The New Sudoku Players' Forum

Website · by **denis_berthier** » Mon Aug 01, 2022 4:10 pm

.
[Added for clarity of this thread]: This post marks a new turn in the use of the anti-tridagon pattern; see the post referenced below for the definition of new ORk-chains.

Using the OR-k-Forcing-Whips introduced here http://forum.enjoysudoku.com/or-k-forcing-whips-t40189.html,
here is a solution of min-expand #339, one of the puzzles that still has 4 guardians after applying W7.

Code: Select all: +-------+-------+-------+ ! . . . ! 4 . 6 ! 7 8 . ! ! . . . ! . . . ! 2 . . ! ! . 8 . ! 2 7 . ! . . . ! +-------+-------+-------+ ! 2 . 8 ! 3 4 . ! . . 7 ! ! 3 7 . ! . 6 . ! . . . ! ! . 4 6 ! 8 . 7 ! . . . ! +-------+-------+-------+ ! . 6 . ! . . . ! 1 9 4 ! ! . 3 4 ! . . . ! 5 2 . ! ! . . . ! . . 4 ! . 7 3 ! +-------+-------+-------+ ...4.678.......2...8.27....2.834...737..6.....468.7....6....194.34...52......4.73;111;44367

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 159 1259 12359 ! 4 1359 6 ! 7 8 159 ! ! 145679 159 13579 ! 159 13589 13589 ! 2 13456 1569 ! ! 14569 8 1359 ! 2 7 1359 ! 3469 13456 1569 ! +----------------------+----------------------+----------------------+ ! 2 159 8 ! 3 4 159 ! 69 156 7 ! ! 3 7 159 ! 159 6 1259 ! 489 145 12589 ! ! 159 4 6 ! 8 1259 7 ! 39 135 1259 ! +----------------------+----------------------+----------------------+ ! 578 6 257 ! 57 2358 2358 ! 1 9 4 ! ! 1789 3 4 ! 1679 189 189 ! 5 2 68 ! ! 1589 1259 1259 ! 1569 12589 4 ! 68 7 3 ! +----------------------+----------------------+----------------------+ 184 candidates

Code: Select all: hidden-pairs-in-a-column: c1{n4 n6}{r2 r3} ==> r3c1≠9, r3c1≠5, r3c1≠1, r2c1≠9, r2c1≠7, r2c1≠5, r2c1≠1 hidden-single-in-a-block ==> r2c3=7 biv-chain[3]: r4c7{n9 n6} - b9n6{r9c7 r8c9} - c9n8{r8 r5} ==> r5c9≠9 z-chain[3]: c4n6{r9 r8} - r8n7{c4 c1} - r8n1{c1 .} ==> r9c4≠1 z-chain[3]: c4n6{r9 r8} - r8n7{c4 c1} - r8n9{c1 .} ==> r9c4≠9 biv-chain[4]: r4c7{n9 n6} - b9n6{r9c7 r8c9} - c9n8{r8 r5} - b6n2{r5c9 r6c9} ==> r6c9≠9 whip[1]: c9n9{r3 .} ==> r3c7≠9 biv-chain[4]: r8c9{n6 n8} - b6n8{r5c9 r5c7} - c7n4{r5 r3} - r3c1{n4 n6} ==> r3c9≠6 z-chain[5]: c1n7{r8 r7} - c1n8{r7 r9} - b9n8{r9c7 r8c9} - r8n6{c9 c4} - r8n7{c4 .} ==> r8c1≠1, r8c1≠9 whip[1]: r8n9{c6 .} ==> r9c5≠9 whip[1]: r8n1{c6 .} ==> r9c5≠1 +-------------------+-------------------+-------------------+ ! 159 1259 12359 ! 4 1359 6 ! 7 8 159 ! ! 46 159 7 ! 159 13589 13589 ! 2 13456 1569 ! ! 46 8 1359 ! 2 7 1359 ! 346 13456 159 ! +-------------------+-------------------+-------------------+ ! 2 159 8 ! 3 4 159 ! 69 156 7 ! ! 3 7 159 ! 159 6 1259 ! 489 145 1258 ! ! 159 4 6 ! 8 1259 7 ! 39 135 125 ! +-------------------+-------------------+-------------------+ ! 578 6 25 ! 57 2358 2358 ! 1 9 4 ! ! 78 3 4 ! 1679 189 189 ! 5 2 68 ! ! 1589 1259 1259 ! 56 258 4 ! 68 7 3 ! +-------------------+-------------------+-------------------+ OR4-anti-tridagon[12] (type diag) for digits 1, 5 and 9 in blocks: b1, with cells: r1c1, r2c2, r3c3 b2, with cells: r1c5, r2c4, r3c6 b4, with cells: r6c1, r4c2, r5c3 b5, with cells: r6c5, r4c6, r5c4 with 4 guardians: n3r1c5 n3r3c3 n3r3c6 n2r6c5

Based on this OR4 relation, several OR4-Forcing-Whips will allow an easy solution.

Code: Select all: OR4-forcing-whip-elim[4] based on OR4-anti-tridagon[12] for n3r3c6, n3r1c5, n3r3c3 and n2r6c5: || n3r3c6 - || n3r1c5 - || n3r3c3 - partial-whip[1]: b3n3{r3c8 r2c8} - || n2r6c5 - partial-whip[2]: c6n2{r5 r7} - r7n3{c6 c5} - ==> r2c5≠3 t-whip[5]: c6n8{r8 r2} - r2n3{c6 c8} - r2n4{c8 c1} - r2n6{c1 c9} - r8c9{n6 .} ==> r8c5≠8 OR4-forcing-whip-elim[5] based on OR4-anti-tridagon[12] for n3r3c6, n3r3c3, n3r1c5 and n2r6c5: || n3r3c6 - || n3r3c3 - || n3r1c5 - partial-whip[1]: c3n3{r1 r3} - || n2r6c5 - partial-whip[3]: r5n2{c6 c9} - r5n8{c9 c7} - c7n4{r5 r3} - ==> r3c7≠3 hidden-single-in-a-column ==> r6c7=3 naked-pairs-in-a-row: r3{c1 c7}{n4 n6} ==> r3c8≠6, r3c8≠4 z-chain[5]: c6n2{r5 r7} - r7c3{n2 n5} - r5c3{n5 n9} - r6n9{c1 c5} - b5n2{r6c5 .} ==> r5c6≠1 t-whip[6]: c5n3{r7 r1} - r2n3{c6 c8} - r2n4{c8 c1} - r2n6{c1 c9} - r8n6{c9 c4} - r9c4{n6 .} ==> r7c5≠5 OR4-forcing-whip-elim[7] based on OR4-anti-tridagon[12] for n3r3c6, n3r3c3, n3r1c5 and n2r6c5: || n3r3c6 - partial-whip[1]: r2n3{c6 c8} - || n3r3c3 - partial-whip[1]: b3n3{r3c8 r2c8} - || n3r1c5 - partial-whip[1]: r2n3{c6 c8} - || n2r6c5 - partial-whip[3]: r5n2{c6 c9} - c9n8{r5 r8} - c9n6{r8 r2} - ==> r2c8≠6 hidden-single-in-a-column ==> r4c8=6 naked-single ==> r4c7=9 z-chain[7]: c6n2{r5 r7} - r7c3{n2 n5} - r5c3{n5 n1} - r4n1{c2 c6} - r8c6{n1 n8} - c9n8{r8 r5} - r5n2{c9 .} ==> r5c6≠9 whip[5]: r8c5{n1 n9} - b5n9{r6c5 r5c4} - c4n1{r5 r8} - r8c6{n1 n8} - r2n8{c6 .} ==> r2c5≠1 whip[7]: b3n4{r2c8 r3c7} - c7n6{r3 r9} - c4n6{r9 r8} - c4n1{r8 r5} - b6n1{r5c8 r6c9} - r6n2{c9 c5} - b5n9{r6c5 .} ==> r2c8≠1 whip[7]: b5n9{r6c5 r5c4} - r8n9{c4 c6} - r8c5{n9 n1} - c4n1{r8 r2} - r2c2{n1 n5} - r4n5{c2 c6} - c6n1{r4 .} ==> r2c5≠9 OR4-forcing-whip-elim[7] based on OR4-anti-tridagon[12] for n3r3c6, n3r3c3, n3r1c5 and n2r6c5: || n3r3c6 - partial-whip[1]: r2n3{c6 c8} - || n3r3c3 - partial-whip[1]: b3n3{r3c8 r2c8} - || n3r1c5 - partial-whip[1]: r2n3{c6 c8} - || n2r6c5 - partial-whip[3]: r5n2{c6 c9} - r5n8{c9 c7} - c7n4{r5 r3} - ==> r2c8≠4 [Edit]: better notation for the OR-k-Forcing-Whips (their content is unchanged)

The end is easy:

Code: Select all: singles ==> r3c7=4, r3c1=6, r2c1=4, r5c7=8, r9c7=6, r8c9=8, r8c1=7, r9c4=5, r7c4=7, r2c9=6, r8c4=6, r5c8=4 finned-x-wing-in-rows: n1{r4 r2}{c2 c6} ==> r3c6≠1 biv-chain[2]: r4n1{c2 c6} - c4n1{r5 r2} ==> r2c2≠1 whip[1]: r2n1{c6 .} ==> r1c5≠1 biv-chain[3]: r4n1{c2 c6} - r5c4{n1 n9} - b4n9{r5c3 r6c1} ==> r6c1≠1 biv-chain[4]: c2n2{r1 r9} - r9c5{n2 n8} - r2c5{n8 n5} - r2c2{n5 n9} ==> r1c2≠9 biv-chain[3]: c2n9{r9 r2} - c4n9{r2 r5} - b4n9{r5c3 r6c1} ==> r9c1≠9 biv-chain[4]: b7n9{r9c3 r9c2} - r2c2{n9 n5} - r2c5{n5 n8} - r9c5{n8 n2} ==> r9c3≠2 biv-chain[3]: c1n1{r1 r9} - r9c3{n1 n9} - b4n9{r5c3 r6c1} ==> r1c1≠9 singles ==> r6c1=9, r5c4=9, r2c4=1 finned-x-wing-in-rows: n1{r5 r3}{c3 c9} ==> r1c9≠1 whip[1]: b3n1{r3c9 .} ==> r3c3≠1 biv-chain[3]: r5c3{n1 n5} - r7n5{c3 c1} - r1c1{n5 n1} ==> r1c3≠1 biv-chain[3]: c1n5{r1 r7} - r7c3{n5 n2} - b1n2{r1c3 r1c2} ==> r1c2≠5 biv-chain[3]: c3n2{r7 r1} - r1n3{c3 c5} - b8n3{r7c5 r7c6} ==> r7c6≠2 singles ==> r5c6=2, r6c9=2 finned-x-wing-in-columns: n5{c2 c6}{r4 r2} ==> r2c5≠5 singles ==> r2c5=8, r9c5=2, r7c5=3, r7c6=8, r7c1=5, r1c1=1, r1c2=2, r9c1=8, r7c3=2, r1c3=3 finned-x-wing-in-columns: n5{c3 c9}{r5 r3} ==> r3c8≠5 finned-x-wing-in-columns: n5{c8 c5}{r6 r2} ==> r2c6≠5 finned-x-wing-in-columns: n5{c6 c3}{r3 r4} ==> r4c2≠5 stte

by **yzfwsf** » Mon Aug 01, 2022 5:42 pm

Hidden Text: Show

Code: Select all: Hidden Pair: 46 in r2c1,r3c1 => r2c1<>1579,r3c1<>159 Hidden Single: 7 in r2 => r2c3=7 Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r2c1<>6 3r3c3 - 3r3c78 = 3r2c8 - (3=15896)r2c24569 3r1c5 - (3=15896)r2c24569 3r3c6 - (3=15896)r2c24569 2r6c5 - (2=1594)r5c3468 - 4r2c8 = 4r2c1 Hidden Single: 6 in c1 => r3c1=6 Hidden Single: 4 in c1 => r2c1=4 Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r9c5<>1,r25c4,r7c56,r9c5<>5,r9c5<>9 3r3c3 - (3=14596)b3p36789 - (6=8)r8c9 - (8=16795)b8p14567 3r1c5 - (3=15896)r2c24569 - (6=8)r8c9 - (8=16795)b8p14567 3r3c6 - (3=14596)b3p36789 - (6=8)r8c9 - (8=16795)b8p14567 2r6c5 - (2=15698)r12368c9 - (8=16795)b8p14567 Naked Pair: in r2c4,r5c4 => r8c4<>19,r9c4<>19, Locked Candidates 1 (Pointing): 1 in b8 => r8c1<>1 Locked Candidates 1 (Pointing): 9 in b8 => r8c1<>9 Naked Triple: in r7c5,r7c6,r9c5 => r8c5<>8,r8c6<>8, Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r2c5<>3 3r3c3 - 3r3c78 = 3r2c8 3r1c5 3r3c6 2r6c5 - (2=83)r79c5 UR Forcing Chain: Each true guardian of UR 38{r27c56} will all lead to: r2c6<>3 1r2c5 - (1=5693)r2c2489 5r2c5 - (5=1693)r2c2489 9r2c5 - (9=1563)r2c2489 1r2c6 5r2c6 9r2c6 (2-3)r7c5 = 3r7c6 2r7c6 - (2=1593)r3458c6 Hidden Single: 3 in r2 => r2c8=3 Hidden Single: 6 in r2 => r2c9=6 Hidden Single: 3 in r6 => r6c7=3 Hidden Single: 6 in r8 => r8c4=6 Hidden Single: 7 in r8 => r8c1=7 Hidden Single: 7 in r7 => r7c4=7 Hidden Single: 8 in r8 => r8c9=8 Full House: r9c7=6 Hidden Single: 6 in r4 => r4c8=6 Hidden Single: 8 in r5 => r5c7=8 Hidden Single: 4 in r5 => r5c8=4 Hidden Single: 4 in r3 => r3c7=4 Full House: r4c7=9 Hidden Single: 5 in c4 => r9c4=5 Finned X-Wing:1r24\c26 fr2c45 => r3c6<>1 Finned X-Wing:5r24\c26 fr2c5 => r3c6<>5 Finned Swordfish:1r248\c256 fr2c4 => r1c5<>1 Locked Candidates 1 (Pointing): 1 in b2 => r2c2<>1 Whip[3]: Supposing 9r1c3 would causes 3 to disappear in Row 1 => r1c3<>9 9r1c3 - r2c2(9=5) - 5b2(p6=p2) - 3r1(c5=.) Whip[3]: Supposing 1r6c1 would causes 9 to disappear in Box 4 => r6c1<>1 1r6c1 - 1r4(c2=c6) - r5c4(1=9) - 9b4(p6=.) Whip[4]: Supposing 9r1c2 would causes 5 to disappear in Box 2 => r1c2<>9 9r1c2 - r2c2(9=5) - r1c1(5=1) - r1c9(1=5) - 5b2(p2=.) Whip[3]: Supposing 9r9c1 would causes 9 to disappear in Column 2 => r9c1<>9 9r9c1 - 9r6(c1=c5) - 9c4(r5=r2) - 9c2(r2=.) Whip[4]: Supposing 5r1c9 would causes 5 to disappear in Row 5 => r1c9<>5 5r1c9 - 5r3(c8=c3) - r7c3(5=2) - 2c6(r7=r5) - 5r5(c6=.) Locked Candidates 1 (Pointing): 5 in b3 => r3c3<>5 Whip[4]: Supposing 1r1c3 would causes 1 to disappear in Box 4 => r1c3<>1 1r1c3 - r1c9(1=9) - r1c1(9=5) - 5c2(r2=r4) - 1b4(p2=.) Whip[4]: Supposing 1r5c6 would causes 2 to disappear in Column 6 => r5c6<>1 1r5c6 - r5c4(1=9) - r5c3(9=5) - r7c3(5=2) - 2c6(r7=.) Whip[4]: Supposing 9r1c5 will result in all candidates in cell r2c4 being impossible => r1c5<>9 9r1c5 - r1c9(9=1) - 1r3(c8=c3) - 1r5(c3=c4) - r2c4(1=.) Whip[3]: Supposing 5r2c6 would causes 8 to disappear in Column 6 => r2c6<>5 5r2c6 - r1c5(5=3) - 3r7(c5=c6) - 8c6(r7=.) Locked Candidates 1 (Pointing): 5 in b2 => r6c5<>5 Whip[4]: Supposing 5r1c3 would causes 5 to disappear in Column 2 => r1c3<>5 5r1c3 - r7c3(5=2) - 2c6(r7=r5) - 5c6(r5=r4) - 5c2(r4=.) Whip[3]: Supposing 1r9c2 would causes 1 to disappear in Column 1 => r9c2<>1 1r9c2 - r4c2(1=5) - 5b1(p5=p1) - 1c1(r1=.) Whip[4]: Supposing 5r1c2 would causes 9 to disappear in Box 7 => r1c2<>5 5r1c2 - r2c2(5=9) - r1c1(9=1) - 1r9(c1=c3) - 9b7(p9=.) Whip[3]: Supposing 8r7c5 would causes 3 to disappear in Column 5 => r7c5<>8 8r7c5 - r7c1(8=5) - 5r1(c1=c5) - 3c5(r1=.) Whip[4]: Supposing 9r2c2 would causes 9 to disappear in Box 2 => r2c2<>9 9r2c2 - 5c2(r2=r4) - r4c6(5=1) - r8c6(1=9) - 9b2(p9=.) Hidden Single: 9 in c2 => r9c2=9 Hidden Single: 2 in c2 => r1c2=2 Hidden Single: 1 in c2 => r4c2=1 Full House: r4c6=5 Full House: r2c2=5 Hidden Single: 5 in r1 => r1c5=5 Hidden Single: 3 in r1 => r1c3=3 Hidden Single: 3 in r3 => r3c6=3 Hidden Single: 3 in r7 => r7c5=3 Whip[3]: Supposing 9r2c5 will result in all candidates in cell r8c5 being impossible => r2c5<>9 9r2c5 - r2c4(9=1) - 1c6(r2=r8) - r8c5(1=.) Whip[3]: Supposing 9r5c3 would causes 5 to disappear in Column 3 => r5c3<>9 9r5c3 - r5c6(9=2) - 2r7(c6=c3) - 5c3(r7=.) stte

If i disable UR-Foring chain, the path will be longer.

Hidden Text: Show

Code: Select all: Hidden Pair: 46 in r2c1,r3c1 => r2c1<>1579,r3c1<>159 Hidden Single: 7 in r2 => r2c3=7 Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r2c1<>6 3r3c3 - 3r3c78 = 3r2c8 - (3=15896)r2c24569 3r1c5 - (3=15896)r2c24569 3r3c6 - (3=15896)r2c24569 2r6c5 - (2=1594)r5c3468 - 4r2c8 = 4r2c1 Hidden Single: 6 in c1 => r3c1=6 Hidden Single: 4 in c1 => r2c1=4 Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r9c5<>1,r25c4,r7c56,r9c5<>5,r9c5<>9 3r3c3 - (3=14596)b3p36789 - (6=8)r8c9 - (8=16795)b8p14567 3r1c5 - (3=15896)r2c24569 - (6=8)r8c9 - (8=16795)b8p14567 3r3c6 - (3=14596)b3p36789 - (6=8)r8c9 - (8=16795)b8p14567 2r6c5 - (2=15698)r12368c9 - (8=16795)b8p14567 Naked Pair: in r2c4,r5c4 => r8c4<>19,r9c4<>19, Locked Candidates 1 (Pointing): 1 in b8 => r8c1<>1 Locked Candidates 1 (Pointing): 9 in b8 => r8c1<>9 Naked Triple: in r7c5,r7c6,r9c5 => r8c5<>8,r8c6<>8, Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r2c5<>3 3r3c3 - 3r3c78 = 3r2c8 3r1c5 3r3c6 2r6c5 - (2=83)r79c5 Whip[3]: Supposing 9r5c9 would causes 8 to disappear in Box 6 => r5c9<>9 9r5c9 - r4c7(9=6) - r9c7(6=8) - 8b6(p4=.) Whip[4]: Supposing 9r3c7 would causes 4 to disappear in Box 3 => r3c7<>9 9r3c7 - r4c7(9=6) - 6c8(r4=r2) - 3b3(p5=p8) - 4b3(p8=.) Locked Candidates 2 (Claiming): 9 in c7 => r6c9<>9 Whip[4]: Supposing 1r5c6 would causes 2 to disappear in Column 6 => r5c6<>1 1r5c6 - r5c4(1=9) - r5c3(9=5) - r7c3(5=2) - 2c6(r7=.) Whip[4]: Supposing 9r5c6 would causes 2 to disappear in Column 6 => r5c6<>9 9r5c6 - r5c4(9=1) - r5c3(1=5) - r7c3(5=2) - 2c6(r7=.) Whip[6]: Supposing 1r2c8 would causes 1 to disappear in Column 4 => r2c8<>1 1r2c8 - 3r2(c8=c6) - 8r2(c6=c5) - r9c5(8=2) - 2r6(c5=c9) - 1b6(p9=p6) - 1c4(r5=.) Whip[7]: Supposing 1r2c5 will result in all candidates in cell r8c5 being impossible => r2c5<>1 1r2c5 - r2c4(1=9) - r2c2(9=5) - r2c9(5=6) - 6c8(r2=r4) - r4c7(6=9) - 9c6(r4=r8) - r8c5(9=.) Whip[7]: Supposing 9r2c5 will result in all candidates in cell r8c5 being impossible => r2c5<>9 9r2c5 - r2c4(9=1) - r2c2(1=5) - r2c9(5=6) - 6c8(r2=r4) - 5r4(c8=c6) - 1c6(r4=r8) - r8c5(1=.) Whip[8]: Supposing 1r1c2 will result in all candidates in cell r5c4 being impossible => r1c2<>1 1r1c2 - 2r1(c2=c3) - 3r1(c3=c5) - 3r2(c6=c8) - 6c8(r2=r4) - 1r4(c8=c6) - r8c6(1=9) - 9c5(r8=r6) - r5c4(9=.) Whip[8]: Supposing 9r1c2 will result in all candidates in cell r9c7 being impossible => r1c2<>9 9r1c2 - 2c2(r1=r9) - r9c5(2=8) - r2c5(8=5) - r2c2(5=1) - 1c4(r2=r5) - 1r4(c6=c8) - 6r4(c8=c7) - r9c7(6=.) Whip[8]: Supposing 2r9c2 would causes 1 to disappear in Column 2 => r9c2<>2 2r9c2 - r7c3(2=5) - 5r9(c1=c4) - 6r9(c4=c7) - r4c7(6=9) - 9c2(r4=r2) - 9c4(r2=r5) - r5c3(9=1) - 1c2(r4=.) Hidden Single: 2 in c2 => r1c2=2 Whip[8]: Supposing 2r7c5 would causes 2 to disappear in Box 5 => r7c5<>2 2r7c5 - r7c3(2=5) - r7c4(5=7) - r7c1(7=8) - 8c6(r7=r2) - r2c5(8=5) - 5c2(r2=r4) - 5b5(p3=p6) - 2b5(p6=.) Whip[8]: Supposing 9r1c3 would causes 3 to disappear in Row 2 => r1c3<>9 9r1c3 - 3r1(c3=c5) - r7c5(3=8) - r2c5(8=5) - r2c2(5=1) - 1c4(r2=r5) - 1r4(c6=c8) - 6c8(r4=r2) - 3r2(c8=.) Whip[10]: Supposing 1r1c3 will result in all candidates in cell r1c1 being impossible => r1c3<>1 1r1c3 - 3r1(c3=c5) - r7c5(3=8) - r2c5(8=5) - r2c2(5=9) - r2c4(9=1) - r2c9(1=6) - r8c9(6=8) - r8c1(8=7) - r7c1(7=5) - r1c1(5=.) Whip[10]: Supposing 5r1c3 will result in all candidates in cell r5c4 being impossible => r1c3<>5 5r1c3 - 3r1(c3=c5) - 3r2(c6=c8) - 6c8(r2=r4) - 6c7(r4=r9) - r9c4(6=5) - 5c2(r9=r4) - 1r4(c2=c6) - r8c6(1=9) - 9c5(r8=r6) - r5c4(9=.) Naked Single: r1c3=3 Hidden Single: 3 in c5 => r7c5=3 Whip[10]: Supposing 5r2c8 would causes 5 to disappear in Row 6 => r2c8<>5 5r2c8 - r2c5(5=8) - r9c5(8=2) - 2r6(c5=c9) - 5b6(p9=p6) - 8r5(c9=c7) - 8r9(c7=c1) - r8c1(8=7) - r7c1(7=5) - 5r1(c1=c5) - 5r6(c5=.) Whip[9]: Supposing 5r5c9 will result in all candidates in cell r5c6 being impossible => r5c9<>5 5r5c9 - 5c8(r6=r3) - 4r3(c8=c7) - 3b3(p7=p5) - 6r2(c8=c9) - 6r8(c9=c4) - r9c4(6=5) - 5c3(r9=r7) - 2r7(c3=c6) - r5c6(2=.) Whip[6]: Supposing 5r2c6 would causes 6 to disappear in Column 8 => r2c6<>5 5r2c6 - 3r2(c6=c8) - r3c7(3=4) - 4r5(c7=c8) - 5r5(c8=c3) - 5r4(c2=c8) - 6c8(r4=.) Whip[8]: Supposing 9r2c2 would causes 9 to disappear in Column 3 => r2c2<>9 9r2c2 - 9c4(r2=r5) - 9r4(c6=c7) - 6r4(c7=c8) - 6r2(c8=c9) - 5r2(c9=c5) - 8c5(r2=r9) - 2r9(c5=c3) - 9c3(r9=.) Whip[5]: Supposing 5r2c5 would causes 8 to disappear in Column 5 => r2c5<>5 5r2c5 - r2c2(5=1) - 1c4(r2=r5) - 1c3(r5=r9) - 2r9(c3=c5) - 8c5(r9=.) Naked Single: r2c5=8 Hidden Single: 8 in c6 => r7c6=8 Hidden Single: 2 in r7 => r7c3=2 Hidden Single: 2 in r9 => r9c5=2 Hidden Single: 2 in r6 => r6c9=2 Hidden Single: 2 in r5 => r5c6=2 Locked Candidates 1 (Pointing): 5 in b6 => r3c8<>5 Whip[5]: Supposing 1r2c6 would causes 1 to disappear in Column 4 => r2c6<>1 1r2c6 - 3r2(c6=c8) - r3c7(3=4) - r3c8(4=1) - 1c9(r1=r5) - 1c4(r5=.) Whip[5]: Supposing 5r3c3 would causes 5 to disappear in Column 6 => r3c3<>5 5r3c3 - r2c2(5=1) - 1c4(r2=r5) - r5c3(1=9) - r4c2(9=5) - 5c6(r4=.) Whip[4]: Supposing 5r4c2 would causes 1 to disappear in Column 4 => r4c2<>5 5r4c2 - r2c2(5=1) - r3c3(1=9) - r5c3(9=1) - 1c4(r5=.) Whip[5]: Supposing 5r1c1 would causes 5 to disappear in Box 2 => r1c1<>5 5r1c1 - r2c2(5=1) - 1c4(r2=r5) - 1r4(c6=c8) - 5r4(c8=c6) - 5b2(p9=.) Hidden Single: 5 in b1 => r2c2=5 Whip[5]: Supposing 9r5c3 would causes 9 to disappear in Column 4 => r5c3<>9 9r5c3 - 5r5(c3=c8) - 5r4(c8=c6) - 5r3(c6=c9) - 9r3(c9=c6) - 9c4(r2=.) Whip[3]: Supposing 1r9c1 will result in all candidates in cell r1c1 being impossible => r9c1<>1 1r9c1 - r9c2(1=9) - 9c3(r9=r3) - r1c1(9=.) Whip[6]: Supposing 1r1c5 will result in all candidates in cell r2c4 being impossible => r1c5<>1 1r1c5 - r1c1(1=9) - r1c9(9=5) - 5c5(r1=r6) - 9r6(c5=c7) - 9r5(c7=c4) - r2c4(9=.) Whip[3]: Supposing 9r3c9 would causes 5 to disappear in Box 3 => r3c9<>9 9r3c9 - r3c3(9=1) - 1r1(c1=c9) - 5b3(p3=.) Whip[3]: Supposing 9r4c6 will result in all candidates in cell r5c4 being impossible => r4c6<>9 9r4c6 - r8c6(9=1) - 1c5(r8=r6) - r5c4(1=.) Whip[4]: Supposing 9r1c1 would causes 1 to disappear in Column 1 => r1c1<>9 9r1c1 - 9r3(c3=c6) - r8c6(9=1) - 1c5(r8=r6) - 1c1(r6=.) Hidden Single: 9 in b1 => r3c3=9 Full House: r1c1=1 Whip[3]: Supposing 1r4c6 would causes 5 to disappear in Box 5 => r4c6<>1 1r4c6 - r4c2(1=9) - r6c1(9=5) - 5b5(p8=.) Naked Single: r4c6=5 Hidden Single: 5 in r3 => r3c9=5 Hidden Single: 5 in r1 => r1c5=5 Full House: r1c9=9 X-Wing:1c49\r25 => r5c38<>1 stte

Website · by **denis_berthier** » Mon Aug 01, 2022 5:53 pm

.
Hi yzfwsf

I'm quite surprised that in the few solutions you have given based on your equivalent of my OR3 or OR4 Forcing chains, you have very long cumulated ALS-chains; e.g. for the first in your previous post:

Code: Select all: Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r2c1<>6 3r3c3 - 3r3c78 = 3r2c8 - (3=15896)r2c24569 3r1c5 - (3=15896)r2c24569 3r3c6 - (3=15896)r2c24569 2r6c5 - (2=1594)r5c3468 - 4r2c8 = 4r2c1

Total length: 1 (for the OR4) + 6 + 5 + 5 + 5 = 22
The longest I need is 7.

by **yzfwsf** » Mon Aug 01, 2022 6:28 pm

Hi denis_berthier:
The algorithm of my solver is speed priority, not for rating the puzzle, so the Cell/Region Foring chain does not use the simplest strategy, but a by-product of searching for the Als chain, basically without any additional consumption.If desired, users can use the "Search all possible steps" feature to create their own solution paths.

Website · by **denis_berthier** » Mon Aug 08, 2022 7:36 am

.
A puzzle with lots of guardians: #1182 in mith's list of 63,137 min-expands.

Code: Select all: +-------+-------+-------+ ! . 2 . ! 4 . . ! . . . ! ! . . 7 ! . . . ! . . 6 ! ! 6 . 8 ! . . . ! . 1 5 ! +-------+-------+-------+ ! . . . ! 5 . 4 ! . 6 1 ! ! . . . ! . 9 . ! . . 2 ! ! . 6 . ! . 1 2 ! . 9 . ! +-------+-------+-------+ ! . . . ! . . 5 ! 1 . . ! ! 5 . . ! . . . ! . 2 4 ! ! 9 . 1 ! 2 4 . ! . . . ! +-------+-------+-------+ .2.4.......7.....66.8....15...5.4.61....9...2.6..12.9......51..5......249.124....;236;29694 SER = 11.7

If we try to find an anti-tridagon pattern just after Singles, we find two, one with 9 guardians and one with 12:

Code: Select all: +----------------------+----------------------+----------------------+ ! 13 2 359 ! 4 35678 136789 ! 3789 378 3789 ! ! 134 13459 7 ! 1389 2358 1389 ! 23489 348 6 ! ! 6 349 8 ! 379 237 379 ! 23479 1 5 ! +----------------------+----------------------+----------------------+ ! 2378 3789 239 ! 5 378 4 ! 378 6 1 ! ! 13478 134578 345 ! 3678 9 3678 ! 34578 34578 2 ! ! 3478 6 345 ! 378 1 2 ! 34578 9 378 ! +----------------------+----------------------+----------------------+ ! 23478 3478 2346 ! 36789 3678 5 ! 1 378 3789 ! ! 5 378 36 ! 136789 3678 136789 ! 36789 2 4 ! ! 9 378 1 ! 2 4 3678 ! 35678 3578 378 ! +----------------------+----------------------+----------------------+ OR9-anti-tridagon[12] (type diag) for digits 7, 8 and 3 in blocks: b5, with cells: r4c5, r5c6, r6c4 b6, with cells: r4c7, r5c8, r6c9 b8, with cells: r8c5, r9c6, r7c4 b9, with cells: r8c7, r9c9, r7c8 with 9 guardians: n6r5c6 n4r5c8 n5r5c8 n6r7c4 n9r7c4 n6r8c5 n6r8c7 n9r8c7 n6r9c6 OR12-anti-tridagon[12] (type antidiag) for digits 7, 8 and 3 in blocks: b5, with cells: r4c5, r5c6, r6c4 b6, with cells: r4c7, r5c8, r6c9 b8, with cells: r7c5, r9c6, r8c4 b9, with cells: r7c9, r9c8, r8c7 with 12 guardians: n6r5c6 n4r5c8 n5r5c8 n6r7c5 n9r7c9 n1r8c4 n6r8c4 n9r8c4 n6r8c7 n9r8c7 n6r9c6 n5r9c8

In instead of Singles, we allow Subsets + Finned Fish (IMO, the minimal reasonable thing to do before looking for any anti-tridagon), the situation is hardly better:

Code: Select all: hidden-pairs-in-a-column: c2{n1 n5}{r2 r5} ==> r5c2≠8, r5c2≠7, r5c2≠4, r5c2≠3, r2c2≠9, r2c2≠4, r2c2≠3 +----------------------+----------------------+----------------------+ ! 13 2 359 ! 4 35678 136789 ! 3789 378 3789 ! ! 134 15 7 ! 1389 2358 1389 ! 23489 348 6 ! ! 6 349 8 ! 379 237 379 ! 23479 1 5 ! +----------------------+----------------------+----------------------+ ! 2378 3789 239 ! 5 378 4 ! 378 6 1 ! ! 13478 15 345 ! 3678 9 3678 ! 34578 34578 2 ! ! 3478 6 345 ! 378 1 2 ! 34578 9 378 ! +----------------------+----------------------+----------------------+ ! 23478 3478 2346 ! 36789 3678 5 ! 1 378 3789 ! ! 5 378 36 ! 136789 3678 136789 ! 36789 2 4 ! ! 9 378 1 ! 2 4 3678 ! 35678 3578 378 ! +----------------------+----------------------+----------------------+ OR9-anti-tridagon[12] (type diag) for digits 7, 8 and 3 in blocks: b5, with cells: r4c5, r5c6, r6c4 b6, with cells: r4c7, r5c8, r6c9 b8, with cells: r8c5, r9c6, r7c4 b9, with cells: r8c7, r9c9, r7c8 with 9 guardians: n6r5c6 n4r5c8 n5r5c8 n6r7c4 n9r7c4 n6r8c5 n6r8c7 n9r8c7 n6r9c6 OR12-anti-tridagon[12] (type antidiag) for digits 7, 8 and 3 in blocks: b5, with cells: r4c5, r5c6, r6c4 b6, with cells: r4c7, r5c8, r6c9 b8, with cells: r7c5, r9c6, r8c4 b9, with cells: r7c9, r9c8, r8c7 with 12 guardians: n6r5c6 n4r5c8 n5r5c8 n6r7c5 n9r7c9 n1r8c4 n6r8c4 n9r8c4 n6r8c7 n9r8c7 n6r9c6 n5r9c8

The situation is quite better is one starts with Subsets + Finned Fish + Whips[≤4]: there remains only one anti-tridagon-pattern and it has only 5 guardians:

Code: Select all: hidden-pairs-in-a-column: c2{n1 n5}{r2 r5} ==> r5c2≠8, r5c2≠7, r5c2≠4, r5c2≠3, r2c2≠9, r2c2≠4, r2c2≠3 whip[3]: r1c1{n3 n1} - r2c2{n1 n5} - c5n5{r2 .} ==> r1c5≠3 whip[4]: c5n6{r8 r1} - r1n5{c5 c3} - r2c2{n5 n1} - c4n1{r2 .} ==> r8c4≠6 whip[4]: r3n4{c7 c2} - b1n9{r3c2 r1c3} - b3n9{r1c9 r3c7} - c7n2{r3 .} ==> r2c7≠4 whip[4]: r2c2{n1 n5} - r1n5{c3 c5} - r1n6{c5 c6} - r1n1{c6 .} ==> r2c1≠1 whip[3]: c7n2{r2 r3} - r3n4{c7 c2} - r2c1{n4 .} ==> r2c7≠3 whip[4]: r2n9{c6 c7} - r3n9{c7 c2} - r3n4{c2 c7} - c7n2{r3 .} ==> r1c6≠9 whip[4]: r3n7{c6 c7} - r3n4{c7 c2} - b1n9{r3c2 r1c3} - r1n5{c3 .} ==> r1c5≠7 whip[4]: r6n5{c7 c3} - r1n5{c3 c5} - r1n6{c5 c6} - r9n6{c6 .} ==> r9c7≠5 hidden-single-in-a-block ==> r9c8=5 whip[4]: r1c1{n3 n1} - r2c2{n1 n5} - r1n5{c3 c5} - r1n6{c5 .} ==> r1c6≠3 whip[3]: r3n4{c7 c2} - r2c1{n4 n3} - b2n3{r2c6 .} ==> r3c7≠3 whip[4]: r3n4{c7 c2} - b1n9{r3c2 r1c3} - b3n9{r1c9 r2c7} - c7n2{r2 .} ==> r3c7≠7 whip[1]: r3n7{c6 .} ==> r1c6≠7 whip[4]: r2n2{c7 c5} - c5n5{r2 r1} - r1n8{c5 c6} - r1n6{c6 .} ==> r2c7≠8 whip[4]: r1n6{c6 c5} - r1n5{c5 c3} - r2c2{n5 n1} - b2n1{r2c4 .} ==> r1c6≠8 whip[4]: r2c1{n3 n4} - r2c8{n4 n8} - r1n8{c9 c5} - c5n5{r1 .} ==> r2c5≠3 whip[4]: r1n6{c5 c6} - r1n1{c6 c1} - r2c2{n1 n5} - c5n5{r2 .} ==> r1c5≠8 whip[1]: r1n8{c9 .} ==> r2c8≠8 naked-pairs-in-a-row: r2{c1 c8}{n3 n4} ==> r2c6≠3, r2c4≠3 whip[1]: b2n3{r3c6 .} ==> r3c2≠3 whip[3]: c3n2{r7 r4} - r4n9{c3 c2} - c2n3{r4 .} ==> r7c3≠3 whip[3]: r4n2{c1 c3} - c3n9{r4 r1} - b1n3{r1c3 .} ==> r4c1≠3 whip[4]: c3n6{r8 r7} - c3n2{r7 r4} - r4n9{c3 c2} - c2n3{r4 .} ==> r8c3≠3 singles ==> r8c3=6, r9c7=6 +-------------------+-------------------+-------------------+ ! 13 2 359 ! 4 56 16 ! 3789 378 3789 ! ! 34 15 7 ! 189 258 189 ! 29 34 6 ! ! 6 49 8 ! 379 237 379 ! 249 1 5 ! +-------------------+-------------------+-------------------+ ! 278 3789 239 ! 5 378 4 ! 378 6 1 ! ! 13478 15 345 ! 3678 9 3678 ! 34578 3478 2 ! ! 3478 6 345 ! 378 1 2 ! 34578 9 378 ! +-------------------+-------------------+-------------------+ ! 23478 3478 24 ! 36789 3678 5 ! 1 378 3789 ! ! 5 378 6 ! 13789 378 13789 ! 3789 2 4 ! ! 9 378 1 ! 2 4 378 ! 6 5 378 ! +-------------------+-------------------+-------------------+ OR5-anti-tridagon[12] (type diag) for digits 7, 8 and 3 in blocks: b5, with cells: r4c5, r5c6, r6c4 b6, with cells: r4c7, r5c8, r6c9 b8, with cells: r8c5, r9c6, r7c4 b9, with cells: r8c7, r9c9, r7c8 with 5 guardians: n6r5c6 n4r5c8 n6r7c4 n9r7c4 n9r8c7

Now, the first thing one may want to try is wether the puzzle can be solved in W4+OR5FW4. Unfortunately, the answer is negative.
From this point on, one can progressively increase the max sizes of allowed chains, independently the size of normal chains (bivalue, z-, whips...) and the size of ORk-Forcing-Whips.
For this puzzle, I found a good balance with all chains ≤ 6 and Forcing-Whips ≤ 9:

Code: Select all: hidden-pairs-in-a-column: c2{n1 n5}{r2 r5} ==> r5c2≠8, r5c2≠7, r5c2≠4, r5c2≠3, r2c2≠9, r2c2≠4, r2c2≠3 biv-chain[3]: r1c1{n3 n1} - r2c2{n1 n5} - b2n5{r2c5 r1c5} ==> r1c5≠3 biv-chain[4]: r1c1{n3 n1} - r2c2{n1 n5} - b2n5{r2c5 r1c5} - b2n6{r1c5 r1c6} ==> r1c6≠3 biv-chain[4]: r1n1{c1 c6} - b2n6{r1c6 r1c5} - b2n5{r1c5 r2c5} - r2c2{n5 n1} ==> r2c1≠1 biv-chain[3]: r2c1{n3 n4} - r3n4{c2 c7} - b3n2{r3c7 r2c7} ==> r2c7≠3 whip[3]: r3n4{c7 c2} - r2c1{n4 n3} - b2n3{r2c4 .} ==> r3c7≠3 biv-chain[4]: r1n1{c6 c1} - r2c2{n1 n5} - b2n5{r2c5 r1c5} - b2n6{r1c5 r1c6} ==> r1c6≠7, r1c6≠8, r1c6≠9 biv-chain[3]: r8n1{c4 c6} - r1c6{n1 n6} - b5n6{r5c6 r5c4} ==> r8c4≠6 z-chain[3]: r1n8{c9 c5} - c5n5{r1 r2} - r2n2{c5 .} ==> r2c7≠8 biv-chain[4]: r1n5{c5 c3} - r2c2{n5 n1} - r1n1{c1 c6} - b2n6{r1c6 r1c5} ==> r1c5≠7, r1c5≠8 whip[1]: r1n8{c9 .} ==> r2c8≠8 whip[1]: r1n7{c9 .} ==> r3c7≠7 naked-pairs-in-a-row: r2{c1 c8}{n3 n4} ==> r2c7≠4, r2c6≠3, r2c5≠3, r2c4≠3 whip[1]: b2n3{r3c6 .} ==> r3c2≠3 z-chain[3]: c2n3{r9 r4} - r4n9{c2 c3} - c3n2{r4 .} ==> r7c3≠3 z-chain[3]: b1n3{r2c1 r1c3} - c3n9{r1 r4} - r4n2{c3 .} ==> r4c1≠3 biv-chain[4]: r6n5{c7 c3} - r1n5{c3 c5} - b2n6{r1c5 r1c6} - r9n6{c6 c7} ==> r9c7≠5 hidden-single-in-a-block ==> r9c8=5 z-chain[4]: c2n3{r9 r4} - r4n9{c2 c3} - c3n2{r4 r7} - c3n6{r7 .} ==> r8c3≠3 naked-single ==> r8c3=6 hidden-single-in-a-block ==> r9c7=6 z-chain[5]: b6n5{r6c7 r5c7} - b6n4{r5c7 r5c8} - r2n4{c8 c1} - r6n4{c1 c3} - r6n5{c3 .} ==> r6c7≠3, r6c7≠8, r6c7≠7 whip[6]: r5n6{c4 c6} - r1n6{c6 c5} - r1n5{c5 c3} - r5c3{n5 n4} - r6n4{c3 c7} - r6n5{c7 .} ==> r5c4≠3 +-------------------+-------------------+-------------------+ ! 13 2 359 ! 4 56 16 ! 3789 378 3789 ! ! 34 15 7 ! 189 258 189 ! 29 34 6 ! ! 6 49 8 ! 379 237 379 ! 249 1 5 ! +-------------------+-------------------+-------------------+ ! 278 3789 239 ! 5 378 4 ! 378 6 1 ! ! 13478 15 345 ! 678 9 3678 ! 34578 3478 2 ! ! 3478 6 345 ! 378 1 2 ! 45 9 378 ! +-------------------+-------------------+-------------------+ ! 23478 3478 24 ! 36789 3678 5 ! 1 378 3789 ! ! 5 378 6 ! 13789 378 13789 ! 3789 2 4 ! ! 9 378 1 ! 2 4 378 ! 6 5 378 ! +-------------------+-------------------+-------------------+ OR5-anti-tridagon[12] (type diag) for digits 7, 8 and 3 in blocks: b5, with cells: r4c5, r5c6, r6c4 b6, with cells: r4c7, r5c8, r6c9 b8, with cells: r8c5, r9c6, r7c4 b9, with cells: r8c7, r9c9, r7c8 with 5 guardians: n6r5c6 n4r5c8 n6r7c4 n9r7c4 n9r8c7 OR5-forcing-whip-elim[9] based on OR5-anti-tridagon[12] for n9r8c7, n4r5c8, n9r7c4, n6r5c6 and n6r7c4: || n9r8c7 - || n4r5c8 - partial-whip[1]: c7n4{r6 r3} - || n9r7c4 - partial-whip[1]: c9n9{r7 r1} - || n6r5c6 - partial-whip[3]: r1n6{c6 c5} - r1n5{c5 c3} - b1n9{r1c3 r3c2} - || n6r7c4 - partial-whip[3]: c5n6{r7 r1} - r1n5{c5 c3} - b1n9{r1c3 r3c2} - ==> r3c7≠9 OR5-forcing-whip-elim[9] based on OR5-anti-tridagon[12] for n9r7c4, n9r8c7, n4r5c8, n6r5c6 and n6r7c4: || n9r7c4 - partial-whip[1]: c9n9{r7 r1} - || n9r8c7 - partial-whip[1]: c9n9{r7 r1} - || n4r5c8 - partial-whip[2]: r2n4{c8 c1} - r3c2{n4 n9} - || n6r5c6 - partial-whip[2]: r1n6{c6 c5} - r1n5{c5 c3} - || n6r7c4 - partial-whip[2]: c5n6{r7 r1} - r1n5{c5 c3} - ==> r1c3≠9 singles ==> r3c2=9, r2c1=4, r2c8=3, r3c7=4, r6c7=5, r2c7=2, r3c5=2, r5c8=4, r6c3=4, r7c3=2, r4c1=2, r7c2=4, r4c3=9 z-chain[5]: c5n6{r7 r1} - r1c6{n6 n1} - r1c1{n1 n3} - r7c1{n3 n8} - r7c8{n8 .} ==> r7c5≠7 z-chain[5]: c5n6{r7 r1} - r1c6{n6 n1} - r1c1{n1 n3} - r7c1{n3 n7} - r7c8{n7 .} ==> r7c5≠8 z-chain[4]: r7c5{n3 n6} - r1n6{c5 c6} - c6n1{r1 r2} - c6n9{r2 .} ==> r8c6≠3 z-chain[5]: r3c4{n7 n3} - r6c4{n3 n8} - r4c5{n8 n3} - r7c5{n3 n6} - c4n6{r7 .} ==> r5c4≠7 biv-chain[4]: r5c4{n8 n6} - r7n6{c4 c5} - r1c5{n6 n5} - r2c5{n5 n8} ==> r4c5≠8, r2c4≠8 biv-chain[3]: r8n1{c6 c4} - r2c4{n1 n9} - c6n9{r2 r8} ==> r8c6≠7, r8c6≠8 whip[3]: r4n8{c7 c2} - b7n8{r8c2 r7c1} - c8n8{r7 .} ==> r1c7≠8 biv-chain[3]: r7c8{n7 n8} - r1n8{c8 c9} - c9n9{r1 r7} ==> r7c9≠7 whip[6]: r4n8{c7 c2} - r9n8{c2 c6} - r5n8{c6 c4} - r5n6{c4 c6} - c6n3{r5 r3} - c6n7{r3 .} ==> r8c7≠8 whip[1]: c7n8{r5 .} ==> r6c9≠8 whip[4]: r4n8{c7 c2} - r6n8{c1 c4} - r8n8{c4 c5} - c5n7{r8 .} ==> r4c7≠7 z-chain[3]: b6n7{r5c7 r6c9} - r9n7{c9 c2} - r4n7{c2 .} ==> r5c6≠7 z-chain[3]: c5n3{r8 r4} - b5n7{r4c5 r6c4} - r3c4{n7 .} ==> r8c4≠3, r7c4≠3 whip[4]: r5n7{c1 c7} - r6c9{n7 n3} - b5n3{r6c4 r4c5} - r7n3{c5 .} ==> r5c1≠3 whip[4]: b5n7{r4c5 r6c4} - r6c9{n7 n3} - r7n3{c9 c1} - c2n3{r8 .} ==> r4c5≠3 naked-single ==> r4c5=7 whip[1]: c2n7{r9 .} ==> r7c1≠7 whip[1]: c5n3{r8 .} ==> r9c6≠3 finned-x-wing-in-rows: n3{r9 r4}{c2 c9} ==> r6c9≠3 singles ==> r6c9=7, r5c1=7, r5c2=1, r2c2=5, r1c3=3, r1c1=1, r1c6=6, r1c5=5, r5c3=5, r2c5=8, r8c5=3, r7c5=6, r5c4=6 finned-x-wing-in-rows: n8{r8 r6}{c4 c2} ==> r4c2≠8 stte

Website · by **denis_berthier** » Sun Sep 11, 2022 7:55 am

.
ORk-FW classification results for mith's 63137 min-expand puzzles in T&E(3):

Puzzles solved in SFin+Trid+Wn+ORkFWn

Code: Select all: ----------------------------------------------------------------------- n=3 n=5 n=7 n=8 ----------------------------------------------------------------------- 8 ,196 puzzles solved by SFin+Trid (among 63 137 min-expands) ----------------------------------------------------------------------- k=0 8,137 17,532 21,160 22,332 16,333 9,395 25,728 3,628 29,356 1,172 30,528 ----------------------------------------------------------------------- k=2 3,350 8,713 11,231 12,068 19,683 14,758 34,441 6,146 40,587 2,009 42,596 ----------------------------------------------------------------------- k=3 428 2,255 3,471 20,111 16,585 36,696 7,362 44,058 ----------------------------------------------------------------------- k=4 67 365 540 20,178 16,883 37,061 7,537 44,598 ----------------------------------------------------------------------- k=5 2 28 99 20,180 16,909 37,089 7,608 44,697 ----------------------------------------------------------------------- k=6 0 4 20,180 16,913 37,093 -----------------------------------------------------------------------

Lines are separated by dashes, columns are separated by large white spaces.
Each (k, n) cell has three values in it:
- the main one, in the lower right corner, is the total number of puzzles solved by SFin + Trid + Wn + ORkFWn;
- the value above it is the difference with the previous line; it shows what’s gained by increasing k by 1;
- the value on the left of the main number is the difference with the previous cell; it shows what’s gained by increasing n.

Some general conclusions can be drawn from this table:
• for fixed n, as k increases, the difference between two lines decreases quite fast; this shouldn’t be too surprising, as larger k means more chains have to converge to the same candidate;
• for fixed k, as n increases, the difference between two columns decreases quite fast; this shouldn’t be too surprising either, as it already happens with all the “classical” chains (whips…);
• starting from k=2 and n=3, at any point in the table, it is much more fruitful to increase n than to increase k;

I'll give a similar table for ORk-Contrad-Whips, but the calculations are still running.

[Edit]:added case (k=2, n=8)

Website · by **denis_berthier** » Tue Sep 13, 2022 5:10 am

.
Degenerated trivalue-oddagons: case of 1 decided value

Suppose that, instead of the standard contradictory trivlaue-oddagon pattern, with alll 3 candidates in all 12 cells, one of its cells is a decided value; say r1c1=1.
It is obvious that this pattern is still contradictory. However, it no longer requires T&E(3) to be proven contradictory: this can be done in T&E(2).

The pattern is (modulo isomorphisms):

Code: Select all: +-------------------------------+-------------------------------+-------------------------------+ ! 1 123456789 123456789 ! 123 123456789 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123 123456789 ! 123456789 123 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123 ! 123456789 123456789 123 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+ ! 123 123456789 123456789 ! 123456789 123456789 123 ! 123456789 123456789 123456789 ! ! 123456789 123 123456789 ! 123456789 123 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123 ! 123 123456789 123456789 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+ ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+

Here is a way to prove the contradiction in T&E(2), using SudoRules.
Choose T&E(2) in the configuration file.
If you try to apply function "solve-sukaku-grid" to the above resolution state, computations will take too long.
On the other hand, if you try to use function "solve-k-digit-pattern-string" directly, the candidates n2r1c2 and n3r1c1 will not be deleted before starting.

There is a way out of this. Use the following two commands:

Code: Select all: (bind ?*simulated-eliminations* (create$ 211 311)) (solve-k-digit-pattern-string 3 "100100000010010000001001000100001000010010000001100000000000000000000000000000000")

SudoRules outputs a quick and short proof of the contradiction in T&E(2):

Hidden Text: Show

Code: Select all: *********************************************************************************************** *** SudoRules 20.1.s based on CSP-Rules 2.1.s, config = T&E(BRT, 2) *** Using CLIPS 6.32-r823 *** Running on MacBookPro 16'' M1Max 2021, 64GB LPDDR5, MacOS 12.5 *** Download from: https://github.com/denis-berthier/CSP-Rules-V2.1 *********************************************************************************************** 100100000010010000001001000100001000010010000001100000000000000000000000000000000 Simulated elimination of 311 Simulated elimination of 211 naked-single ==> r1c1=1 Resolution state after Singles: +-------------------------------+-------------------------------+-------------------------------+ ! 1 23456789 23456789 ! 23 23456789 23456789 ! 23456789 23456789 23456789 ! ! 23456789 23 23456789 ! 123456789 123 123456789 ! 123456789 123456789 123456789 ! ! 23456789 23456789 23 ! 123456789 123456789 123 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+ ! 23 123456789 123456789 ! 123456789 123456789 123 ! 123456789 123456789 123456789 ! ! 23456789 123 123456789 ! 123456789 123 123456789 ! 123456789 123456789 123456789 ! ! 23456789 123456789 123 ! 123 123456789 123456789 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+ ! 23456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! ! 23456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! ! 23456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+ 634 candidates, 0 csp-links and 0 links. Density = 0.0% Starting non trivial part of solution. *** STARTING T&E IN CONTEXT 0 at depth 1 with 1 csp-variables solved and 634 candidates remaining *** STARTING PHASE 1 IN CONTEXT 0 with 1 csp-variables solved and 634 candidates remaining GENERATING CONTEXT 1 AT DEPTH 1, SON OF CONTEXT 0, FROM HYPOTHESIS n3r6c4. naked-single ==> r1c4=2 *** STARTING T&E IN CONTEXT 1 at depth 1 with 1 csp-variables solved and 634 candidates remaining *** STARTING PHASE 1 IN CONTEXT 1 AT DEPTH 1, with 1 csp-variables solved and 634 candidates remaining GENERATING CONTEXT 2 AT DEPTH 2, SON OF CONTEXT 1, FROM HYPOTHESIS n2r2c2. naked-single ==> r3c3=3 naked-single ==> r3c6=1 naked-single ==> r2c5=3 naked-single ==> r4c6=2 naked-single ==> r4c1=3 naked-single ==> r5c2=1 NO POSSIBLE VALUE for csp-variable 155 IN CONTEXT 2. RETRACTING CANDIDATE n2r2c2 FROM CONTEXT 1. BACK IN CONTEXT 1 with 1 csp-variables solved and 634 candidates remaining. naked-single ==> r2c2=3 naked-single ==> r3c3=2 naked-single ==> r6c3=1 naked-single ==> r5c2=2 naked-single ==> r5c5=1 NO POSSIBLE VALUE for csp-variable 125 IN CONTEXT 1. RETRACTING CANDIDATE n3r6c4 FROM CONTEXT 0. BACK IN CONTEXT 0 with 1 csp-variables solved and 633 candidates remaining. GENERATING CONTEXT 3 AT DEPTH 1, SON OF CONTEXT 0, FROM HYPOTHESIS n2r6c4. naked-single ==> r1c4=3 *** STARTING T&E IN CONTEXT 3 at depth 1 with 1 csp-variables solved and 633 candidates remaining *** STARTING PHASE 1 IN CONTEXT 3 AT DEPTH 1, with 1 csp-variables solved and 633 candidates remaining GENERATING CONTEXT 4 AT DEPTH 2, SON OF CONTEXT 3, FROM HYPOTHESIS n2r2c2. naked-single ==> r3c3=3 naked-single ==> r6c3=1 naked-single ==> r5c2=3 naked-single ==> r4c1=2 naked-single ==> r5c5=1 NO POSSIBLE VALUE for csp-variable 125 IN CONTEXT 4. RETRACTING CANDIDATE n2r2c2 FROM CONTEXT 3. BACK IN CONTEXT 3 with 1 csp-variables solved and 633 candidates remaining. naked-single ==> r2c2=3 naked-single ==> r3c3=2 naked-single ==> r3c6=1 naked-single ==> r4c6=3 naked-single ==> r5c5=1 naked-single ==> r5c2=2 NO POSSIBLE VALUE for csp-variable 141 IN CONTEXT 3. RETRACTING CANDIDATE n2r6c4 FROM CONTEXT 0. BACK IN CONTEXT 0 with 1 csp-variables solved and 632 candidates remaining. naked-single ==> r6c4=1 GENERATING CONTEXT 5 AT DEPTH 1, SON OF CONTEXT 0, FROM HYPOTHESIS n3r6c3. naked-single ==> r3c3=2 naked-single ==> r2c2=3 naked-single ==> r4c1=2 naked-single ==> r5c2=1 naked-single ==> r4c6=3 naked-single ==> r5c5=2 naked-single ==> r2c5=1 NO POSSIBLE VALUE for csp-variable 136 IN CONTEXT 5. RETRACTING CANDIDATE n3r6c3 FROM CONTEXT 0. BACK IN CONTEXT 0 with 2 csp-variables solved and 612 candidates remaining. naked-single ==> r6c3=2 naked-single ==> r3c3=3 naked-single ==> r2c2=2 naked-single ==> r4c1=3 naked-single ==> r4c6=2 naked-single ==> r3c6=1 naked-single ==> r2c5=3 PUZZLE 0 HAS NO SOLUTION : NO CANDIDATE FOR RC-CELL r5c5 MOST COMPLEX RULE TRIED = NS Puzzle 100100000010010000001001000100001000010010000001100000000000000000000000000000000 : init-time = 0.0s, solve-time = 0.11s, total-time = 0.11s

.

Website · by **denis_berthier** » Tue Sep 13, 2022 5:16 am

.
Degenerated trivalue-oddagons: case of 1 missing candidate

Suppose that, instead of the standard contradictory trivlaue-oddagon pattern, with alll 3 candidates in all 12 cells, one of these candidates is missing; say n3r1c1.
It is obvious that this pattern is still contradictory. However, it no longer requires T&E(3) to be proven contradictory: this can be done in T&E(2).

The pattern is (modulo isomorphisms):

Code: Select all: +-------------------------------+-------------------------------+-------------------------------+ ! 12 123456789 123456789 ! 123 123456789 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123 123456789 ! 123456789 123 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123 ! 123456789 123456789 123 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+ ! 123 123456789 123456789 ! 123456789 123456789 123 ! 123456789 123456789 123456789 ! ! 123456789 123 123456789 ! 123456789 123 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123 ! 123 123456789 123456789 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+ ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+

Here is a way to prove the contradiction in T&E(2), using SudoRules, in the same way as before.
Choose T&E(2) in the configuration file.
Use the following two commands:

Code: Select all: (bind ?*simulated-eliminations* (create$ 311)) (solve-k-digit-pattern-string 3 "100100000010010000001001000100001000010010000001100000000000000000000000000000000")

SudoRules still outputs a quick and short proof (though not as short as before) of the contradiction in T&E(2):

Hidden Text: Show

Code: Select all: *********************************************************************************************** *********************************************************************************************** *** SudoRules 20.1.s based on CSP-Rules 2.1.s, config = T&E(BRT, 2) *** Using CLIPS 6.32-r823 *** Running on MacBookPro 16'' M1Max 2021, 64GB LPDDR5, MacOS 12.5 *** Download from: https://github.com/denis-berthier/CSP-Rules-V2.1 *********************************************************************************************** 100100000010010000001001000100001000010010000001100000000000000000000000000000000 Simulated elimination of 311 Resolution state after Singles: +-------------------------------+-------------------------------+-------------------------------+ ! 12 123456789 123456789 ! 123 123456789 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123 123456789 ! 123456789 123 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123 ! 123456789 123456789 123 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+ ! 123 123456789 123456789 ! 123456789 123456789 123 ! 123456789 123456789 123456789 ! ! 123456789 123 123456789 ! 123456789 123 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123 ! 123 123456789 123456789 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+ ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+ 656 candidates, 0 csp-links and 0 links. Density = 0.0% Starting non trivial part of solution. *** STARTING T&E IN CONTEXT 0 at depth 1 with 0 csp-variables solved and 656 candidates remaining *** STARTING PHASE 1 IN CONTEXT 0 with 0 csp-variables solved and 656 candidates remaining GENERATING CONTEXT 1 AT DEPTH 1, SON OF CONTEXT 0, FROM HYPOTHESIS n3r6c4. *** STARTING T&E IN CONTEXT 1 at depth 1 with 0 csp-variables solved and 656 candidates remaining *** STARTING PHASE 1 IN CONTEXT 1 AT DEPTH 1, with 0 csp-variables solved and 656 candidates remaining GENERATING CONTEXT 2 AT DEPTH 2, SON OF CONTEXT 1, FROM HYPOTHESIS n1r1c1. naked-single ==> r1c4=2 NO CONTRADICTION FOUND IN CONTEXT 2. BACK IN CONTEXT 1 with 0 csp-variables solved and 656 candidates remaining. GENERATING CONTEXT 3 AT DEPTH 2, SON OF CONTEXT 1, FROM HYPOTHESIS n2r1c1. naked-single ==> r1c4=1 NO CONTRADICTION FOUND IN CONTEXT 3. BACK IN CONTEXT 1 with 0 csp-variables solved and 656 candidates remaining. GENERATING CONTEXT 4 AT DEPTH 2, SON OF CONTEXT 1, FROM HYPOTHESIS n1r1c4. naked-single ==> r1c1=2 NO CONTRADICTION FOUND IN CONTEXT 4. BACK IN CONTEXT 1 with 0 csp-variables solved and 656 candidates remaining. GENERATING CONTEXT 5 AT DEPTH 2, SON OF CONTEXT 1, FROM HYPOTHESIS n2r1c4. naked-single ==> r1c1=1 NO CONTRADICTION FOUND IN CONTEXT 5. BACK IN CONTEXT 1 with 0 csp-variables solved and 656 candidates remaining. GENERATING CONTEXT 6 AT DEPTH 2, SON OF CONTEXT 1, FROM HYPOTHESIS n1r2c2. naked-single ==> r1c1=2 naked-single ==> r1c4=1 naked-single ==> r3c3=3 naked-single ==> r3c6=2 naked-single ==> r2c5=3 naked-single ==> r4c6=1 naked-single ==> r4c1=3 naked-single ==> r5c2=2 NO POSSIBLE VALUE for csp-variable 155 IN CONTEXT 6. RETRACTING CANDIDATE n1r2c2 FROM CONTEXT 1. BACK IN CONTEXT 1 with 0 csp-variables solved and 656 candidates remaining. GENERATING CONTEXT 7 AT DEPTH 2, SON OF CONTEXT 1, FROM HYPOTHESIS n2r2c2. naked-single ==> r1c1=1 naked-single ==> r1c4=2 naked-single ==> r3c3=3 naked-single ==> r3c6=1 naked-single ==> r2c5=3 naked-single ==> r4c6=2 naked-single ==> r4c1=3 naked-single ==> r5c2=1 NO POSSIBLE VALUE for csp-variable 155 IN CONTEXT 7. RETRACTING CANDIDATE n2r2c2 FROM CONTEXT 1. BACK IN CONTEXT 1 with 0 csp-variables solved and 656 candidates remaining. naked-single ==> r2c2=3 GENERATING CONTEXT 8 AT DEPTH 2, SON OF CONTEXT 1, FROM HYPOTHESIS n1r2c5. naked-single ==> r5c5=2 naked-single ==> r4c6=1 naked-single ==> r5c2=1 naked-single ==> r6c3=2 naked-single ==> r3c3=1 naked-single ==> r1c1=2 NO POSSIBLE VALUE for csp-variable 114 IN CONTEXT 8. RETRACTING CANDIDATE n1r2c5 FROM CONTEXT 1. BACK IN CONTEXT 1 with 0 csp-variables solved and 656 candidates remaining. naked-single ==> r2c5=2 naked-single ==> r5c5=1 naked-single ==> r5c2=2 naked-single ==> r6c3=1 naked-single ==> r4c1=3 naked-single ==> r3c3=2 naked-single ==> r1c1=1 NO POSSIBLE VALUE for csp-variable 114 IN CONTEXT 1. RETRACTING CANDIDATE n3r6c4 FROM CONTEXT 0. BACK IN CONTEXT 0 with 0 csp-variables solved and 655 candidates remaining. GENERATING CONTEXT 9 AT DEPTH 1, SON OF CONTEXT 0, FROM HYPOTHESIS n2r6c4. *** STARTING T&E IN CONTEXT 9 at depth 1 with 0 csp-variables solved and 655 candidates remaining *** STARTING PHASE 1 IN CONTEXT 9 AT DEPTH 1, with 0 csp-variables solved and 655 candidates remaining GENERATING CONTEXT 10 AT DEPTH 2, SON OF CONTEXT 9, FROM HYPOTHESIS n1r1c1. naked-single ==> r1c4=3 NO CONTRADICTION FOUND IN CONTEXT 10. BACK IN CONTEXT 9 with 0 csp-variables solved and 655 candidates remaining. GENERATING CONTEXT 11 AT DEPTH 2, SON OF CONTEXT 9, FROM HYPOTHESIS n2r1c1. NO CONTRADICTION FOUND IN CONTEXT 11. BACK IN CONTEXT 9 with 0 csp-variables solved and 655 candidates remaining. GENERATING CONTEXT 12 AT DEPTH 2, SON OF CONTEXT 9, FROM HYPOTHESIS n1r1c4. naked-single ==> r1c1=2 NO CONTRADICTION FOUND IN CONTEXT 12. BACK IN CONTEXT 9 with 0 csp-variables solved and 655 candidates remaining. GENERATING CONTEXT 13 AT DEPTH 2, SON OF CONTEXT 9, FROM HYPOTHESIS n3r1c4. NO CONTRADICTION FOUND IN CONTEXT 13. BACK IN CONTEXT 9 with 0 csp-variables solved and 655 candidates remaining. GENERATING CONTEXT 14 AT DEPTH 2, SON OF CONTEXT 9, FROM HYPOTHESIS n1r2c2. naked-single ==> r1c1=2 naked-single ==> r3c3=3 naked-single ==> r6c3=1 naked-single ==> r4c1=3 naked-single ==> r4c6=1 naked-single ==> r3c6=2 naked-single ==> r2c5=3 NO POSSIBLE VALUE for csp-variable 155 IN CONTEXT 14. RETRACTING CANDIDATE n1r2c2 FROM CONTEXT 9. BACK IN CONTEXT 9 with 0 csp-variables solved and 655 candidates remaining. GENERATING CONTEXT 15 AT DEPTH 2, SON OF CONTEXT 9, FROM HYPOTHESIS n2r2c2. naked-single ==> r1c1=1 naked-single ==> r1c4=3 naked-single ==> r2c5=1 naked-single ==> r3c6=2 naked-single ==> r5c5=3 naked-single ==> r4c6=1 naked-single ==> r5c2=1 naked-single ==> r6c3=3 NO POSSIBLE VALUE for csp-variable 133 IN CONTEXT 15. RETRACTING CANDIDATE n2r2c2 FROM CONTEXT 9. BACK IN CONTEXT 9 with 0 csp-variables solved and 655 candidates remaining. naked-single ==> r2c2=3 GENERATING CONTEXT 16 AT DEPTH 2, SON OF CONTEXT 9, FROM HYPOTHESIS n1r2c5. naked-single ==> r5c5=3 naked-single ==> r4c6=1 naked-single ==> r1c4=3 naked-single ==> r3c6=2 naked-single ==> r3c3=1 naked-single ==> r1c1=2 naked-single ==> r4c1=3 NO POSSIBLE VALUE for csp-variable 163 IN CONTEXT 16. RETRACTING CANDIDATE n1r2c5 FROM CONTEXT 9. BACK IN CONTEXT 9 with 0 csp-variables solved and 655 candidates remaining. naked-single ==> r2c5=2 GENERATING CONTEXT 17 AT DEPTH 2, SON OF CONTEXT 9, FROM HYPOTHESIS n1r3c3. naked-single ==> r6c3=3 naked-single ==> r3c6=3 naked-single ==> r1c4=1 naked-single ==> r4c6=1 naked-single ==> r4c1=2 NO POSSIBLE VALUE for csp-variable 111 IN CONTEXT 17. RETRACTING CANDIDATE n1r3c3 FROM CONTEXT 9. BACK IN CONTEXT 9 with 0 csp-variables solved and 655 candidates remaining. naked-single ==> r3c3=2 naked-single ==> r1c1=1 naked-single ==> r1c4=3 naked-single ==> r3c6=1 naked-single ==> r4c6=3 naked-single ==> r5c5=1 naked-single ==> r5c2=2 NO POSSIBLE VALUE for csp-variable 141 IN CONTEXT 9. RETRACTING CANDIDATE n2r6c4 FROM CONTEXT 0. BACK IN CONTEXT 0 with 0 csp-variables solved and 654 candidates remaining. naked-single ==> r6c4=1 GENERATING CONTEXT 18 AT DEPTH 1, SON OF CONTEXT 0, FROM HYPOTHESIS n3r6c3. *** STARTING T&E IN CONTEXT 18 at depth 1 with 1 csp-variables solved and 633 candidates remaining *** STARTING PHASE 1 IN CONTEXT 18 AT DEPTH 1, with 1 csp-variables solved and 633 candidates remaining GENERATING CONTEXT 19 AT DEPTH 2, SON OF CONTEXT 18, FROM HYPOTHESIS n1r1c1. naked-single ==> r4c1=2 naked-single ==> r4c6=3 naked-single ==> r5c5=2 naked-single ==> r5c2=1 naked-single ==> r3c3=2 naked-single ==> r2c2=3 naked-single ==> r2c5=1 NO POSSIBLE VALUE for csp-variable 136 IN CONTEXT 19. RETRACTING CANDIDATE n1r1c1 FROM CONTEXT 18. BACK IN CONTEXT 18 with 1 csp-variables solved and 633 candidates remaining. naked-single ==> r1c1=2 naked-single ==> r4c1=1 naked-single ==> r5c2=2 naked-single ==> r5c5=3 naked-single ==> r4c6=2 naked-single ==> r3c3=1 naked-single ==> r3c6=3 NO POSSIBLE VALUE for csp-variable 114 IN CONTEXT 18. RETRACTING CANDIDATE n3r6c3 FROM CONTEXT 0. BACK IN CONTEXT 0 with 1 csp-variables solved and 632 candidates remaining. naked-single ==> r6c3=2 GENERATING CONTEXT 20 AT DEPTH 1, SON OF CONTEXT 0, FROM HYPOTHESIS n3r5c5. naked-single ==> r4c6=2 naked-single ==> r5c2=1 naked-single ==> r4c1=3 *** STARTING T&E IN CONTEXT 20 at depth 1 with 2 csp-variables solved and 612 candidates remaining *** STARTING PHASE 1 IN CONTEXT 20 AT DEPTH 1, with 2 csp-variables solved and 612 candidates remaining GENERATING CONTEXT 21 AT DEPTH 2, SON OF CONTEXT 20, FROM HYPOTHESIS n1r1c1. naked-single ==> r3c3=3 naked-single ==> r2c2=2 naked-single ==> r2c5=1 NO POSSIBLE VALUE for csp-variable 136 IN CONTEXT 21. RETRACTING CANDIDATE n1r1c1 FROM CONTEXT 20. BACK IN CONTEXT 20 with 2 csp-variables solved and 612 candidates remaining. naked-single ==> r1c1=2 naked-single ==> r2c2=3 naked-single ==> r3c3=1 naked-single ==> r3c6=3 NO POSSIBLE VALUE for csp-variable 114 IN CONTEXT 20. RETRACTING CANDIDATE n3r5c5 FROM CONTEXT 0. BACK IN CONTEXT 0 with 2 csp-variables solved and 611 candidates remaining. naked-single ==> r5c5=2 naked-single ==> r4c6=3 naked-single ==> r4c1=1 naked-single ==> r1c1=2 naked-single ==> r1c4=3 naked-single ==> r2c5=1 naked-single ==> r2c2=3 PUZZLE 0 HAS NO SOLUTION : NO CANDIDATE FOR RC-CELL r5c2 MOST COMPLEX RULE TRIED = NS Puzzle 100100000010010000001001000100001000010010000001100000000000000000000000000000000 : init-time = 0.0s, solve-time = 0.25s, total-time = 0.25s s

.

Website · by **denis_berthier** » Fri Sep 16, 2022 4:19 pm

.
ORk-CW classification results for mith's 63137 min-expand puzzles in T&E(3):

Here are results for ORk-Contrad-Whips similar to those for ORk-Forcing-Whips in the first post of this page.

Puzzles solved in SFin+Trid+Wn+ORkCWn

Code: Select all: ----------------------------------------------------------------------- n=3 n=5 n=7 n=8 ----------------------------------------------------------------------- 8 ,196 puzzles solved by SFin+Trid (among 63 137 min-expands) ----------------------------------------------------------------------- k=0 8,137 17,532 21,160 22,332 16,333 9,395 25,728 3,628 29,356 1,172 30,528 ----------------------------------------------------------------------- k=2 1,700 6,276 8,863 9,944 18,033 13,971 32,004 6,215 38,219 2,253 40,472 ----------------------------------------------------------------------- k=3 286 1,379 2,413 18,319 15,064 33,383 7,249 40,632 ----------------------------------------------------------------------- k=4 49 319 478 18,368 15,534 33,702 7,408 41,110 ----------------------------------------------------------------------- k=5 6 25 111 18,374 15,353 33,727 7,494 41,221 -----------------------------------------------------------------------

Same conventions as above:
Lines are separated by dashes, columns are separated by large white spaces.
Each (k, n) cell has three values in it:
- the main one, in the lower right corner, is the total number of puzzles solved by SFin + Trid + Wn + ORkFWn;
- the value above it is the difference with the previous line; it shows what’s gained by increasing k by 1;
- the value on the left of the main number is the difference with the previous cell; it shows what’s gained by increasing n.

And also same general conclusions:
• for fixed n, as k increases, the difference between two lines decreases quite fast; this shouldn’t be too surprising, as larger k means more chains have to converge to the same candidate;
• for fixed k, as n increases, the difference between two columns decreases quite fast; this shouldn’t be too surprising either, as it already happens with all the “classical” chains (whips…);
• starting from k=2 and n=3, at any point in the table, it is much more fruitful to increase n than to increase k;

Plus a new one:
For any fixed k and n, ORk-Forcing-Whips are more powerful than ORk-Contrad-Whips

Website · by **denis_berthier** » Wed Oct 26, 2022 7:14 am

.
ORk-W classification results for mith's 63137 min-expand puzzles in T&E(3):

I've almost completed calculations similar to the above ones, for the classification of the 63137 min-expand database:
1) using ORk-whips instead of ORk-forcing-whips or ORk-contrad-whips;
2) using all the ORk-chains together.

The global results and general conclusions are similar to the above ones, with some additional conclusions:
- for any k and n, there is a large overlap between what can be solved with ORk-whips[n] and what can be solved with ORk-forcing-whips[n];
- for any k and n, ORk-whips[n] (which include ORk-contrad-whips as a special case) have a greater resolution power than ORk-forcing-whips[n];
- most of the puzzles that have a solution with ORk-forcing-whips[n] also have one with ORk-whips[n] but the converse is not true: for instance, only 96 puzzles can be solved in SFin+Trid+W5+OR5FW5 but not in SFin+Trid+W5+OR5W5; whereas 1894 can be solved in SFin+Trid+W5+OR5W5 but not in SFin+Trid+W5+ OR5FW5; and the difference is still larger for n=7;
- the difference between ORk-forcing-whips and ORk-whips is not well compensated by increasing the FW lengths: for instance, only 55 puzzles can be solved in SFin+Trid+W5+OR5FW5 but still not in SFin+Trid+W7+OR5W7; whereas 683 can be solved in SFin+Trid+W5+OR5W5 but still not in SFin+Trid+W7+OR5FW7.

It is so complicated to format a table as those I've posted before that I don't try to do it for the ORk-whips. You will find all the details in the forthcoming new version of CSP-Rules manual.

I consider these results as important when one wants to choose which kinds of rules to use.

Notice also that:
- as much as 13% of the puzzles can be solved using only SFin+Trid (Subsets + Finned Fish + the elementary tridagon elimination rule with only 1 guardian); this may give the wrong idea that puzzles with the anti-tridagon pattern can easily be reduced to easy puzzles;
- BUT, even with whips[≤8] and all the ORk-chains[≤8], a noticeable proportion of the puzzles (21%) remains unsolved.
This is to be compared with the result that +99,97% of all the Sudoku puzzles (unbiased statistics - see [PBCS]) can be solved by whips[≤8].

This should be food for thought for people who might have had the idea that finding it was all there is to the anti-tridagon pattern.

As an aside remark, the above results also show that there are several different rating systems for puzzles in T&E(3). Even in case we choose the system based on all the types of ORk-chains, k remains as a parameter.

Website · by **denis_berthier** » Thu Dec 01, 2022 4:06 am

.
I don't plan to analyse mith's extended database of 158,276 min-expand puzzles (http://forum.enjoysudoku.com/t-e-3-puzzles-split-from-hardest-sudokus-thread-t40514.html) in as much detail as I've analysed the previous 83,177 one (mainly in this thread).
But here is an important result that remains valid after its publication:

All the known 9x9 sudoku puzzles in T&E(3) are indeed at most in T&E(W2, 2).
With the large number of minimal puzzles involved (847,778), this seems to entrench a new frontier of complexity.

(More detail here: http://forum.enjoysudoku.com/t-e-3-puzzles-split-from-hardest-sudokus-thread-t40514-18.html)

Website · by **denis_berthier** » Sat Apr 08, 2023 6:17 am

.
As a result of my previous analyses, the tridagon related ORk-chain rules are not enough to solve all the puzzles in T&E(3). How much resolution power does adding other impossible patterns bring? Do we need to use a lot more impossible patterns?

In this thread: http://forum.enjoysudoku.com/how-to-deal-with-large-numbers-of-patterns-t40889.html, I've analysed how to deal with large numbers of patterns.

As reported here: http://forum.enjoysudoku.com/csp-rules-sudorules-kakurules-t38200-80.html, I have found that, among the 630 impossible patterns in two bands or two stacks published by eleven, a few impossible patterns appear to be useful in T&E(3) puzzles with much higher frequencies than all the rest.
Most of these patterns are "close" to Tridagon.

The question that naturally arises now is: do these few patterns allow to solve most of the puzzles that could be solved with all the 630 impossible patterns?
This is of interest for manual solvers (who can't obviously learn 630 patterns) and for programmers (who may want to code only a few patterns).

I'll compare 4 sets of patterns (each pattern described in detail in the Augmented User Manual for CSP-Rules):
1) Tridagon
2) Imp630-Select1 (i.e. Trid + EL13c290 EL14c30 EL14c159 EL14c13 EL14c1)
3) Imp630-Select2 ( (i.e. Imp630-Select1 + EL13c30 EL10c28 EL13c179 EL13c176 EL13c234 EL13c171 EL10c6)
4) Imp630-all

In the 4 cases, the following ORk-chains are active: ORk-whips[n], with k≤5 and n≤8. This choice for k and n is based on previous analyses.
The analysis currently bears only on the first 10,000 puzzles in mith's list of 158,276 T&E(3) min-expands; but all my previous analyses on this list have shown little variation among the different slices of the whole list.

Two questions can be asked:
1) does going from one set of impossible patterns to the next allow to lower the ratings of puzzles?
It does in the following numbers of puzzles:

Code: Select all: Trid to Select1 = 1941 Select1 to Select2 = 396 Select2 to all = 215

Note that in most solvable cases, the difference in ratings is low (1 or 2).

2) does going from one set of impossible patterns to the next allow to solve more puzzles?
It does in the following numbers of puzzles:

Code: Select all: Trid to Select1 = 1171 Select1 to Select2 = 127 Select2 to all = 60

Note in particular that only 60 puzzles (in 10,000) can be solved when the 630 patterns are allowed but not when only Select2 is allowed.

It seems to me that Select2 is the right balance between number of patterns used and resolution power.

.

Website · by **denis_berthier** » Wed May 03, 2023 6:20 am

.
This post elaborates the results of the previous one, with the sets Select1, Select2 and Imp630 as above.
All the results are based on the first 60,000 puzzles of mith's list of 158,276 T&E(3) min-expands.
(Because nothing guarantees any kind of randomness in the 158,276 min-expand collection, the results may differ slightly if one uses the full collection instead. However, the computations have been extended progressively by slices of 10000 and they show no significant variations among the slices.)

The following table compares the cumulative distribution functions (expressed in %) of the S + W + T-OR5-W ratings for 4 sets T of impossible patterns (rows are for the sets of rules in the 1st column, columns are for n).
columns n = 1 and n = 2 have 0s because of the priority assigned to the detection of impossible patterns: immediately after S3.

Code: Select all: n -> 1 2 3 4 5 6 7 8 rest Trid 0 0 32.54 48.46 61.75 69.64 74.19 78.35 21.65 Select1 0 0 38.29 58.74 74.43 83.12 87.80 91.15 8.85 Select2 0 0 39.46 60.53 76.08 84.46 88.83 91.95 8.05 Imp630 0 0 40.23 61.74 77.32 85.55 89.58 92.45 7.55

The table clearly shows that:
– for any of the four T’s, increasing n brings diminishing returns – which we already knew from previous tables in the T = Trid case;
– going from Trid to Select1 allows to solve significantly more puzzles, be it globally or at any length ≤ 6;
– going from Select1 to Select2 has a much smaller impact;
– going from Select2 to all of Imp630 doesn’t bring much.

The results also show that increasing the maximal allowed chain length (n) by one is statistically much more useful than passing from Select1 to all of Imp630.

I think the above results fully justify my global approach of how to deal with large sets of impossible patterns and my general guideline for puzzles in T&E(3): first try a solution with no additional pattern and then, if not enough, with only a small set of them (Select1).

If a short summary is needed: starting from 630 impossible patterns, I've shown that, apart from rare cases (1.3%), only the five ones in Select1 are really useful.

So, I now consider that the right balance between resolution power and number of patterns is Select1 (not the larger Select2, as I said in my previous post).
.

Website · by **denis_berthier** » Sun May 14, 2023 11:55 am

.
More classification results.
Let's forget the 630 impossible patterns for a while and get back to the tridagon pattern and to mith's large collection of 158,276 T&E(3) min-expand puzzles.
Until now, I had provided statistical analyses only for the smaller collection of 63,137 ones.

My stats for the larger collection are based on a different approach: instead of concentrating on which puzzles are solvable depending on n and k, I used the previous result that, for the tridagon pattern, k=5 (the max number of guardians) is a good balance between resolution power and complexity and I computed the W+Trid-OR5W ratings (it doesn't make much difference with the previous tables for k=5, but it avoids redundant calculations).
The table below shows the distribution of the W+Trid-OR5W rating (with all the ultra-persistency and splitting rules active) for increasing slices of the larger collection. The last line is for the full collection. Column 0 is the slice size; columns 1 and 2 have zeroes because the tridagon pattern is looked for only after S3 and no puzzle in the list can be solved without a tridagon.

Code: Select all: rating -> 1 2 3 4 5 6 7 8 >8 20,000 0 0 30.51 14.67 12.46 8.14 4.78 4.16 25.28 40,000 0 0 31.08 14.14 12.73 7.82 4.71 4.21 24.31 60,000 0 0 32.54 15.92 13.29 7.89 4.55 4.17 21.65 80,000 0 0 30.72 15.61 13.36 8.14 4.82 4.28 23.07 100,000 0 0 30.49 15.00 13.01 7.98 4.85 4.23 24.44 120,000 0 0 30.89 14.95 12.99 7.98 4.82 4.20 24.17 140,000 0 0 31.41 14.71 12.75 7.84 4.86 4.21 24.22 158,276 0 0 31.60 15.00 13.15 7.94 4.86 4.20 23.25

The table shows slight fluctuations (to be expected, considering the way the collection was generated, by vicinity search) from one slice to the next larger one, with no systematic trend, in particular no noticeable trend for puzzles being harder or easier when one goes from the first ones to the full list.

This observation also provides some justification for my approach of eleven's 630 impossible patterns:
1) I used only the first 50,000 puzzles to define four increasing sets (Select1 ... Select4) of most "useful" patterns;
2) for the patterns thus found, I analysed their additional resolution power wrt tridagon and their missing resolution power wrt to the full 630 list;
3) the above analyses are progressively being extended from the initial sublist of 50,000 puzzles (when I and my Mac have free time); the progressive extension has currently reached 100,000 puzzles and it doesn't show any noticeable difference with the results for the first 60,000 reported in my previous posts. Considering this result, I may decide to stop the extension at this point.
.

Website · by **denis_berthier** » Thu Jun 01, 2023 5:28 am

.
Remember my old comparisons of ratings reported in [PBCS 1 to 3]
(https://www.researchgate.net/publication/280301697_Pattern-Based_Constraint_Satisfaction_and_Logic_Puzzles,...,
https://www.researchgate.net/publication/356313228_Pattern-Based_Constraint_Satisfaction_and_Logic_Puzzles_Third_Edition)
and recalled here:
http://forum.enjoysudoku.com/pattern-based-constraint-satisfaction-2nd-3rd-eds-t32567-11.html?

in particular those for the W and gW ratings, based on the 21,375 first puzzles of the controlled-bias collection (cbg-000 ):

denis_berthier wrote:W vs gW
Note that one must always have W ≥ gW
In reality, there are only 48 differences, i.e. a proportion of 0,22% differences.
And there are only 5 cases with difference > 1 (0,023%) and only one with difference > 2.

The question here is: what about the puzzles in T&E(3)? As this is a completely different collection - strongly biased, with all the puzzles having a particular pattern (tridagon) -, there's no a priori reason that would allow to extend the previous results to it. And indeed, they can't be extended. The differences are much higher.
To be more precise, consider the W+Trid-OR5W and the gW+Trid-OR5gW ratings (with ?*max-guardians* set to 8 and with all the ORk-consistency and splitting rules active). From my experience with puzzles in T&E(3), this is a good basis for comparison.
The comparison is here restricted to the first 10,000 puzzles in mith's collection of 158,276 min-expands.

Of course, one always has: gW+Trid-OR5gW ≤ W+Trid-OR5W
In reality, the W+Trid-OR5W and gW+Trid-OR5gW ratings are different in 3.69% of the T&E(3) puzzles (instead of 0.22% for W vs gW in cbg-000).
Also, 1.12% of the puzzles that are not solved in W8+OR5W8 get solved in gW8+OR5gW8

Whether this larger difference is totally due to the puzzles being much harder than those in cbg-000 or whether this is somehow specifically related to the presence of the anti-tridagon pattern remains undecided at this point.

For two examples of puzzles with different ratings, see:
- http://forum.enjoysudoku.com/1418-in-158-276-t-e-3-min-expands-t41482.html
- http://forum.enjoysudoku.com/5383-in-158-276-t-e-3-min-expands-t41504.html
The difference can appear with the presence of ordinary g-whips, ORk-gwhips or both.
.

The New Sudoku Players' Forum

The tridagon rule

Re: The tridagon rule

Re: The tridagon rule

Re: The tridagon rule

Re: The tridagon rule

Re: The tridagon rule

Re: The tridagon rule

Re: The tridagon rule

Re: The tridagon rule

Re: The tridagon rule

Re: The tridagon rule

Re: The tridagon rule

Re: The tridagon rule

Re: The tridagon rule

Re: The tridagon rule

Re: The tridagon rule