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Website · by **denis_berthier** » Wed Mar 15, 2023 3:57 am

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As all the puzzles I'm currently proposing, this is intended to illustrate the most frequent impossible 3-digit patterns in eleven's list of 630.
I choose them so that they can be solved with short chains and the main challenge is to find the patterns, not the chains.
To be more specific, I choose puzzles that can't be solved in Trid+W8+OR5W8 but that can be solved in Trid+PPP+Wn+OR5Wn, where PPP is 1 or 2 or 3 of the most frequent patterns and n (the max length of chains) is as small as possible. The smallest such n I've found up to now is 3.

Code: Select all: +-------+-------+-------+ ! 1 . . ! . 5 6 ! . . 9 ! ! . . . ! 1 . 9 ! . . 6 ! ! 6 9 . ! 2 7 . ! . . . ! +-------+-------+-------+ ! 2 1 . ! 7 . 5 ! 9 . . ! ! . 7 . ! . . . ! 1 2 . ! ! 9 . . ! . 2 1 ! . . . ! +-------+-------+-------+ ! . . . ! . . 7 ! 6 9 . ! ! 5 . 9 ! . . . ! 3 . . ! ! 7 . 1 ! . . . ! 8 4 . ! +-------+-------+-------+ 1...56..9...1.9..669.27....21.7.59...7....12.9...21........769.5.9...3..7.1...84.;3360;245372 SER = 10.4

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 1 2348 23478 ! 348 5 6 ! 247 378 9 ! ! 348 5 23478 ! 1 348 9 ! 247 378 6 ! ! 6 9 348 ! 2 7 348 ! 45 1358 1348 ! +-------------------+-------------------+-------------------+ ! 2 1 3468 ! 7 348 5 ! 9 368 348 ! ! 348 7 5 ! 34689 34689 348 ! 1 2 348 ! ! 9 348 3468 ! 348 2 1 ! 457 35678 3478 ! +-------------------+-------------------+-------------------+ ! 348 2348 2348 ! 3458 1348 7 ! 6 9 125 ! ! 5 2468 9 ! 468 1468 248 ! 3 17 127 ! ! 7 236 1 ! 3569 369 23 ! 8 4 25 ! +-------------------+-------------------+-------------------+ 160 candidates.

[Edit]: corrected # of puzzle to #14987

by **eleven** » Wed Mar 15, 2023 10:02 am

Code: Select all: +----------------------+----------------------+----------------------+ | 1 2348 23478 | 348 5 6 | 247 d378 9 | | 348 5 23478 | 1 348 9 | 247 d378 6 | | 6 9 348 | 2 7 348 | 45 1358 1348 | +----------------------+----------------------+----------------------+ | 2 1 3468 | 7 348 5 | 9 368 a348 | | 348 7 5 | 69 69 348 | 1 2 a348 | | 9 348 3468 | 348 2 1 | 457 35678 a3478 | +----------------------+----------------------+----------------------+ | 348 2348 2348 | 3458 1348 7 | 6 9 125 | | 5 2468 9 | 468 1468 248 | 3 c17 b127 | | 7 236 1 | 3569 369 23 | 8 4 25 | +----------------------+----------------------+----------------------+

(348=7)r456c9 - r8c9 = r8c8 - (7=38)r12c8 => -38r3c9

Code: Select all: +----------------------+----------------------+----------------------+ | 1 *348+2 23478 |*348 5 6 | 247 378 9 | |*348 5 23478 | 1 *348 9 | 247 378 6 | | 6 9 *348 | 2 7 *348 | 45 1358 14 | +----------------------+----------------------+----------------------+ | 2 1 *348 | 7 *348 5 | 9 6 348 | |*348 7 5 | 69 69 *348 | 1 2 348 | | 9 *348 6 |*348 2 1 | 457 57 3478 | +----------------------+----------------------+----------------------+ | 348 2348 2348 | 3458 1348 7 | 6 9 125 | | 5 2468 9 | 468 1468 248 | 3 17 127 | | 7 236 1 | 3569 369 23 | 8 4 25 | +----------------------+----------------------+----------------------+

TH (*) => 2r1c2, RT 348 r1c4,r6c24

Code: Select all: +-------------------+-------------------+-------------------+ | 1 2 3478 |R348 5 6 | 47 378 9 | | 348 5 3478 | 1 348 9 | 2 378 6 | | 6 9 348 | 2 7 348 | 45 1358 14 | +-------------------+-------------------+-------------------+ | 2 1 348 | 7 348 5 | 9 6 348 | | 348 7 5 | 69 69 348 | 1 2 348 | | 9 R348 6 |R348 2 1 | 457 57 3478 | +-------------------+-------------------+-------------------+ |b348 a348 2 | 5-348 1348 7 | 6 9 15 | | 5 a468 9 | 468 1468 248 | 3 17 127 | | 7 a36 1 | 3569 369 23 | 8 4 25 | +-------------------+-------------------+-------------------+

xyz is 348: xr6c2 goes to r7c1 (xr6c2 - r789c2 = xr7c1) and by the remote triple yz to r16c4 => -348r7c4

Code: Select all: +----------------+----------------+----------------+ | 1 2 b348 |a348 5 6 | 7 38 9 | | 348 5 7 | 1 38-4 9 | 2 38 6 | | 6 9 38 | 2 7 38 | 5 1 4 | +----------------+----------------+----------------+ | 2 1 c348 | 7 d348 5 | 9 6 38 | | 348 7 5 | 69 69 348 | 1 2 38 | | 9 38 6 | 38 2 1 | 4 5 7 | +----------------+----------------+----------------+ | 38 348 2 | 5 348 7 | 6 9 1 | | 5 468 9 | 468 1 48 | 3 7 2 | | 7 36 1 | 369 69 2 | 8 4 5 | +----------------+----------------+----------------+

skyscraper 4 r14 => -4r2c5, stte

Website · by **denis_berthier** » Fri Mar 17, 2023 4:32 am

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Solved with two of the 5 most frequent impossible patterns: EL14c30 and EL14c13.

hidden-pairs-in-a-row: r5{n6 n9}{c4 c5} ==> r5c5≠8, r5c5≠4, r5c5≠3, r5c4≠8, r5c4≠4, r5c4≠3

Code: Select all: OR2-anti-tridagon[12] for digits 3, 4 and 8 in blocks: b1, with cells (marked #): r1c2, r2c1, r3c3 b2, with cells (marked #): r1c4, r2c5, r3c6 b4, with cells (marked #): r6c2, r5c1, r4c3 b5, with cells (marked #): r6c4, r5c6, r4c5 with 2 guardians (in cells marked @): n2r1c2 n6r4c3 +----------------------+----------------------+----------------------+ ! 1 2348#@ 23478 ! 348# 5 6 ! 247 378 9 ! ! 348# 5 23478 ! 1 348# 9 ! 247 378 6 ! ! 6 9 348# ! 2 7 348# ! 45 1358 1348 ! +----------------------+----------------------+----------------------+ ! 2 1 3468#@ ! 7 348# 5 ! 9 368 348 ! ! 348# 7 5 ! 69 69 348# ! 1 2 348 ! ! 9 348# 3468 ! 348# 2 1 ! 457 35678 3478 ! +----------------------+----------------------+----------------------+ ! 348 2348 2348 ! 3458 1348 7 ! 6 9 125 ! ! 5 2468 9 ! 468 1468 248 ! 3 17 127 ! ! 7 236 1 ! 3569 369 23 ! 8 4 25 ! +----------------------+----------------------+----------------------+ EL14c30s-OR4-relation for digits: 3, 4 and 8 in cells (marked #): (r7c1 r7c4 r2c1 r2c5 r1c3 r1c4 r3c3 r3c6 r5c1 r5c6 r4c3 r4c5 r6c2 r6c4) with 4 guardians (in cells marked @) : n5r7c4 n2r1c3 n7r1c3 n6r4c3 +-------------------------+-------------------------+-------------------------+ ! 1 2348 23478#@ ! 348# 5 6 ! 247 378 9 ! ! 348# 5 23478 ! 1 348# 9 ! 247 378 6 ! ! 6 9 348# ! 2 7 348# ! 45 1358 1348 ! +-------------------------+-------------------------+-------------------------+ ! 2 1 3468#@ ! 7 348# 5 ! 9 368 348 ! ! 348# 7 5 ! 69 69 348# ! 1 2 348 ! ! 9 348# 3468 ! 348# 2 1 ! 457 35678 3478 ! +-------------------------+-------------------------+-------------------------+ ! 348# 2348 2348 ! 3458#@ 1348 7 ! 6 9 125 ! ! 5 2468 9 ! 468 1468 248 ! 3 17 127 ! ! 7 236 1 ! 3569 369 23 ! 8 4 25 ! +-------------------------+-------------------------+-------------------------+ EL14c13s-OR4-relation for digits: 3, 4 and 8 in cells (marked #): (r7c1 r7c4 r1c4 r3c3 r3c6 r2c3 r2c1 r2c5 r5c1 r5c6 r4c3 r4c5 r6c2 r6c4) with 4 guardians (in cells marked @) : n5r7c4 n2r2c3 n7r2c3 n6r4c3 +-------------------------+-------------------------+-------------------------+ ! 1 2348 23478 ! 348# 5 6 ! 247 378 9 ! ! 348# 5 23478#@ ! 1 348# 9 ! 247 378 6 ! ! 6 9 348# ! 2 7 348# ! 45 1358 1348 ! +-------------------------+-------------------------+-------------------------+ ! 2 1 3468#@ ! 7 348# 5 ! 9 368 348 ! ! 348# 7 5 ! 69 69 348# ! 1 2 348 ! ! 9 348# 3468 ! 348# 2 1 ! 457 35678 3478 ! +-------------------------+-------------------------+-------------------------+ ! 348# 2348 2348 ! 3458#@ 1348 7 ! 6 9 125 ! ! 5 2468 9 ! 468 1468 248 ! 3 17 127 ! ! 7 236 1 ! 3569 369 23 ! 8 4 25 ! +-------------------------+-------------------------+-------------------------+

biv-chain[3]: r9c6{n3 n2} - r9c9{n2 n5} - b8n5{r9c4 r7c4} ==> r7c4≠3
z-chain[3]: c8n5{r6 r3} - r3n1{c8 c9} - c9n3{r3 .} ==> r6c8≠3
z-chain[3]: c8n5{r6 r3} - r3n1{c8 c9} - c9n8{r3 .} ==> r6c8≠8
z-chain[4]: b6n8{r6c9 r4c8} - c8n6{r4 r6} - c8n5{r6 r3} - r3n1{c8 .} ==> r3c9≠8
whip[1]: c9n8{r6 .} ==> r4c8≠8
z-chain[4]: b6n3{r6c9 r4c8} - c8n6{r4 r6} - c8n5{r6 r3} - r3n1{c8 .} ==> r3c9≠3
whip[1]: c9n3{r6 .} ==> r4c8≠3
singles ==> r4c8=6, r6c3=6

Code: Select all: +-------------------+-------------------+-------------------+ ! 1 2348 23478 ! 348 5 6 ! 247 378 9 ! ! 348 5 23478 ! 1 348 9 ! 247 378 6 ! ! 6 9 348 ! 2 7 348 ! 45 1358 14 ! +-------------------+-------------------+-------------------+ ! 2 1 348 ! 7 348 5 ! 9 6 348 ! ! 348 7 5 ! 69 69 348 ! 1 2 348 ! ! 9 348 6 ! 348 2 1 ! 457 57 3478 ! +-------------------+-------------------+-------------------+ ! 348 2348 2348 ! 458 1348 7 ! 6 9 125 ! ! 5 2468 9 ! 468 1468 248 ! 3 17 127 ! ! 7 236 1 ! 3569 369 23 ! 8 4 25 ! +-------------------+-------------------+-------------------+ At least one candidate of a previous Trid-OR2-relation between candidates n2r1c2 n6r4c3 has just been eliminated. There remains a Trid-OR1-relation between candidates: n2r1c2

Trid-ORk-relation with only one candidate => r1c2=2

Code: Select all: singles ==> r7c3=2, r2c7=2 +----------------+----------------+----------------+ ! 1 2 3478 ! 348 5 6 ! 47 378 9 ! ! 348 5 3478 ! 1 348 9 ! 2 378 6 ! ! 6 9 348 ! 2 7 348 ! 45 1358 14 ! +----------------+----------------+----------------+ ! 2 1 348 ! 7 348 5 ! 9 6 348 ! ! 348 7 5 ! 69 69 348 ! 1 2 348 ! ! 9 348 6 ! 348 2 1 ! 457 57 3478 ! +----------------+----------------+----------------+ ! 348 348 2 ! 458 1348 7 ! 6 9 15 ! ! 5 468 9 ! 468 1468 248 ! 3 17 127 ! ! 7 36 1 ! 3569 369 23 ! 8 4 25 ! +----------------+----------------+----------------+ At least one candidate of a previous EL14c30s-OR4-relation between candidates n5r7c4 n2r1c3 n7r1c3 n6r4c3 has just been eliminated. There remains an EL14c30s-OR2-relation between candidates: n5r7c4 n7r1c3 +----------------+----------------+----------------+ ! 1 2 3478 ! 348 5 6 ! 47 378 9 ! ! 348 5 3478 ! 1 348 9 ! 2 378 6 ! ! 6 9 348 ! 2 7 348 ! 45 1358 14 ! +----------------+----------------+----------------+ ! 2 1 348 ! 7 348 5 ! 9 6 348 ! ! 348 7 5 ! 69 69 348 ! 1 2 348 ! ! 9 348 6 ! 348 2 1 ! 457 57 3478 ! +----------------+----------------+----------------+ ! 348 348 2 ! 458 1348 7 ! 6 9 15 ! ! 5 468 9 ! 468 1468 248 ! 3 17 127 ! ! 7 36 1 ! 3569 369 23 ! 8 4 25 ! +----------------+----------------+----------------+ At least one candidate of a previous EL14c13s-OR4-relation between candidates n5r7c4 n2r2c3 n7r2c3 n6r4c3 has just been eliminated. There remains an EL14c13s-OR2-relation between candidates: n5r7c4 n7r2c3

EL14c30s-OR2-whip[3]: OR2{{n7r1c3 | n5r7c4}} - r7c9{n5 n1} - r8c8{n1 .} ==> r1c8≠7
EL14c13s-OR2-whip[3]: OR2{{n7r2c3 | n5r7c4}} - r7c9{n5 n1} - r8c8{n1 .} ==> r2c8≠7
singles ==> r1c7=7, r2c3=7

Code: Select all: +----------------+----------------+----------------+ ! 1 2 348 ! 348 5 6 ! 7 38 9 ! ! 348 5 7 ! 1 348 9 ! 2 38 6 ! ! 6 9 348 ! 2 7 348 ! 45 1358 14 ! +----------------+----------------+----------------+ ! 2 1 348 ! 7 348 5 ! 9 6 348 ! ! 348 7 5 ! 69 69 348 ! 1 2 348 ! ! 9 348 6 ! 348 2 1 ! 45 57 3478 ! +----------------+----------------+----------------+ ! 348 348 2 ! 458 1348 7 ! 6 9 15 ! ! 5 468 9 ! 468 1468 248 ! 3 17 127 ! ! 7 36 1 ! 3569 369 23 ! 8 4 25 ! +----------------+----------------+----------------+ At least one candidate of a previous EL14c30s-OR2-relation between candidates n5r7c4 n7r1c3 has just been eliminated. There remains an EL14c30s-OR1-relation between candidates: n5r7c4

EL14c30s-ORk-relation with only one candidate => r7c4=5
S2-tte

by **yzfwsf** » Fri Mar 17, 2023 7:51 am

Code: Select all: Locked Candidates 1 (Pointing): 6 in b7 => r6c2<>6 Locked Candidates 1 (Pointing): 5 in b9 => r3c9<>5,r5c9<>5,r6c9<>5 Hidden Single: 5 in r5 => r5c3=5 Hidden Single: 5 in c2 => r2c2=5 Locked Candidates 2 (Claiming): 6 in r5 => r4c5<>6,r6c4<>6 Hidden Pair: 69 in r5c4,r5c5 => r5c4<>348,r5c5<>348 Cell Forcing Chain: Each candidate in r3c7 true in turn will all lead to: r789c2,r12c3,r7c9<>2,r7c3<>3,r7c3<>4,r7c3<>8 4r3c7 - (4=27)r12c7 - (7=386)r124c8 - 6r4c3 = 2r1c2 - 2r12c3 = 2r7c3 5r3c7 - (5=13786)r12348c8 - 6r4c3 = 2r1c2 - 2r12c3 = 2r7c3 Hidden Single: 2 in r2 => r2c7=2 Hidden Single: 2 in r1 => r1c2=2 Hidden Single: 2 in r7 => r7c3=2 Grouped X-Cycle with Triplet Oddagon: 4r7c1 = r78c2 - r6c2 = 4r16c4 => r7c4<>4 UR Forcing Chain: Each true guardian of UR 69{r59c45} will all lead to: r8c25,r9c45,r5c4<>6,r8c4<>8 3r9c4 - (3=486)r168c4 5r9c4 - (5=3486)r1678c4 3r9c5 - (3=481)r247c5 - (1=5)r7c9 - (5=3486)r1678c4 Hidden Single: 6 in r5 => r5c5=6 Hidden Single: 9 in r5 => r5c4=9 Hidden Single: 6 in r8 => r8c4=6 Hidden Single: 6 in r9 => r9c2=6 Hidden Single: 9 in r9 => r9c5=9 Locked Candidates 1 (Pointing): 3 in b7 => r7c4<>3,r7c5<>3 ALS Discontinuous Nice Loop with Triplet Oddagon: (5=8)r7c4 - r16c4 = r6c2 - (8=4)r8c2 - (4=385)r7c124 => r7c4=5 stte

Website · by **denis_berthier** » Fri Mar 17, 2023 9:23 am

yzfwsf wrote:
Code: Select all
Cell Forcing Chain: 4r3c7 - (4=27)r12c7 - (7=386)r124c8 - 6r4c3 = 2r1c2 - 2r12c3 = 2r7c3 5r3c7 - (5=13786)r12348c8 - 6r4c3 = 2r1c2 - 2r12c3 = 2r7c3

total length = 15

yzfwsf wrote:
Code: Select all
UR Forcing Chain: 3r9c4 - (3=486)r168c4 5r9c4 - (5=3486)r1678c4 3r9c5 - (3=481)r247c5 - (1=5)r7c9 - (5=3486)r1678c4

total length = 16

Are you sure that, even after making the assumption of uniqueness, you need so long chains?

In any case, thanks for this solution. It allows to compare a classical SER or Hodoku style solution [for a puzzle one can consider as easy in T&E(3) - SER = 10.4] to a solution obtained using the impossible patterns (max chain length = 3).
Complexirty inherent in the puzzle can be found either in long chains or in the ORk-patterns.
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The New Sudoku Players' Forum

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Re: #14987 in mith's 158,276 T&E(3) min-expands

Re: #14987 in mith's 158,276 T&E(3) min-expands

Re: #14987 in mith's 158,276 T&E(3) min-expands