The hardest sudokus (new thread)

Everything about Sudoku that doesn't fit in one of the other sections

Re: The hardest sudokus (new thread)

Postby mith » Sat May 18, 2024 10:07 pm

The entire list (which includes puzzles I would consider degenerate, but which you would not) is here: drive_link

And yes, the vast majority are from Paquita (838544) and all of those are 11.0-11.1 except the single 11.3. There are a few older ones from dobrichev (23), and a few from champagne (4), and then the ones I contributed in between ph1910 and ph2010 (273).

I appreciate that it's a surprising finding, but my script wasn't just spitting our random puzzles. :) My current script is working through these and giving the exact details (digits, cells, number of guardian cells/candidates) for the non-degenerate cases, so I'll provide that when it's done (about 5% so far, it'll be a few days).

I'm not sure there necessarily is a deeper explanation other than no one had hit the right neighborhood for the highest SER puzzles at these clue counts (and as you point out in your recent post, puzzles at higher clue counts tend to have lots of similar puzzles, whether close neighbors or minimals of the same expanded puzzle). At the time, there was a very strong peak in the 21c-23c range. At 28c there are no puzzles at 11.2 except for minimals of Mauricio's 37c, and at 29c+ 11.1 is the highest. There are higher SER puzzles in the 25c-27c range, but they are quite infrequent. Basic neighborhood searching is just more expensive at higher clue counts, and all of the 11.8-11.9 SER puzzles were in the 20c-24c range, so there was little incentive to search outside that range for those looking for the hardest (by SER) puzzles.
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Re: The hardest sudokus (new thread)

Postby denis_berthier » Sun May 19, 2024 3:22 am

.
In the night, my Mac has found 42,097 more tridagons between 100,001 and 200,000, all from Paquita.

I agree with your explanations about the clue count.

At first, I thought of a bug in your program: I had found no non-degenerate tridagon in eleven's gotchi collection, at any SER, and only one in ph2010 at any SER ≥ 11.2.
I thought it would require a miracle to find many ones at lower SER. But what we finally have is an anti-miracle: a pattern that has been sitting there in massive amounts, unnoticed for 15 years, and secretly contributing to produce a totally biased database. (I've noticed elsewhere that once found, the pattern is hard to get rid of by vicinity search.)

It may now be very hard to find the first unnoticed apparition of the pattern and to track its origin in puzzles that didn't have it.
.
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Re: The hardest sudokus (new thread)

Postby mith » Sun May 19, 2024 6:53 pm

I already posted one of the first appearances of the pattern in the ph databases. It is from the batch of puzzles from dobrichev added and/or provided to champagne on December 3, 2012, which includes the puzzle I posted previously. There are four from that batch, and I don't know which was actually found first (not sure if there was any sorting by SER before they were added; the first two IDs are 11.0, the other two 10.9).

They are in the file above, but here they are again on their own:

Code: Select all
........1.....2.3....14..5...678...2.18.2467.72.6.1.4..61.7....2.4.18...87.4.6...;11.00;1.20;1.20;dob;12_12_03;534519;32;*
..............1234..2.5.6....3...76..47.3...262..78..3.764....828.....7.3.4..7.26;11.00;1.20;1.20;dob;12_12_03;536614;31;*
..............1..2..3.4..15.341....6.7.38..418.1.6...7.478.....16.4.3.7.3.8.76...;10.90;1.20;1.20;dob;12_12_03;545447;31;*
..............1..2..3.4..15.341.6..7.7.38..418.1.7...6.478.....16.4.3.7.3.8.67...;10.90;1.20;1.20;dob;12_12_03;545448;32;*


Obviously the last two are closely related. All four have single guardian non-degenerate trivalue oddagons.

The 23 puzzles from dobrichev in the above list were all added in Dec. 2012 or Jan. 2013. champagne's 4 were added Dec. 2018 or Jan. 2019. Paquita's were in the second half of 2019, while mine were from 2020.

So there's a roughly 6 year gap between the first puzzles with the pattern and the next appearances. We probably can't establish definitively whether there is a direct line from the dobrichev puzzles to the later ones in ph2010, never mind to the first T&E(3) puzzles (which still seem to be from August or September 2021, though those scripts haven't finished yet). But I will still at some point attempt to trace back using neighborhood searches as a guide.

------

[edit]I should clarify/reiterate that the list of 838844 puzzles linked above is generated without actually checking for trivalue oddagons at all. The first pass filter only checks for the conditions guaranteeing that it is not a degenerate tridagon by Denis' definition; namely, that there are at least three digits which appear at most once as givens with those appearances confined to a single box. The non-degenerate trivalue oddagon, if it exists, is in the four boxes which do not share a band or stack with the box containing those digits as givens.

This has the advantage of being very fast, but obviously it is possible for a puzzle to meet the first pass conditions and not actually have a non-degenerate tridagon because of the placement of the other digits in the other boxes. For example:

Code: Select all
2...8.....3...9......7...6....456.....6...7.9.......8..89.4567.4..6.79.8..789..45


would make it through the first pass filter, but box 5 cannot have three cells containing 123 in a diagonal pattern.

That said, the vast majority of the list do have non-degenerate trivalue oddagons (even by my stronger restriction on "non-degenerate") - this may be due to the bias toward high SER, or it may just be more common in unique puzzles with this property, I'm not sure. Once the current script is finished, I'll recheck the ones marked as degenerate (again, by my definition, not Denis').
Last edited by mith on Mon May 20, 2024 2:37 am, edited 1 time in total.
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Re: The hardest sudokus (new thread)

Postby mith » Mon May 20, 2024 1:18 am

mith wrote:Based on the 27c database, the first depth 3 puzzle was probably found in August 2021.

Code: Select all
1771897
~ August 13, 2021 (post 308061, first puzzle)

2009985
.............12.34..135.6.2....34.65...2....1.5.16..2...5.4....6.7.....38.9....4.  ED=10.5/1.2/1.2 (DCFC+FC), depth 3

2667557
~ September 29, 2021 (post 310474, first puzzle)


If the first depth 3 puzzles are in similar positions for other clue counts, I may not be able to identify definitively which came first. But we'll see.


Increasingly likely that this was the first T&E(3) puzzle found. The first 28c was sometime after November 3, 2021; the first 29c was January 25-26, 2022; the first 30c was March 5-8, 2022.

24c-26c are still in progress, but the 25c and 26c are well past September 2021 at this point (we know we'll hit one eventually in 26c because of Loki, of course), and I doubt there's one in 24c that far before 25c.
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Re: The hardest sudokus (new thread)

Postby denis_berthier » Mon May 20, 2024 3:05 am

mith wrote:I already posted one of the first appearances of the pattern in the ph databases. It is from the batch of puzzles from dobrichev added and/or provided to champagne on December 3, 2012, which includes the puzzle I posted previously. There are four from that batch, [...]
Code: Select all
........1.....2.3....14..5...678...2.18.2467.72.6.1.4..61.7....2.4.18...87.4.6...;11.00;1.20;1.20;dob;12_12_03;534519;32;*
..............1234..2.5.6....3...76..47.3...262..78..3.764....828.....7.3.4..7.26;11.00;1.20;1.20;dob;12_12_03;536614;31;*
..............1..2..3.4..15.341....6.7.38..418.1.6...7.478.....16.4.3.7.3.8.76...;10.90;1.20;1.20;dob;12_12_03;545447;31;*
..............1..2..3.4..15.341.6..7.7.38..418.1.7...6.478.....16.4.3.7.3.8.67...;10.90;1.20;1.20;dob;12_12_03;545448;32;*


It's great to have found the (likely) first cases of a non-degenerate tridagon with a single guardian.

Interestingly, the second puzzle has both this tridagon and a degenerate-cyclic one (in different blocks) with only 5 guardians:

Code: Select all
   +----------------------+----------------------+----------------------+
   ! 1479   1369   189    ! 236789 24689  23469  ! 1589   1589   1579   !
   ! 579    569    589    ! 6789   689    1      ! 2      3      4      !
   ! 1479   139    2      ! 3789   5      349    ! 6      189    179    !
   +----------------------+----------------------+----------------------+
   ! 8      159#   3      ! 1259   1249   2459   ! 7      6      159#   !
   ! 159#   4      7      ! 1569   3      569    ! 1589   1589#@ 2      !
   ! 6      2      159#   ! 159    7      8      ! 159#   4      3      !
   +----------------------+----------------------+----------------------+
   ! 159#   7      6      ! 4      129    259    ! 3      159#   8      !
   ! 2      8      159#   ! 13569  169    3569   ! 4      7      159#   !
   ! 3      159#   4      ! 1589   189    7      ! 159#   2      6      !
   +----------------------+----------------------+----------------------+

tridagon for digits 1, 5 and 9 in blocks:
        b6, with cells (marked #): r5c8 (target cell, marked @), r6c7, r4c9
        b4, with cells (marked #): r5c1, r6c3, r4c2
        b9, with cells (marked #): r7c8, r9c7, r8c9
        b7, with cells (marked #): r7c1, r9c2, r8c3
 ==> r5c8≠1,5,9

singles ==> r5c8=8, r1c7=8, r2c3=8, r3c4=8, r9c5=8

Degen-Cycl-Trid-OR5-relation for digits 1, 5 and 9 in blocks:
        b4, with cells (marked #): r4c2, r5c1, r6c3
        b5, with cells (marked #): r4c5, r5c6, r6c4
        b7, with cells (marked #): r9c2, r7c1, r8c3
        b8, with cells (marked #): r9c4, r7c6, r8c5
with 5 guardians (in cells marked @): n2r4c5 n4r4c5 n6r5c6 n2r7c6 n6r8c5

   +----------------------+----------------------+----------------------+
   ! 1479   1369   19     ! 23679  2469   23469  ! 8      159    1579   !
   ! 579    569    8      ! 679    69     1      ! 2      3      4      !
   ! 1479   139    2      ! 8      5      349    ! 6      19     179    !
   +----------------------+----------------------+----------------------+
   ! 8      159#   3      ! 1259   1249#@ 2459   ! 7      6      159    !
   ! 159#   4      7      ! 1569   3      569#@  ! 159    8      2      !
   ! 6      2      159#   ! 159#   7      8      ! 159    4      3      !
   +----------------------+----------------------+----------------------+
   ! 159#   7      6      ! 4      129    259#@  ! 3      159    8      !
   ! 2      8      159#   ! 13569  169#@  3569   ! 4      7      159    !
   ! 3      159#   4      ! 159#   8      7      ! 159    2      6      !
   +----------------------+----------------------+----------------------+

.
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Re: The hardest sudokus (new thread)

Postby mith » Mon May 20, 2024 3:42 pm

Here's one with three non-degenerate trivalue oddagons:

Code: Select all
98.7..6....75...94..5.6....7..8......6...5.7..5....4.94.......3.9...8......94...2;11.00;1.20;1.20;PAQ;2019_10_1110_090;2899835;25;*;3
8;11;n123;b1p357+b2p267+b4p267+b5p348
8;9;n123;b1p357+b2p267+b4p249+b5p249
8;9;n123;b1p357+b2p267+b4p249+b5p357
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Re: The hardest sudokus (new thread)

Postby denis_berthier » Mon May 20, 2024 3:57 pm

.
After applying Singles and two whips[2]
Code: Select all
whip[2]: r6n8{c3 c8} - r7n8{c8 .} ==> r5c3≠8
whip[2]: r5n4{c4 c3} - r1n4{c3 .} ==> r4c6≠4, r3c4≠4

I find only one non-degenerate:

Code: Select all
Trid-OR4-relation for digits 2, 3 and 1 in blocks:
        b1, with cells (marked #): r1c3, r2c2, r3c1
        b2, with cells (marked #): r1c5, r2c6, r3c4
        b4, with cells (marked #): r6c3, r4c2, r5c1
        b5, with cells (marked #): r6c4, r4c6, r5c5
with 4 guardians (in cells marked @): n6r4c6 n8r5c1 n8r6c3 n6r6c4

   +----------------------+----------------------+----------------------+
   ! 9      8      123#   ! 7      123#   4      ! 6      1235   15     !
   ! 6      123#   7      ! 5      8      123#   ! 123    9      4      !
   ! 123#   4      5      ! 123#   6      9      ! 12378  1238   178    !
   +----------------------+----------------------+----------------------+
   ! 7      123#   4      ! 8      9      1236#@ ! 1235   12356  156    !
   ! 1238#@ 6      9      ! 4      123#   5      ! 1238   7      18     !
   ! 1238   5      1238#@ ! 1236#@ 1237   12367  ! 4      12368  9      !
   +----------------------+----------------------+----------------------+
   ! 4      127    1268   ! 126    1257   1267   ! 9      1568   3      !
   ! 1235   9      1236   ! 1236   12357  8      ! 157    4      1567   !
   ! 1358   137    1368   ! 9      4      1367   ! 1578   1568   2      !
   +----------------------+----------------------+----------------------+


but 5 degenerate-cyclic:

Code: Select all
Degen-Cycl-Trid-OR20-relation for digits 1, 3 and 6 in blocks:
        b5, with cells (marked #): r4c6, r5c5, r6c4
        b6, with cells (marked #): r4c9, r5c7, r6c8
        b8, with cells (marked #): r9c6, r8c5, r7c4
        b9, with cells (marked #): r9c7, r8c9, r7c8
with 20 guardians (in cells marked @): n2r4c6 n5r4c9 n2r5c5 n2r5c7 n8r5c7 n2r6c4 n2r6c8 n8r6c8 n2r7c4 n5r7c8 n8r7c8 n2r8c5 n5r8c5 n7r8c5 n5r8c9 n7r8c9 n7r9c6 n5r9c7 n7r9c7 n8r9c7

   +-------------------------+-------------------------+-------------------------+
   ! 9       8       123     ! 7       123     4       ! 6       1235    15      !
   ! 6       123     7       ! 5       8       123     ! 123     9       4       !
   ! 123     4       5       ! 123     6       9       ! 12378   1238    178     !
   +-------------------------+-------------------------+-------------------------+
   ! 7       123     4       ! 8       9       1236#@  ! 1235    12356   156#@   !
   ! 1238    6       9       ! 4       123#@   5       ! 1238#@  7       18      !
   ! 1238    5       1238    ! 1236#@  1237    12367   ! 4       12368#@ 9       !
   +-------------------------+-------------------------+-------------------------+
   ! 4       127     1268    ! 126#@   1257    1267    ! 9       1568#@  3       !
   ! 1235    9       1236    ! 1236    12357#@ 8       ! 157     4       1567#@  !
   ! 1358    137     1368    ! 9       4       1367#@  ! 1578#@  1568    2       !
   +-------------------------+-------------------------+-------------------------+

Degen-Cycl-Trid-OR20-relation for digits 1, 2 and 6 in blocks:
        b5, with cells (marked #): r4c6, r5c5, r6c4
        b6, with cells (marked #): r4c9, r5c7, r6c8
        b8, with cells (marked #): r9c6, r8c5, r7c4
        b9, with cells (marked #): r9c7, r8c9, r7c8
with 20 guardians (in cells marked @): n3r4c6 n5r4c9 n3r5c5 n3r5c7 n8r5c7 n3r6c4 n3r6c8 n8r6c8 n5r7c8 n8r7c8 n3r8c5 n5r8c5 n7r8c5 n5r8c9 n7r8c9 n3r9c6 n7r9c6 n5r9c7 n7r9c7 n8r9c7

   +-------------------------+-------------------------+-------------------------+
   ! 9       8       123     ! 7       123     4       ! 6       1235    15      !
   ! 6       123     7       ! 5       8       123     ! 123     9       4       !
   ! 123     4       5       ! 123     6       9       ! 12378   1238    178     !
   +-------------------------+-------------------------+-------------------------+
   ! 7       123     4       ! 8       9       1236#@  ! 1235    12356   156#@   !
   ! 1238    6       9       ! 4       123#@   5       ! 1238#@  7       18      !
   ! 1238    5       1238    ! 1236#@  1237    12367   ! 4       12368#@ 9       !
   +-------------------------+-------------------------+-------------------------+
   ! 4       127     1268    ! 126#    1257    1267    ! 9       1568#@  3       !
   ! 1235    9       1236    ! 1236    12357#@ 8       ! 157     4       1567#@  !
   ! 1358    137     1368    ! 9       4       1367#@  ! 1578#@  1568    2       !
   +-------------------------+-------------------------+-------------------------+

Degen-Cycl-Trid-OR9-relation for digits 1, 2 and 3 in blocks:
        b1, with cells (marked #): r1c3, r2c2, r3c1
        b2, with cells (marked #): r1c5, r2c6, r3c4
        b7, with cells (marked #): r8c3, r7c2, r9c1
        b8, with cells (marked #): r8c5, r7c4, r9c6
with 9 guardians (in cells marked @): n7r7c2 n6r7c4 n6r8c3 n5r8c5 n7r8c5 n5r9c1 n8r9c1 n6r9c6 n7r9c6

   +-------------------------+-------------------------+-------------------------+
   ! 9       8       123#    ! 7       123#    4       ! 6       1235    15      !
   ! 6       123#    7       ! 5       8       123#    ! 123     9       4       !
   ! 123#    4       5       ! 123#    6       9       ! 12378   1238    178     !
   +-------------------------+-------------------------+-------------------------+
   ! 7       123     4       ! 8       9       1236    ! 1235    12356   156     !
   ! 1238    6       9       ! 4       123     5       ! 1238    7       18      !
   ! 1238    5       1238    ! 1236    1237    12367   ! 4       12368   9       !
   +-------------------------+-------------------------+-------------------------+
   ! 4       127#@   1268    ! 126#@   1257    1267    ! 9       1568    3       !
   ! 1235    9       1236#@  ! 1236    12357#@ 8       ! 157     4       1567    !
   ! 1358#@  137     1368    ! 9       4       1367#@  ! 1578    1568    2       !
   +-------------------------+-------------------------+-------------------------+

Degen-Cycl-Trid-OR9-relation for digits 1, 2 and 3 in blocks:
        b1, with cells (marked #): r1c3, r2c2, r3c1
        b2, with cells (marked #): r1c5, r2c6, r3c4
        b7, with cells (marked #): r9c3, r7c2, r8c1
        b8, with cells (marked #): r9c6, r7c5, r8c4
with 9 guardians (in cells marked @): n7r7c2 n5r7c5 n7r7c5 n5r8c1 n6r8c4 n6r9c3 n8r9c3 n6r9c6 n7r9c6

   +----------------------+----------------------+----------------------+
   ! 9      8      123#   ! 7      123#   4      ! 6      1235   15     !
   ! 6      123#   7      ! 5      8      123#   ! 123    9      4      !
   ! 123#   4      5      ! 123#   6      9      ! 12378  1238   178    !
   +----------------------+----------------------+----------------------+
   ! 7      123    4      ! 8      9      1236   ! 1235   12356  156    !
   ! 1238   6      9      ! 4      123    5      ! 1238   7      18     !
   ! 1238   5      1238   ! 1236   1237   12367  ! 4      12368  9      !
   +----------------------+----------------------+----------------------+
   ! 4      127#@  1268   ! 126    1257#@ 1267   ! 9      1568   3      !
   ! 1235#@ 9      1236   ! 1236#@ 12357  8      ! 157    4      1567   !
   ! 1358   137    1368#@ ! 9      4      1367#@ ! 1578   1568   2      !
   +----------------------+----------------------+----------------------+

Degen-Cycl-Trid-OR9-relation for digits 1, 2 and 3 in blocks:
        b1, with cells (marked #): r1c3, r2c2, r3c1
        b2, with cells (marked #): r1c5, r2c6, r3c4
        b7, with cells (marked #): r7c3, r9c2, r8c1
        b8, with cells (marked #): r7c4, r9c6, r8c5
with 9 guardians (in cells marked @): n6r7c3 n8r7c3 n6r7c4 n5r8c1 n5r8c5 n7r8c5 n7r9c2 n6r9c6 n7r9c6

   +-------------------------+-------------------------+-------------------------+
   ! 9       8       123#    ! 7       123#    4       ! 6       1235    15      !
   ! 6       123#    7       ! 5       8       123#    ! 123     9       4       !
   ! 123#    4       5       ! 123#    6       9       ! 12378   1238    178     !
   +-------------------------+-------------------------+-------------------------+
   ! 7       123     4       ! 8       9       1236    ! 1235    12356   156     !
   ! 1238    6       9       ! 4       123     5       ! 1238    7       18      !
   ! 1238    5       1238    ! 1236    1237    12367   ! 4       12368   9       !
   +-------------------------+-------------------------+-------------------------+
   ! 4       127     1268#@  ! 126#@   1257    1267    ! 9       1568    3       !
   ! 1235#@  9       1236    ! 1236    12357#@ 8       ! 157     4       1567    !
   ! 1358    137#@   1368    ! 9       4       1367#@  ! 1578    1568    2       !
   +-------------------------+-------------------------+-------------------------+
denis_berthier
2010 Supporter
 
Posts: 4018
Joined: 19 June 2007
Location: Paris

Re: The hardest sudokus (new thread)

Postby mith » Mon May 20, 2024 5:07 pm

denis_berthier wrote:.
After applying Singles and two whips[2]
Code: Select all
whip[2]: r6n8{c3 c8} - r7n8{c8 .} ==> r5c3≠8
whip[2]: r5n4{c4 c3} - r1n4{c3 .} ==> r4c6≠4, r3c4≠4


Yeah, my code (actually rangsk's code, I'm getting the pencilmark grid from his solver) only applies singles, subsets, whips[1]; the eliminations of 4 place several digits by singles in b45 which removes the other two non-degenerate options.
mith
 
Posts: 981
Joined: 14 July 2020

Re: The hardest sudokus (new thread)

Postby Paquita » Thu Jun 13, 2024 3:35 pm

denis_berthier wrote:
Hi Paquita,
Have you kept a trace of the seed which each of these puzzles originates from?
I'm particularly interested in the following 4:
Code: Select all
98.76....7.....9...54......6...3.8....5....2......4.1..7.38.......9..3.......1..2 ED=11.7/11.7/2.6
98.7.....7..86......5..4...3..6..9....2.............51.7..8.3.......67.......1.42 ED=11.6/1.2/1.2
98.7.....7..86......5..4...3..6..9....2.............51.7..8.3.......67.......1.24 ED=11.6/1.2/1.2
98765.......4..........3...8.62..7..2......9..95...8.276..2.9.85......6..2.....75 ED=11.5/1.2/1.2


More precisely, did they originate from puzzles in the pre-tridagon collections?
.


Hi Denis

A late reply, are you still interested in these origins? I may or may not be able to find it but it wil take some time if I can.

(And Hendrik, you replied that the first was by mith (that is correct) and that the second and third are the same and by mith; they are not the same, different last 2 digits; I can't find those in miths collection, can you point out what they are, if duplicates?)

At 2019, I was the only one generating larger collections of puzzles between 10.5 and 11.5 so it is not a big surprise that those non-degenerate tridagon puzzles are all mine. I must have come across the tridagon and vicinity search creates this effect. I don't know if I can find the origin there, but I used the then current database as seed so it may have come from dobrichevs puzzles.
I do wonder if the finding of those tridagons did bias the collection of puzzles - is it possible that this is in fact a large part of all the hardest puzzles?
I think it may be interesting to collect the non-tridagon seeds and try to generate new puzzles there. Is there such a collection available now, or else, a collection of all the T&E(2) that are tridagons, so I can distract those?
Paquita
 
Posts: 73
Joined: 11 November 2018

Re: The hardest sudokus (new thread)

Postby denis_berthier » Thu Jun 13, 2024 3:56 pm

.
Hi Paquita
Yes, I'm still interested in the origins, if it doesn't take too much time.

There's undoubtedly a strong bias in all the collections that have tridagons. The problem I've already mentioned is, once you have a tridagon in the puzzles used as seeds, the likelihood of having it in a large proportion of its neighbours is very high. The more steps of vicinity searche you do, the more bias this will introduce.
This high persistence of the tridagon is unique to this pattern.

As for the T&E(2) collections, I know only 2:
eleven's gotchi has no (non-degenerate) tridagon
- ph2010 has a lot of them, but only one for SER 11.3 and the rest for SER ≤ 11.1. If you're interested, I have a list of the puzzles (only among the first 300,000) that have the pattern. I think mith has checked the whole collection for it.
denis_berthier
2010 Supporter
 
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Joined: 19 June 2007
Location: Paris

Re: The hardest sudokus (new thread)

Postby denis_berthier » Thu Jun 13, 2024 3:58 pm

Here's the start of the list (the numbers are the positions in the ph2010 collection):
Hidden Text: Show
18678
67684 67748 67751 67752 67753 67754 67932 68008 68146 68215 68216 68262 68393 68394 68395 68396 68397 68398 68399 68400 68401 68402 68403 68404 68405 68406 68407 68408 68409 68410 68411 68414 68415 68416 68417 68418 68419 68420 68421 68422 68423 68424 68425 68426 68427 68428 68429 68430 68431 68432 68433 68434 68435 68436 68437 68438 68439 68440 68441 68442 68443 68444 68445 68446 68447 68448 68449 68450 68451 68452 68453 68454 68455 68456 68457 68458 68459 68460 68461 68462 68463 68464 68465 68466 68467 68468 68469 68470 68471 68472 68473 68474 68476 68477 68478 68479 68480 68481 68482 68484 68485 68486 68487 68488 68489 68490 68491 68492 68493 68494 68495 68496 68497 68498 68499 68500 68501 68502 68503 68504 68506 68507 68509 68510 68511 68512 68513 68514 68515 68516 68517 68518 68519 68520 68521 68522 68523 68524 68525 68526 68527 68528 68529 68530 68531 68532 68533 68534 68535 68536 68537 68538 68539 68540 68541 68542 68543 68544 68545 68546 68547 68548 68549 68550 68551 68552 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denis_berthier
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