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 ghfick » Sun Oct 03, 2021 5:32 pm

I tried the three puzzles using YZF_Sudoku. I changed the settings so that MSLS and Junior Exocets are listed before Dynamic Chains.
The first puzzle has a long solution path but after the MSLS, the next hardest step is a Death Blossom :

Code: Select all

..3....8..5....2.17...........5.8..6.9.12....8....3....6.9....5..4....7.....1.6.2

Locked Candidates 1 (Pointing): 7 in b3 => r1c4<>7,r1c5<>7,r1c6<>7
Locked Candidates 1 (Pointing): 9 in b5 => r1c5<>9,r2c5<>9,r3c5<>9

MSLS:16 Cells r2579c1368, 16 Links 69r2,56r5,12r7,59r9,34c1,78c3,47c6,34c8
18 Eliminations: r7c7<>1,r4c18,r3c8,r8c1<>3,r346c8,r1c16,r3c6, r4c1<>4,r5c7<>5,r2c45<>6,r46c3<>7,r3c3<>8

Locked Pair: in r4c1,r4c3 => r4c2<>12,r6c2<>12,r6c3<>12,r4c2<>12,r4c7<>1,r4c8<>12,
Hidden Single: 2 in b6 => r6c8=2
Hidden Single: 1 in b6 => r6c7=1
Hidden Single: 1 in b9 => r7c8=1
Hidden Single: 5 in b6 => r5c8=5
Hidden Single: 5 in b4 => r6c3=5
Naked Single: r4c8=9
Hidden Single: 9 in b5 => r6c5=9
Hidden Single: 6 in r6 => r6c4=6
Naked Single: r3c8=6
Locked Candidates 1 (Pointing): 9 in b9 => r8c1<>9
Empty Rectangle : 4 in b5 connected by c1 => r2c5 <> 4
Finned X-Wing:4c48\r29 fr13c4 => r2c6<>4
Finned Swordfish:4c168\r259 fr7c6 => r9c4<>4
Locked Candidates 2 (Claiming): 4 in c4 => r1c5<>4,r3c5<>4
AIC Type 2: 3r4c2 = r5c1 - (3=2)r7c1 - (2=1)r4c1 - r8c1 = 1r8c2 => r8c2<>3

MSLS:13 Cells r48c457+r123c45,r4c2,r8c9,13 Links 347r4,389r8,2c4,56c5,3478b2
2 Eliminations: r2c6<>7,r8c2<>8

Almost Locked Set XZ-Rule: A=r45c7,r6c9 {3478},B=r7c7,r9c8 {348}, X=8, Z=3 => r8c7<>3
Almost Locked Set XY-Wing: A=r1c24{124}, B=r48c1{125}, C=r8c245679{1235689}, X,Y=1, 5, Z=2 =>  r1c1<>2
Almost Locked Set XY-Wing: A=r1c24{124}, B=r579c6{2457}, C=r8c245679{1235689}, X,Y=1, 5, Z=2 =>  r1c6<>2
Death Blossom Complex Type 2: Set have degrees of freedom of 0-23478{r7c13567} => r4c2<>4

3r7c1-(3=12478){r13689c2}
3r7c5-(3=45678){r12348c5}
3r7c7-(3=47){r4c57}
W-Wing: 47 in r5c6,r6c9 connected by 4r4 => r5c79<>7
Grouped 2-String Kite: 7 in r5c6,r9c2 connected by r46c2,r5c3 => r9c6 <> 7

Death Blossom Complex Type 2: Set have degrees of freedom of 2-3469{r2c18} => r5c9<>4
6r2c1,9r2c1-(69=12345){r145789c1}
3r2c8-(3=479){r136c9}

Region Forcing Chain: Each 4 in r1 true in turn will all lead r1c2<>1
(4-1)r1c2
(4-2)r1c4 = (2-1)r1c2
4r1c7 - r2c8 = r9c8 - (4=5)r9c6 - r8c56 = (5-1)r8c1 = r8c2 - 1r1c2
4r1c9 - r2c8 = r9c8 - (4=5)r9c6 - r8c56 = (5-1)r8c1 = r8c2 - 1r1c2

Naked Pair: in r1c2,r1c4 => r1c7<>4,r1c9<>4,
WXYZ-Wing: 5679 in r1c579,r2c6,Pivot Cell Is r1c5 => r1c6<>9
Almost Locked Set XY-Wing: A=r1c1579{15679}, B=r9c23468{345789}, C=r4789c1{12359}, X,Y=1, 9, Z=5 =>  r1c6<>5

Region Forcing Chain: Each 7 in r9 true in turn will all lead r2c5<>3
7r9c2 - r6c2 = (7-4)r6c9 = r3c9 - (4=3)r2c8 - 3r2c5
(7-9)r9c3 = (9-5)r9c1 = (5-4)r9c6 = r9c8 - (4=3)r2c8 - 3r2c5
7r9c4 - r2c4 = (7-3)r2c5

Death Blossom Complex Type 2: Set have degrees of freedom of 1-34578{r2347c5} => r1c7<>9
3r3c5-(3=479){r136c9}
5r3c5-(5=169){r1c156}
3r7c5-(3=125689){r8c124567}

AIC Type 2: 9r1c1 = (9-7)r1c9 = (7-4)r6c9 = r6c2 - r5c1 = 4r2c1 => r2c1<>9
AIC Type 2: 5r8c1 = (5-9)r9c1 = r1c1 - (9=7)r1c9 - (7=5)r1c7 - (5=6)r1c5 - r8c5 = 6r8c6 => r8c6<>5

Death Blossom Complex Type 2: Set have degrees of freedom of 2-12358{r8c124} => r2c8<>4
5r8c1-(5=34789){r9c12348}
3r8c4-(3=46789){r2c13456}
8r8c4-(8=349){r8c79,r9c8}

Hidden Single: 4 in c8 => r9c8=4
Full House: r2c8=3
Naked Single: r9c6=5
Hidden Single: 5 in b7 => r8c1=5
Hidden Single: 1 in b7 => r8c2=1
Locked Candidates 1 (Pointing): 4 in b3 => r3c2<>4,r3c4<>4
Locked Candidates 1 (Pointing): 2 in b7 => r7c6<>2
Locked Candidates 2 (Claiming): 2 in c2 => r3c3<>2
Hidden Pair: 38 in r5c9,r8c9 => r8c9<>9
Hidden Single: 9 in b9 => r8c7=9
AIC Type 2: 2r1c2 = r1c4 - r3c6 = (2-6)r8c6 = r8c5 - (6=5)r1c5 - (5=7)r1c7 - r1c9 = (7-4)r6c9 = 4r6c2 => r1c2<>4
Hidden Single: 4 in b1 => r2c1=4
Hidden Single: 4 in b4 => r6c2=4
Full House: r6c9=7
Hidden Single: 7 in b3 => r1c7=7
Hidden Single: 5 in b3 => r3c7=5
Hidden Single: 4 in b3 => r3c9=4
Full House: r1c9=9
Hidden Single: 4 in b2 => r1c4=4
Hidden Single: 5 in b2 => r1c5=5
Hidden Single: 6 in c5 => r8c5=6
Hidden Single: 9 in c1 => r9c1=9
Hidden Single: 2 in r1 => r1c2=2
Naked Single: r3c2=8
Naked Single: r3c5=3
Naked Single: r3c4=2
Hidden Single: 2 in b8 => r8c6=2
Skyscraper : 3 in r4c2,r7c1 connected by r47c7 => r5c1,r9c2 <> 3
[stte]


The second puzzle is essentially cracked with an early MSLS :

Code: Select all

.2...67..4...8......9.......3.....7.5.8....4..1.3....2....9..5....6.1..3...2..6.7

Hidden Single: 3 in b6 => r5c7=3
Locked Candidates 1 (Pointing): 2 in b4 => r4c5<>2,r4c6<>2
Locked Candidates 2 (Claiming): 4 in c2 => r7c3<>4,r8c3<>4,r9c3<>4

MSLS:16 Cells r1689c1358, 16 Links 13r1,67r6,27r8,13r9,89c1,45c3,45c5,89c8
19 Eliminations: r1c49<>1,r8c7<>2,r9c6<>3,r346c5,r4c3,r6c6<>4,r34c5,r2c3<>5, r6c6,r8c2<>7,r3c18,r7c1<>8,r2c8,r4c1<>9

Hidden Single: 4 in b4 => r6c3=4
Locked Pair: in r4c1,r4c3 => r5c2<>6,r6c1<>6,r4c5<>6,r4c9<>6,
Naked Single: r4c5=1
Hidden Single: 1 in b6 => r5c9=1
Hidden Single: 6 in b6 => r6c8=6
Hidden Single: 6 in b5 => r5c5=6
Hidden Single: 2 in b5 => r5c6=2
Hidden Single: 2 in b2 => r3c5=2
2-String Kite: 8 in r6c7,r7c4 connected by r4c4,r6c6 => r7c7 <> 8
Empty Rectangle : 7 in b8 connected by r6 => r7c1 <> 7
Sue de Coq: r23c7 - {124589} (r23c8 - {123}, r468c7 -{4589}) =>  r1c8<>1 r1c8<>3 r7c7<>4
Locked Candidates 2 (Claiming): 1 in r1 => r2c3<>1,r3c1<>1
AIC Type 1: 4r3c7 = r8c7 - (4=8)r7c9 - r7c4 = (8-4)r4c4 = 4r4c6 => r3c6<>4
AIC Type 2: 4r3c7 = r8c7 - (4=8)r7c9 - r7c4 = r4c4 - r6c6 = 8r6c7 => r3c7<>8
AIC Type 1: 4r4c6 = (4-8)r4c4 = r7c4 - (8=4)r7c9 => r7c6<>4
Grouped AIC Type 2: 9r2c6 = r46c6 - (9=7)r5c4 - (7=5)r6c5 - r89c5 = 5r9c6 => r2c6<>5
Grouped AIC Type 2: 4r4c6 = (4-5)r9c6 = r89c5 - (5=7)r6c5 - (7=9)r5c4 => r4c6<>9
Almost Locked Set XY-Wing: A=r7c49{478}, B=r469c6{4589}, C=r5c4{79}, X,Y=7, 9, Z=8 =>  r7c6<>8
AIC Type 2: 3r1c5 = r9c5 - (3=7)r7c6 - r8c5 = r6c5 - (7=9)r6c1 - r6c6 = 9r2c6 => r2c6<>3
Almost Locked Set XY-Wing: A=r19c3{135}, B=r7c7{12}, C=r2c234679{1235679}, X,Y=3, 2, Z=1 =>  r7c3<>1
Almost Locked Set XY-Wing: A=r2c236789{1235679}, B=r23469c6{345789}, C=r3c8{13}, X,Y=1, 3, Z=7 =>  r2c4<>7
Almost Locked Set XY-Wing: A=r6c167{5789}, B=r123c4,r1c5,r3c6{134579}, C=r5c4,r6c5{579}, X,Y=5, 9, Z=7 =>  r3c1<>7
X-Wing:7c15\r68  => r8c3<>7
AIC Type 2: 1r1c3 = r9c3 - r9c8 = (1-2)r7c7 = r8c8 - (2=5)r8c3 => r1c3<>5
Locked Candidates 1 (Pointing): 5 in b1 => r8c2<>5,r9c2<>5
AIC Type 2: 5r2c2 = (5-8)r3c2 = (8-6)r3c9 = 6r2c9 => r2c9<>5r2c2<>6
AIC Type 1: (2=6)r4c1 - (6=3)r3c1 - r2c3 = (3-2)r2c8 = 2r8c8 => r8c1<>2
WXYZ-Wing: 4789 in r8c12,r6c1,r9c2,Pivot Cell Is r8c1 => r9c1<>9
Hidden Pair: 79 in r6c1,r8c1 => r8c1<>8
Uniqueness Test 7: 26 in r47c13; 2*biCell + 1*conjugate pairs(2c1) => r7c3 <> 6
Sashimi X-Wing:8r38\c29 fr8c78 => r7c9<>8
Naked Single: r7c9=4
Hidden Single: 4 in b3 => r3c7=4
Hidden Pair: 12 in r2c7,r7c7 => r2c7<>59
Locked Candidates 2 (Claiming): 5 in c7 => r4c9<>5
AIC Type 2: 4r1c4 = (4-3)r1c5 = r9c5 - (3=7)r7c6 - (7=9)r2c6 => r1c4<>9
Locked Candidates 2 (Claiming): 9 in r1 => r2c9<>9
Naked Single: r2c9=6
Hidden Single: 6 in c3 => r4c3=6
Hidden Single: 2 in b4 => r4c1=2
AIC Type 2: (8=9)r1c8 - r9c8 = r9c2 - (9=7)r8c1 - r7c3 = (7-3)r2c3 = (3-2)r2c8 = 2r8c8 => r8c8<>8
X-Wing:8c18\r19  => r9c26,r1c9<>8
Hidden Single: 8 in b8 => r7c4=8
W-Wing: 45 in r1c4,r9c6 connected by 4r4 => r3c6<>5
Naked Pair: in r3c6,r7c6 => r2c6<>7,
Naked Single: r2c6=9
Locked Candidates 2 (Claiming): 7 in r2 => r3c2<>7
XYZ-Wing: 489 in r8c2 r8c7 r9c2 => r8c1 <> 9
[stte]



The third puzzle has an early MSLS but remains unsolved without 'Brute Force'.
ghfick
 
Posts: 149
Joined: 06 April 2016

Error Launching Web Browsers On Smart Tv

Postby Krist_snowflakeFausy » Mon Oct 04, 2021 9:48 pm

Please tell me, can anyone come across such an error
on Smart.
When you open web Browsers, it writes like this (photo in the attached file)
... What can you do ... Thank you
In search of love, xxx me somebody :))
User avatar
Krist_snowflakeFausy
 
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Joined: 01 October 2021
Location: Lithuania

Re: The hardest sudokus (new thread)

Postby mith » Tue Oct 05, 2021 6:21 pm

Finished the last bit of rating I was doing for the update. I'm going to go through the thread in the next couple days to get puzzles from others.
mith
 
Posts: 586
Joined: 14 July 2020

Re: The hardest sudokus (new thread)

Postby mith » Wed Oct 06, 2021 5:35 pm

I believe these are all the puzzles posted by someone other than me since the 2020-10 update:

jco
98.7.....7.6...9...5.......4..8..6......5..3......2..1..89..4......1...2.....3.5. ER/EP/ED=11,8/11,8/10,4
98.7.....7.6...8....5......4..6..9......3..5......2..1.9.4..7......1..3......5..2 ER/EP/ED=11,8/11,8/10,3
98.7.....7.6...8....5......4..8..9......3..5......2..1.9.6..4......5...2.....1.3. ER/EP/ED=11,8/11,8/3,4
98.7.....6.....5....4......7..8..9......3..4......2..1.5.9..6......4..3......1..2 ER/EP/ED=11,8/11,8/3,4
98.7.....7.6...8....5......8..6..4......5..3......2..1.7.4..9......1..2......3..5 ER/EP/ED=11,7/11,7/11,1
98.7.....7.6...9...5.......4..6..8......5..3......2..1..94..7......3...5.....1.2. ER/EP/ED=11,7/11,7/11,1
98.7.....7.6...8...5.......4..9..6......5..3......2..1..78..4......1..5......3..2 ER/EP/ED=11,7/11,7/10,7 (*)
98.7.....7.6...8....5......4..8..9......3..5......2..1.7.6..4......5...3.....1.2. ER/EP/ED=11,7/11,7/3,4
98.7.....7.6...8...5.......4..8..9......5..3......2..1..74..6......1..5......3..2 (ER/EP/ED=11,7/11,7/11,3) (*)
98.7.....7.6...5....4......5..6..7......4..3......2..1.7.8..9......1..2......3..4 (ER/EP/ED=11,7/11,7/10,6)
(these were "generated" with YZF and are just morphs of puzzles already in the database, but I will double-check that)

hendrik_monrad
98.7.....6...5.8....4....3.7..9..5....3....4......2..1.6...1..2..9.........89.6.. 11.8/1.2/1.2
98.7.....6...8......5..4...3..8..9...7....86.....4...2.9.6..3....1.5.........2..1 11.7/11.7/8.0
98.7.....6..85......4..3....9..8.7.......2.1.........2.5..7.6....9...5....1....43 11.7/1.2/1.2
98.7..6..5..69.....4...8...3.....56..2......8..4.....2...97........3.7.......2.31 11.7/11.7/2.6
98.76.5..54.........75.4.9.7..4.9.5..9...5..3....2....4......81.7.9.8.4.......... 11.6/1.2/1.2
98.76.5..75.........4..5.978......3..4...9.58........24....8.7..9.2..38.....9.1.. 11.6/1.2/1.2
98.7.....7.6.5......4..3...2.5...4...9.....82...2....1.7.1...2......63.........19 11.6/11.6/2.6
98.7.....6..85......4..3....9..8.6....2.............32.5.6..7.......85.......1.24 11.6/1.2/1.2
98.7..6..5...9..84.........46..7.8....86....3..5..4....4...79....2.....8....8..1. 11.6/1.2/1.2
98.7..6..5...9..4......8...46..7.8....86...3...5..4....4...79....2....8.....8...1 11.6/1.2/1.2
98.7..6....7.5..98.........76.9....5..4..........37...5..6...8...2...7.6....715.. 11.6/1.2/1.2
98.76....5....49....3......4....5.9..7......5..5...2.41....9..2..6.8.......3...1. 11.6/11.6/2.6
98.76....5.....7.......5.987.....4...9.....83..32.....3....9.57.5..7..3.......1.. 11.6/1.2/1.2
98.76.5..54....7.....5...848..9....7.7..5..........32..9.4...75....1.2........... 11.6/1.2/1.2
98.76....54....7.....5...848..9...57.7..5..........32..9.4...75....1.2........... 11.6/1.2/1.2
98.7..6..5.46.........9..837..9..5......2...7.....6.4.1....9.7..5..671.....1....5 11.6/1.2/1.2
98.7.......6.5.4.......6...8..4...7..7......3..2..71...9.3....4..1.6.........25.. SER = 11.7/11.7/4.3
98.7..6..7...9......5..4...3...7.8....9..6.42.....1....9..6.3....2.............15 SER = 11.6/11.6/2.6
98.7..6....5.9...........4.79....3....6..3........7.6236...98....1.8.......6..... SER = 11.6/1.2/1.2
98.7..6..7..6..5......4..8.5..9..7...7...3....6......21.......5.9...71.....1...6. SER = 11.6/1.2/1.2
98.7..6..7..6..5......4..8.5..9..7...6......3.....2...1.......5.9...71....71...6. SER = 11.6/1.2/1.2

JPF
........1....12.....34...5.....6...2.7....53..8.5..4...378.....1....9...94....7..;11.8;1.2;1.2
........1....21.....34...5.....6...2.7....53..8.5..4...378.....6....9...94....7..;11.7;11.7;2.6
........1....21.....34...5.....6...2.7....53..8.5..4...378.....1....9...94....7..;11.3;1.2;1.2
........1....23.....45...6.....7...3.5....84..9.8..6...469.....2....1...18....4..;11.3;1.2;1.2
........1....21.....34...5.....6...2.7....53..8.3..4...378.....6....9...94....7..;11.3;1.2;1.2
........1....12.....34...5.....6...7.8....43..9.3..5...598.....2....7...73....9..;11.2;1.2;1.2
........1....23.....45...6.....7...2.5....48..8.4..6...659.....3....1...14....8..;11.1;1.2;1.2
........1....21.....34...5.....6...2.7....53..8.5..4...378.....9....6...64....7..;11.0;11.0;2.6
........1....12.....34...5.....6...2.4....37..7.3..5...548.....1....9...93....7..;11.0;1.2;1.2

999_Springs
(several non-minimals, will check all the minimals against the database)

Some of the above are already in my local database, but if I hadn't posted them here already I will update the "creator" field accordingly. :)

Will scrape the patterns game later.
mith
 
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Joined: 14 July 2020

Re: The hardest sudokus (new thread)

Postby yzfwsf » Wed Oct 06, 2021 8:15 pm

mith wrote:(these were "generated" with YZF and are just morphs of puzzles already in the database, but I will double-check that)

All these puzzles are taken from the database, so you don’t need to recheck .These puzzles are only provided for the convenience of players to learn JE, MSLS and other technologies.
yzfwsf
 
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Joined: 16 April 2019

Re: The hardest sudokus (new thread)

Postby mith » Wed Oct 06, 2021 9:20 pm

That's what I suspected, but thank you for the confirmation :)
mith
 
Posts: 586
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Re: The hardest sudokus (new thread)

Postby jco » Wed Oct 06, 2021 9:55 pm

mith wrote:That's what I suspected, but thank you for the confirmation :)

I should have deleted that listing as soon as mith made me aware that they were all known puzzles.
My interest at that time was on SK-loops (practicing to identify them), but got distracted/curious with the unusually
high (for me, at that time) ratings of that puzzles. Anyway, It was a mistake to post them and I should have deleted them as soon as I read mith's observation. Now I deleted that content (except the one after mith's observation, explaining basically the same
as above). My apologies.
JCO
jco
 
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Re: The hardest sudokus (new thread)

Postby mith » Thu Oct 07, 2021 2:34 pm

No worries!
mith
 
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Re: The hardest sudokus (new thread)

Postby mith » Thu Oct 14, 2021 6:58 pm

Our modem got zapped by lightning a few nights ago, so I've been occupied getting our internet back up. Will get back on this soon.
mith
 
Posts: 586
Joined: 14 July 2020

Re: The hardest sudokus (new thread)

Postby mith » Tue Oct 26, 2021 8:27 pm

Haven't forgotten about this! I will definitely be getting the update out before next Thursday (having foot surgery and will have trouble getting to this computer for a few weeks after), but hopefully sooner.
mith
 
Posts: 586
Joined: 14 July 2020

Re: The hardest sudokus (new thread)

Postby mith » Wed Nov 03, 2021 9:42 pm

I should stop committing to dates, life keeps getting in the way. :)

In the meantime, I went ahead and pulled all my 10.6+ puzzles (which weren't in a previous update) along with 10.2+ for 19c, 35c, and 36c. You can view them here: https://drive.google.com/file/d/1DEYdwZ ... sp=sharing

Since I'm having surgery tomorrow and won't be able to get up to this computer for a while, I'm just going to go ahead and start my scripts running again and maybe there will be some nice surprises waiting when I check in a few weeks.
mith
 
Posts: 586
Joined: 14 July 2020

Re: The hardest sudokus (new thread)

Postby mith » Wed Nov 03, 2021 10:06 pm

The above is a lot of puzzles (394844)! If all you care about is the 11.6+ puzzles, here are those (535): https://drive.google.com/file/d/1xjnXn1 ... sp=sharing
mith
 
Posts: 586
Joined: 14 July 2020

Re: The hardest sudokus (new thread)

Postby mith » Wed Nov 03, 2021 10:37 pm

Here are all of the 11.6+ puzzles now in my database (1161 puzzles): https://drive.google.com/file/d/1_N6oN4 ... sp=sharing

(the last two numbers in each file are gsf's q1 and q2 respectively)
mith
 
Posts: 586
Joined: 14 July 2020

Re: The hardest sudokus (new thread)

Postby mith » Thu Nov 04, 2021 4:09 pm

marek stefanik wrote:Hello everyone,

I came here looking for new sudoku techniques.

Very interesting puzzles, make me wonder what kind of magic would one have to use to prove this contradiction using truths and links (it's in every single one of these 30+-clue puzzles and I'm slowly going insane thinking about it :D).

Code: Select all
123   .    .  |  .    .   123
 .   123   .  |  .   123   .
 .    .   123 | 123   .    .
––––––––––––––+––––––––––––––
 .    .   123 | 123   .    .
 .   123   .  |  .    .   123
123   .    .  |  .   123   .


In sukaku format: Show
123000000000456789000456789000456789000456789123000000123456789123456709123456780000456789123000000000456789000456789123000000000456789123456789123456709123456780000456789000456789123000000123000000000456789000456789123456789123456709123456780000456789000456789123000000123000000000456789000456789123456789123456709123456780000456789123000000000456789000456789000456789123000000123456789123456709123456780123000000000456789000456789000456789123000000000456789123456789123456709123456780123456789123456789123456789123456789123456789123456789123456700123456700123456700123456709123456709123456709123456709123456709123456709123456700000000080123456700123456780123456780123456780123456780123456780123456780123456700123456700000000009


Marek


Updating on this technique, ryokousha of the CTC discord came up with an elegant permutation parity argument, which generalizes nicely to other triplet patterns, and rangsk recorded a follow-up video to our series (the last episode of which featured a puzzle with this property), explaining the new approach (my approach in the episode amounted to “however one digit is placed, you are left with a bivalue oddagon on the other digits): https://youtu.be/V7RC1hJ8vZ8
mith
 
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