The New Sudoku Players' Forum

by **udosuk** » Fri Oct 31, 2008 5:14 pm

Okay, I see the problem of the confusion here. Your interpretation works perfectly from a computer program's point of view, but is counter-intuitive for how a human player thinks (well, at least to me). From now on I'll specify it in this format:

r123564897: re-arrange the original rows in this order (e.g. the old r5 becomes the 4th row)
c456789123: re-arrange the original columns in this order (e.g. the old c4 becomes new 1st column)
d123456789 -> d312645978: map the digits in this manner (e.g. 1 becomes 3, 2 becomes 1)

That's because I think re-arranging rows/columns and mapping digits are two different operations for the human brain, I'm trying to use the most intuitive specification for these two operations respectively.

Also, for you new elimination technique, I haven't described the whole of it:

Eleven's elimination for minirow shifting cyclic triple

If there is minirow shifting within the band (e.g. r7c123 -> r8c456), and there is a cyclic triple of digits {xyz} (x -> y -> z ->x) then x can't appear on a minirow cyclically above or cyclically on the left of a minirow containing y. Ditto for y,z and z,x.

Example of minirow relationships:

r7c123 is cyclically above r8c123, and is cyclically on the left of r7c456.
r9c789 is cyclically above r7c789, and is cyclically on the left of r9c123.

With this technique Mauricio's puzzle #1 can be solved without any Gludo or chains:

Code: Select all: +-------------------+-------------------+-------------------+ | 1 5689 5789 | 2 4679 6789 | 3 4578 4789 | | 679 2 5789 | 478 3 6789 | 589 1 4789 | | 379 389 4 | 178 179 5 | 289 278 6 | +-------------------+-------------------+-------------------+ | 5 389 2389 | 6 179 1379 | 4 278 1278 | | 2346 7 23 | 1345 8 13 | 1256 9 12 | | 469 4689 1 | 457 4579 2 | 568 5678 3 | +-------------------+-------------------+-------------------+ | 2347 1345 6 | 9 1257 1378 | 128 2348 1248 | | 239 1359 2359 | 1358 1256 4 | 7 2368 1289 | | 8 1349 2379 | 137 1267 1367 | 1269 2346 5 | +-------------------+-------------------+-------------------+

Hidden triple @ r5: r5c147={456}
2 @ c1,b7 locked @ r78c1 => r8c1 can't be 3
3 @ c4,b8 locked @ r89c4 => r9c4 can't be 1
1 @ c7,b9 locked @ r79c7 => r7c7 can't be 2
7 @ b7 locked @ r7c1+r9c3 => r9c3 can't be 9
Symmetry: r7c6 can't be 7, r8c9 can't be 8
5 @ r7 locked @ r7c25 => r7c2 can't be 4
Symmetry: r8c5 can't be 5, r9c8 can't be 6
Naked pair @ r7: r7c67={18}
Naked pair @ r8: r8c19={29}
Naked pair @ b7: r7c2+r8c3={35}

All naked singles from here.

:idea:

by **eleven** » Tue Nov 04, 2008 9:54 am

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

This seems to be harder.

by **eleven** » Wed Nov 05, 2008 12:34 am

For those, it was too hard, here is a puzzle with ER 9.8, which can be solved quickly with the technique above and singles.

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

Thanks to my friend, who let his program run for hours to generate a couple of highrated puzzles with this symmetry.

by **Glyn** » Wed Nov 05, 2008 4:29 am

My walkthrough for the first puzzle is a bit long winded so it needs reviewing.

The second one is nice and easy

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

Code: Select all: .------------------.------------------.------------------. | 6 2 13459| 5789 1589 157 | 3789 479 348 | | 1789 578 159 | 4 3 12567| 6789 2679 268 | | 4789 3478 349 | 2789 689 267 | 5 1 23468| :------------------+------------------+------------------: | 1278 9 1236 | 2358 158 1235 | 4 2567 12356| | 5 3468 12346| 2389 7 1234 | 1369 269 1236 | | 1247 347 1234 | 6 1459 12345| 1379 8 1235 | :------------------+------------------+------------------: | 24 456 7 | 1 456 9 | 68 3 4568 | | 49 1 4569 | 35 456 8 | 2 456 7 | | 3 456 8 | 57 2 4567 | 16 456 9 | '------------------'------------------'------------------' We can immediately see using Gludo that all these are true 1) r1c3=9,r2c6=7,r3c9=8 2) r4c3=6,r5c6=4,r6c9=5 3) r5c2=8,r6c5=9,r4c8=7 Singles to finish

by **eleven** » Wed Nov 05, 2008 8:52 am

I needed 3 short (6 cell?) chains for the first.

This one is even higher rated, but very easy either:

Code: Select all: +-------+-------+-------+ | . . . | . . . | . 2 6 | | . 3 4 | . . . | . . . | | . . . | . 1 5 | . . . | +-------+-------+-------+ | . . . | 9 . 4 | . . . | | . . . | . . . | 7 . 5 | | 8 . 6 | . . . | . . . | +-------+-------+-------+ | 3 . . | 7 . 2 | . 9 1 | | . 7 2 | 1 . . | 8 . 3 | | 9 . 1 | . 8 3 | 2 . . | +-------+-------+-------+ ER 10.6

by **Glyn** » Wed Nov 05, 2008 11:17 am

eleven This was my effort on the first. Initial PM

Code: Select all: .---------------------.---------------------.---------------------. | 238 1 2368 | 7 3589 359 | 269 2569 4 | | 347 3467 5 | 139 2 1349 | 8 1679 167 | | 9 2478 248 | 158 1458 6 | 127 3 1257 | :---------------------+---------------------+---------------------: | 1 2378 2389 | 389 389 379 | 5 4 2678 | | 6 5 3489 | 2 1389 1379 | 179 179 178 | | 278 278 289 | 4 6 1579 | 3 1279 1278 | :---------------------+---------------------+---------------------: | 2345 2346 7 | 13569 13459 123459| 1246 8 12356 | | 2345 9 12346 | 1356 1345 8 | 12467 12567 123567| | 23458 23468 123468| 1356 7 12345 | 1246 1256 9 | '---------------------'---------------------'---------------------'

Rows 456 completed minirows must be {456},{123},{789}

Code: Select all: .------------------.------------------.------------------. | 238 1 2368 | 7 3589 39 | 269 2569 4 | | 347 3467 5 | 139 2 1349 | 8 1679 17 | | 9 2478 28 | 158 1458 6 | 127 3 1257 | :------------------+------------------+------------------: | 1 23 23 | 89 89 7 | 5 4 6 | | 6 5 4 | 2 13 13 | 79 79 8 | | 78 78 9 | 4 6 5 | 3 12 12 | :------------------+------------------+------------------: | 2345 2346 7 | 13569 13459 12349| 1246 8 1235 | | 2345 9 1236 | 1356 1345 8 | 12467 12567 12357| | 23458 23468 12368| 1356 7 1234 | 1246 1256 9 | '------------------'------------------'------------------'

Minirow r1c123 is either {123} or {168} => r1c3<>8,r2c6<>9,r3c9<>7
Minirow r2c123 is either {357} or {456} => r2c2<>4,r3c5<>5,r1c8<>6
Minirow r3c123 is either {789} or {249} => r3c2<>2,8;r1c5<>3,9;r2c8<>1,7
Now it gets nasty.

Code: Select all: .------------------.------------------.------------------. | 238 1 236 | 7 58 39 | 269 259 4 | | 347 367 5 | 139 2 134 | 8 69 17 | | 9 47 28 | 158 148 6 | 127 3 125 | :------------------+------------------+------------------: | 1 23 23 | 89 89 7 | 5 4 6 | | 6 5 4 | 2 13 13 | 79 79 8 | | 78 78 9 | 4 6 5 | 3 12 12 | :------------------+------------------+------------------: | 2345 2346 7 | 13569 13459 12349| 1246 8 1235 | | 2345 9 1236 | 1356 1345 8 | 12467 12567 12357| | 23458 23468 12368| 1356 7 1234 | 1246 1256 9 | '------------------'------------------'------------------'

Either r8c3=1 => r9c6=2,r7c9=3,r7c12=2,r7c45=1
or r9c3=1 => r7c6=2,r8c9=3,r8c45=1,r7c12=3,r7c79=1
This is enough to partition {123} in row 7.
therefore r7c456<>3,r7c79<>2
Following the symmetry r8c789<>1,r8c13<>3 and r9c123<>2,r9c46<>1

Code: Select all: .---------------.---------------.---------------. | 238 1 236 | 7 58 39 | 269 259 4 | | 347 367 5 | 139 2 134 | 8 69 17 | | 9 47 28 | 158 148 6 | 127 3 125 | :---------------+---------------+---------------: | 1 23 23 | 89 89 7 | 5 4 6 | | 6 5 4 | 2 13 13 | 79 79 8 | | 78 78 9 | 4 6 5 | 3 12 12 | :---------------+---------------+---------------: | 2345 2346 7 | 1569 1459 1249| 146 8 135 | | 245 9 126 | 1356 1345 8 | 2467 2567 2357| | 3458 3468 1368| 356 7 234 | 1246 1256 9 | '---------------'---------------'---------------'

Either r1c1=8 or r3c3=8 the latter => r3c2=7(Minirow requirement),r6c2=8 therefore r6c1<>8.
Immediately r6c1=7,r4c4=8,r5c7=9 and the resultant singles give.

Code: Select all: .---------------.---------------.---------------. | 238 1 236 | 7 58 39 | 26 259 4 | | 34 367 5 | 139 2 134 | 8 69 17 | | 9 47 28 | 15 148 6 | 127 3 125 | :---------------+---------------+---------------: | 1 23 23 | 8 9 7 | 5 4 6 | | 6 5 4 | 2 13 13 | 9 7 8 | | 7 8 9 | 4 6 5 | 3 12 12 | :---------------+---------------+---------------: | 2345 2346 7 | 1569 145 1249| 146 8 135 | | 245 9 126 | 1356 1345 8 | 2467 256 2357| | 3458 346 1368| 356 7 234 | 1246 1256 9 | '---------------'---------------'---------------'

Minirow r2c123 requirements => r2c2<>3 also r3c5<>1,r1c8<>2

Code: Select all: .---------------.---------------.---------------. | 238 1 236 | 7 58 39 | 26 59 4 | | 34 67 5 | 139 2 134 | 8 69 17 | | 9 47 28 | 15 48 6 | 127 3 125 | :---------------+---------------+---------------: | 1 23 23 | 8 9 7 | 5 4 6 | | 6 5 4 | 2 13 13 | 9 7 8 | | 7 8 9 | 4 6 5 | 3 12 12 | :---------------+---------------+---------------: | 2345 2346 7 | 1569 145 1249| 146 8 135 | | 245 9 126 | 1356 1345 8 | 2467 256 2357| | 3458 346 1368| 356 7 234 | 1246 1256 9 | '---------------'---------------'---------------'

If r7c9=5 then minirow r7c789 ={158} => r7c5=4 and r8c8=5 by symmetry. Contradiction 2 5's in box 9. So r7c9<>5 and by symmetry r8c3<>6,r9c6<>4.
Only valid candidate for minrow r7c789 from (456) is in r7c7 therefore r7c7<>1,r8c1<>2,r9c4<>3.

Code: Select all: .---------------.---------------.---------------. | 238 1 236 | 7 58 39 | 26 59 4 | | 34 67 5 | 139 2 134 | 8 69 17 | | 9 47 28 | 15 48 6 | 127 3 125 | :---------------+---------------+---------------: | 1 23 23 | 8 9 7 | 5 4 6 | | 6 5 4 | 2 13 13 | 9 7 8 | | 7 8 9 | 4 6 5 | 3 12 12 | :---------------+---------------+---------------: | 2345 2346 7 | 1569 145 1249| 46 8 13 | | 45 9 12 | 1356 1345 8 | 2467 256 2357| | 3458 346 1368| 56 7 23 | 1246 1256 9 | '---------------'---------------'---------------'

Either r2c6=4 or r2c1=4 leading to r8c1=5 and r7c7=4 by symmetry. Therefore r7c6<>4.
Singles to finish.

by **eleven** » Wed Nov 05, 2008 11:40 am

Nice, it looks similar to my solution (in the complexity and number of steps), but you seem to use more Gludo moves

Hope i can post my solution tomorrow.

by **udosuk** » Wed Nov 05, 2008 12:09 pm

For the first puzzle...

Glyn wrote:Now it gets nasty.
Code: Select all
.------------------.------------------.------------------. | 238 1 236 | 7 58 39 | 269 259 4 | | 347 367 5 | 139 2 134 | 8 69 17 | | 9 47 28 | 158 148 6 | 127 3 125 | :------------------+------------------+------------------: | 1 23 23 | 89 89 7 | 5 4 6 | | 6 5 4 | 2 13 13 | 79 79 8 | | 78 78 9 | 4 6 5 | 3 12 12 | :------------------+------------------+------------------: | 2345 2346 7 | 13569 13459 12349| 1246 8 1235 | | 2345 9 1236 | 1356 1345 8 | 12467 12567 12357| | 23458 23468 12368| 1356 7 1234 | 1246 1256 9 | '------------------'------------------'------------------'

It doesn't need to get nasty...

From now on I'll denote eleven's cyclic triple minirow trick as eleveno.

Using a combination of gludo & eleveno you can solve this puzzle without chains.

First notice in r789, you can't have the minirows formulated as {123}, {456} & {789}. Therefore using gludo we can conclude each minirow must be consisted of one each member from those 3 sets. Particularly, each minirow must have one member of {123}.

Now notice r7c123 can only have 2 or 3. Using eleveno, we can conclude r7c456 & r8c123 can't have 3.

Symmetry: r8c789 and r9c456 can't have 1, r9c123 & r7c789 can't have 2.

Code: Select all: +-------------------+-------------------+-------------------+ | 238 1 236 | 7 58 39 | 269 259 4 | | 347 367 5 | 139 2 134 | 8 69 17 | | 9 *47 28 | 158 148 6 |*127 3 125 | +-------------------+-------------------+-------------------+ | 1 23 23 | 89 89 7 | 5 4 6 | | 6 5 4 | 2 13 13 |#79 #79 8 | |#78 #78 9 | 4 6 5 | 3 12 12 | +-------------------+-------------------+-------------------+ | 2345 2346 7 | 1569 1459 1249 | 146 8 135 | | 245 9 126 | 1356 1345 8 | 2467 2567 2357 | | 3458 3468 1368 | 356 7 234 | 1246 1256 9 | +-------------------+-------------------+-------------------+

Next, notice the 7 @ r3 is locked @ r3c27.

Therefore r5c7+r6c2 can't be both 7.

Symmetry: r5c78+r6c12 can't be [79]+[87].

Hence r5c78 must be [97], r6c12=[78], r4c45=[89].

Code: Select all: +-------------------+-------------------+-------------------+ | 238 1 236 | 7 58 39 | 26 259 4 | | 34 367 5 | 139 2 134 | 8 69 17 | | 9 47 28 | 15 148 6 | 127 3 125 | +-------------------+-------------------+-------------------+ | 1 23 23 | 8 9 7 | 5 4 6 | | 6 5 4 | 2 13 13 | 9 7 8 | | 7 8 9 | 4 6 5 | 3 12 12 | +-------------------+-------------------+-------------------+ | 2345 2346 7 | 1569 145 1249 | 146 8 135 | | 245 9 126 | 1356 1345 8 | 2467 256 2357 | |-3458 *346 1368 |*356 7 234 | 1246 1256 9 | +-------------------+-------------------+-------------------+

Now focus on r9.

r9c24 can't be both 3, so either r9c2=4|6 or r9c4=5|6.

But r9c12 can't be [54|56] (gludo) and r9c14 can't be [56] (eleveno).

Therefore r9c1 can't be 5.

Now a turbot fish on 5 reduces the puzzle to singles. :idea:

by **Glyn** » Wed Nov 05, 2008 12:47 pm

Very nice udosuk. A much tidier description of the reduction of 1,2,3 in rows 789. The elimination of 5 at r9c1 was very neat. The gludo is obvious but the alternative eleveno producing pincer 5's is something i wouldn't have spotted.

by **Glyn** » Wed Nov 05, 2008 12:57 pm

eleven wrote:This one is even higher rated, but very easy either:
Code: Select all
+-------+-------+-------+ | . . . | . . . | . 2 6 | | . 3 4 | . . . | . . . | | . . . | . 1 5 | . . . | +-------+-------+-------+ | . . . | 9 . 4 | . . . | | . . . | . . . | 7 . 5 | | 8 . 6 | . . . | . . . | +-------+-------+-------+ | 3 . . | 7 . 2 | . 9 1 | | . 7 2 | 1 . . | 8 . 3 | | 9 . 1 | . 8 3 | 2 . . | +-------+-------+-------+ ER 10.6

It is a pussy cat gludo to complete the 3 minirows in rows 123. Then either a naked pair or gludo to complete any of the minirows in rows 456, brings it to singles.

by **eleven** » Thu Nov 06, 2008 5:47 am

Though i only looked at puzzles with ER 9+, most are pretty simple to solve directly with gludo and eleveno.

Perhaps this is a bit more tricky also:

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

by **Glyn** » Thu Nov 06, 2008 8:25 am

eleven Starting PM for puzzle in the previous post

Code: Select all: .---------------------.---------------------.---------------------. | 1679 679 8 | 5 3467 347 | 1349 2 139 | | 1257 3 127 | 2478 478 9 | 6 1458 158 | | 4 2569 269 | 2368 1 238 | 3589 589 7 | :---------------------+---------------------+---------------------: | 12679 8 5 | 2347 347 12347 | 139 169 12369 | | 127 247 12347 | 23478 9 6 | 1358 158 12358 | | 1269 269 12369 | 238 358 12358 | 13589 7 4 | :---------------------+---------------------+---------------------: | 567 1 467 | 9 345678 34578 | 2 4568 568 | | 3 4569 469 | 468 2 458 | 7 145689 15689 | | 8 245679 24679 | 1 4567 457 | 459 3 569 | '---------------------'---------------------'---------------------'

Applying gludo to the whole puzzle we get to here.

Code: Select all: .------------------.------------------.------------------. | 179 679 8 | 5 367 347 | 134 2 139 | | 125 3 127 | 278 478 9 | 6 148 158 | | 4 259 269 | 236 1 238 | 389 589 7 | :------------------+------------------+------------------: | 12 8 5 | 2347 347 12347| 139 169 12369| | 127 247 12347| 23 9 6 | 1358 158 12358| | 1269 269 12369| 238 358 12358| 13 7 4 | :------------------+------------------+------------------: | 567 1 467 | 9 3456 345 | 2 4568 568 | | 3 4569 469 | 468 2 458 | 7 1456 156 | | 8 2456 246 | 1 4567 457 | 459 3 569 | '------------------'------------------'------------------'

Either we have naked pair (37)r1c56
or by gludo we have r1c5=6 => r8c4=6, r7c1=5 (by cyclic rotation) => naked triplet (127)r245c1.
Either way r1c1<>7. By rotation r2c4<>8,r3c7<>9.

Gludo to Rows 123 => r1c2<>9,r2c5<>7,r3c8<>8 (this move is not necessary the next one clinches it).

If r1c2=6 => r2c3=7,r7c1=7 => r8c4=8 by cyclic rotation
If r1c5=6 => r8c4=6.
Either way r8c4<>4. Rotation gives r9c7<>5,r7c1<>6

Singles to follow

by **udosuk** » Thu Nov 06, 2008 4:07 pm

Glyn wrote:eleven Starting PM for puzzle in the previous post
Code: Select all
.---------------------.---------------------.---------------------. | 1679 679 8 | 5 3467 347 | 1349 2 139 | | 1257 3 127 | 2478 478 9 | 6 1458 158 | | 4 2569 269 | 2368 1 238 | 3589 589 7 | :---------------------+---------------------+---------------------: | 12679 8 5 | 2347 347 12347 | 139 169 12369 | | 127 247 12347 | 23478 9 6 | 1358 158 12358 | | 1269 269 12369 | 238 358 12358 | 13589 7 4 | :---------------------+---------------------+---------------------: | 567 1 467 | 9 345678 34578 | 2 4568 568 | | 3 4569 469 | 468 2 458 | 7 145689 15689 | | 8 245679 24679 | 1 4567 457 | 459 3 569 | '---------------------'---------------------'---------------------'

Applying gludo to the whole puzzle we get to here.
Code: Select all
.------------------.------------------.------------------. | 179 679 8 | 5 367 347 | 134 2 139 | | 125 3 127 | 278 478 9 | 6 148 158 | | 4 259 269 | 236 1 238 | 389 589 7 | :------------------+------------------+------------------: | 12 8 5 | 2347 347 12347| 139 169 12369| | 127 247 12347| 23 9 6 | 1358 158 12358| | 1269 269 12369| 238 358 12358| 13 7 4 | :------------------+------------------+------------------: | 567 1 467 | 9 3456 345 | 2 4568 568 | | 3 4569 469 | 468 2 458 | 7 1456 156 | | 8 2456 246 | 1 4567 457 | 459 3 569 | '------------------'------------------'------------------'

From here you can apply eleveno @ r456 to get a lot of eliminations, but they don't help much to solve the puzzle.

My critical step is like this:

By gludo, r3c23 must be {29} or [56], so must have a 5 or a 9.

Therefore r12c1 can't be [95], must have one of {127}.

Thus r12c1 and r45c1 form a "killer naked triple" of {127} @ c1 (this is a killer sudoku technique but shouldn't be too hard to see for vanilla players).

Hence r67c1 can't have {127}.

Hidden single: r7c3=7

Gludo: r2c123 from {1235} must be {123}

All singles from here. :idea:

by **Glyn** » Thu Nov 06, 2008 6:10 pm

udosuk Nice moves introducing a killer technique. I rechecked and found the primary elimination of the two chains r1c1<>7 and r8c4<>4 were enough the cyclic rotation wasn't required, but because they are so trivial I just applied them without thinking.

by **Mauricio** » Mon Nov 24, 2008 5:25 pm

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..

Each one has 6 distinct automorphisms (one of them being the trivial one). See if you can find them all.

Surely a Gludo can kill them.

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