## Down Under Upside Down - a Sudoku puzzle

Post puzzles for others to solve here.
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.

udosuk

Posts: 2698
Joined: 17 July 2005

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.
eleven

Posts: 1907
Joined: 10 February 2008

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.
eleven

Posts: 1907
Joined: 10 February 2008

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 true1) r1c3=9,r2c6=7,r3c9=82) r4c3=6,r5c6=4,r6c9=53) r5c2=8,r6c5=9,r4c8=7Singles to finish`
Glyn

Posts: 357
Joined: 26 April 2007

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`
eleven

Posts: 1907
Joined: 10 February 2008

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.
Glyn

Posts: 357
Joined: 26 April 2007

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.
eleven

Posts: 1907
Joined: 10 February 2008

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.
udosuk

Posts: 2698
Joined: 17 July 2005

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.
Glyn

Posts: 357
Joined: 26 April 2007

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.
Glyn

Posts: 357
Joined: 26 April 2007

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 . | +-------+-------+-------+`
eleven

Posts: 1907
Joined: 10 February 2008

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
Glyn

Posts: 357
Joined: 26 April 2007

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.
udosuk

Posts: 2698
Joined: 17 July 2005

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.
Glyn

Posts: 357
Joined: 26 April 2007

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.
Mauricio

Posts: 1174
Joined: 22 March 2006

PreviousNext