Mauricio** » Fri Dec 05, 2008 4:50 am

Directly from Red Ed's class 10

Code: Select all: +-------+-------+-------+ | . . . | . . 1 | . . 2 | | . . . | 3 . . | 4 . . | | . . . | . 5 . | . 6 . | +-------+-------+-------+ | . . 1 | . . 7 | . 8 . | | 3 . . | 9 . . | . . 7 | | . 5 . | . 8 . | 9 . . | +-------+-------+-------+ | . . 4 | 7 . . | . . 1 | | 6 . . | . 9 . | 3 . . | | . 2 . | . . 8 | . 5 . | +-------+-------+-------+

Code: Select all: +-------+-------+-------+ | . . 1 | . . 2 | . . 3 | | 2 . . | 4 . . | 5 . . | | . 4 . | . 1 . | . 6 . | +-------+-------+-------+ | . . 3 | . . 7 | . 2 . | | 5 . . | 8 . . | . . 4 | | . 6 . | . 9 . | 1 . . | +-------+-------+-------+ | . . 4 | . . 9 | . . 8 | | 1 . . | 7 . . | 9 . . | | . 2 . | . 8 . | . 7 . | +-------+-------+-------+

by **udosuk** » Sun Dec 07, 2008 4:41 am

Mauricio wrote:
Code: Select all
+-------+-------+-------+ | . . 1 | . . 2 | . . 3 | | 2 . . | 4 . . | 5 . . | | . 4 . | . 1 . | . 6 . | +-------+-------+-------+ | . . 3 | . . 7 | . 2 . | | 5 . . | 8 . . | . . 4 | | . 6 . | . 9 . | 1 . . | +-------+-------+-------+ | . . 4 | . . 9 | . . 8 | | 1 . . | 7 . . | 9 . . | | . 2 . | . 8 . | . 7 . | +-------+-------+-------+

This (2nd) one is very easy. Just move the top band to the bottom (i.e. rearrange the rows into r456789123), then you have reflective symmetry about the non-leading diagonal. Very easy to solve after that diagonal is filled in.

However, the 1st one is not an automorphic puzzle I'm afraid. The solution might be highly automorphic, but the puzzle doesn't match any automorphism. I've tried the 6 main types (180, 90, diagonal, sticks, bands, blocks), none of them works. Perhaps a mistake from Mauricio?

by **eleven** » Sun Dec 07, 2008 6:59 am

The symmetry both puzzles have, is new for me. Just move the rows in the bands down cyclically and the columns in the stacks right, i.e.
r123456789 -> r231564897
c123456789 -> c231564897
The number cycles are (134)(246)(798) and (124)(356)(789) resp.

But for the first puzzle i only found one elimination using the symmetry (r1c7<>7, because r1c7=7 places all 7's and 9's, but with a contradiction in box 1).

by **udosuk** » Sun Dec 07, 2008 8:05 pm

Thanks eleven! I totally missed that obvious automorphism. Was too keen on morphing for "easy to use" symmetries.

I guess that makes the 7th main type of automorphism. Perhaps we could call it "mini-diagonal" (although it also includes "mini-broken-diagonals"

).

eleven wrote:But for the first puzzle i only found one elimination using the symmetry (r1c7<>7, because r1c7=7 places all 7's and 9's, but with a contradiction in box 1).

I don't see any obvious contradiction. Could you elaborate more please?

From what I see placing 7 @ r1c7 enables one to place all {789} with 2 choices on c123. However the eventual contradictions all need to bring {135} or {246} into consideration, and occur after a whole lot of other placements, that it's very hard not to be considered as "major T&E".

On the other hand, I have this approach (which I haven't confirmed to work yet):

Code: Select all: Rearrange the rows to r231564897: +-------+-------+-------+ | . . . | 3 . . | 4 . . | | . . . | . 5 . | . 6 . | | . . . | . . 1 | . . 2 | +-------+-------+-------+ | 3 . . | 9 . . | . . 7 | | . 5 . | . 8 . | 9 . . | | . . 1 | . . 7 | . 8 . | +-------+-------+-------+ | 6 . . | . 9 . | 3 . . | | . 2 . | . . 8 | . 5 . | | . . 4 | 7 . . | . . 1 | +-------+-------+-------+

You can see the clue pattern has "diagonal reflective symmetry" now. However, that doesn't make it a puzzle with "diagonal reflective automorphism". Anyway, if we reflect the puzzle along the leading diagonal, only 6 clues are changed, which are all from {246}. That prompts me to think the key to crack this puzzle might be on the digits {246}. Anyway, after peeking at the solution (yes I know this is cheating) I found that the 3 missing digits on the leading diagonal are indeed {246}. If only we can develop a sound logical argument that it is a direct consequence of the "pattern symmetry" above... :idea:

by **eleven** » Mon Dec 08, 2008 12:47 am

udosuk wrote:
eleven wrote:But for the first puzzle i only found one elimination using the symmetry (r1c7<>7, because r1c7=7 places all 7's and 9's, but with a contradiction in box 1).

I don't see any obvious contradiction. Could you elaborate more please?

Sorry, no, i managed to make the same mistake twice

On the other hand, I have this approach (which I haven't confirmed to work yet)...

Interesting, but from my feeling i cant believe, that it works. I guess, that partial symmetry does not say anything about the rest of the grid.

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