The New Sudoku Players' Forum

by **Mathimagics** » Fri Jan 17, 2020 6:16 pm

A sample puzzle from my catalog - it is challenging, but it is still definitely P&P solvable.

Due to the size and nature of the grid, you will probably need some hours to complete it (and lots of tea!).

Hopefully someone will enjoy puzzles like this, as I do. If so I can keep them coming.

Puzzle image: Show

RB0000_1 Solution: Show

Website · by **denis_berthier** » Sat Jan 18, 2020 11:41 pm

Hi Mathimagics,

I don't have time to solve so large puzzles manually. Most large puzzles have many obvious elementary steps, which makes them very boring. This is no exception.
Some preliminary analysis of the data shows a great number of magic sectors:

Hidden Text: Show

Here is now KakuRules pure logic solution.
The resolution rules used are the general whips and g-whips defined in my book "Pattern-Based Constraint Satisfaction and Logic Puzzles (available at https://arxiv.org/abs/1304.1628). The resolution strategy is the default "simple-first" one.
This puzzle is one more example of the power of g-whips in Kakuro.
In my rating system, it is in gW8.

Hidden Text: Show

Code: Select all: *********************************************************************************************** *** KakuRules 2.0.s based on CSP-Rules 2.0.s, config = gW+S *** using CLIPS 6.31-r738 *********************************************************************************************** Uppermost (black) row and leftmost (black) column have index 1 naked-single ==> r24c14 = 3 naked-single ==> r24c13 = 1 naked-single ==> r24c10 = 6 naked-single ==> r23c3 = 8 naked-single ==> r23c2 = 9 naked-single ==> r22c8 = 9 naked-single ==> r23c8 = 7 naked-single ==> r14c2 = 4 naked-single ==> r12c3 = 7 naked-single ==> r12c2 = 9 naked-single ==> r10c8 = 7 naked-single ==> r9c8 = 9 naked-single ==> r5c10 = 7 naked-single ==> r6c10 = 9 naked-single ==> r6c8 = 8 naked-single ==> r7c8 = 9 naked-single ==> r5c4 = 5 naked-single ==> r3c14 = 1 naked-single ==> r3c13 = 2 naked-single ==> r3c3 = 9 naked-single ==> r3c2 = 8 naked-single ==> r2c13 = 9 naked-single ==> r2c14 = 8 naked-single ==> vr3c4 = 59 naked-single ==> r4c4 = 9 naked-single ==> hr5c8 = 123467 naked-single ==> hr10c6 = 127 naked-single ==> vr10c3 = 12347 naked-single ==> hr14c1 = 24 naked-single ==> r14c3 = 2 naked-single ==> vr22c10 = 16 naked-single ==> r23c10 = 1 hidden-single-for-magic-digit-in-horiz-sector ==> r4c5 = 8 cell-to-horiz-ctr ==> hr13c1 ≠ 28 ctr-to-horiz-sector ==> r13c2 ≠ 8 cell-to-horiz-ctr ==> hr13c1 ≠ 19 ctr-to-horiz-sector ==> r13c3 ≠ 1 cell-to-horiz-ctr ==> hr11c1 ≠ 123459 ctr-to-horiz-sector ==> r11c4 ≠ 9 ctr-to-horiz-sector ==> r11c5 ≠ 9 ctr-to-horiz-sector ==> r11c6 ≠ 9 ctr-to-horiz-sector ==> r11c7 ≠ 9 cell-to-horiz-ctr ==> hr4c6 ≠ 279 ctr-to-horiz-sector ==> r4c7 ≠ 2 ctr-to-horiz-sector ==> r4c9 ≠ 2 cell-to-horiz-ctr ==> hr4c6 ≠ 459 cell-to-horiz-ctr ==> hr4c6 ≠ 468 ctr-to-horiz-sector ==> r4c9 ≠ 4 ctr-to-horiz-sector ==> r4c7 ≠ 4 cell-to-horiz-ctr ==> hr4c6 ≠ 567 ctr-to-horiz-sector ==> r4c9 ≠ 5 ctr-to-horiz-sector ==> r4c7 ≠ 5 cell-to-horiz-ctr ==> hr17c5 ≠ 24569 cell-to-horiz-ctr ==> hr17c5 ≠ 24578 cell-to-horiz-ctr ==> hr18c1 ≠ 24689 cell-to-horiz-ctr ==> hr18c1 ≠ 25679 cell-to-horiz-ctr ==> hr21c8 ≠ 345679 cell-to-verti-ctr ==> vr18c7 ≠ 123579 cell-to-verti-ctr ==> vr18c9 ≠ 124567 cell-to-verti-ctr ==> vr10c14 ≠ 24689 cell-to-verti-ctr ==> vr10c14 ≠ 25679 cell-to-verti-ctr ==> vr10c13 ≠ 13567 cell-to-verti-ctr ==> vr10c13 ≠ 23467 cell-to-verti-ctr ==> vr10c13 ≠ 12568 cell-to-verti-ctr ==> vr10c13 ≠ 13468 cell-to-verti-ctr ==> vr10c13 ≠ 23458 cell-to-verti-ctr ==> vr14c4 ≠ 25789 ctr-to-verti-sector ==> r15c4 ≠ 2 ctr-to-verti-sector ==> r17c4 ≠ 2 ctr-to-verti-sector ==> r18c4 ≠ 2 ctr-to-verti-sector ==> r19c4 ≠ 2 cell-to-verti-ctr ==> vr14c4 ≠ 45679 horiz-sector-to-ctr ==> hr2c1 ≠ 68 naked-single ==> hr2c1 = 59 naked-single ==> r2c3 = 5 naked-single ==> r2c2 = 9 horiz-sector-to-ctr ==> hr6c12 ≠ 23 naked-single ==> hr6c12 = 14 naked-single ==> r6c14 = 4 naked-single ==> r7c14 = 3 naked-single ==> r5c14 = 6 naked-single ==> r6c13 = 1 naked-single ==> r7c13 = 4 naked-single ==> r5c13 = 3 cell-to-horiz-ctr ==> hr9c11 ≠ 489 ctr-to-horiz-sector ==> r9c12 ≠ 4 cell-to-verti-ctr ==> vr3c12 ≠ 37 ctr-to-verti-sector ==> r4c12 ≠ 7 cell-to-verti-ctr ==> vr6c12 ≠ 34568 horiz-sector-to-ctr ==> hr20c12 ≠ 14 naked-single ==> hr20c12 = 23 naked-single ==> r20c14 = 2 naked-single ==> r22c14 = 5 naked-single ==> r20c13 = 3 naked-single ==> r22c13 = 2 naked-single ==> r22c12 = 1 naked-single ==> r22c11 = 3 naked-single ==> r21c12 = 2 ctr-to-horiz-sector ==> r21c11 ≠ 1 ctr-to-horiz-sector ==> r21c9 ≠ 1 hidden-single-for-magic-digit-in-verti-sector ==> r20c9 = 1 hidden-single-for-magic-digit-in-verti-sector ==> r23c9 = 2 verti-sector-to-ctr ==> vr18c8 ≠ 19 ctr-to-verti-sector ==> r19c8 ≠ 9 ctr-to-verti-sector ==> r20c8 ≠ 9 verti-sector-to-ctr ==> vr4c6 ≠ 48 ctr-to-verti-sector ==> r5c6 ≠ 4 ctr-to-verti-sector ==> r6c6 ≠ 4 verti-sector-to-ctr ==> vr4c6 ≠ 39 naked-single ==> vr4c6 = 57 naked-single ==> r5c6 = 7 naked-single ==> r6c6 = 5 1433 candidates, 5203 csp-links and 9993 links. Density = 0.973950028654189% naked-pairs-in-horiz-sector: r21{c4 c6}{n8 n9} ==> r21c7 ≠ 9 naked-pairs-in-horiz-sector: r21{c4 c6}{n8 n9} ==> r21c7 ≠ 8 naked-pairs-in-horiz-sector: r21{c4 c6}{n8 n9} ==> r21c5 ≠ 9 naked-pairs-in-horiz-sector: r21{c4 c6}{n8 n9} ==> r21c5 ≠ 8 naked-pairs-in-horiz-sector: r21{c4 c6}{n8 n9} ==> r21c2 ≠ 8 x-wing-in-horiz-sectors: n9{r13 r14}{c13 c14} ==> r15c14 ≠ 9 x-wing-in-horiz-sectors: n9{r13 r14}{c13 c14} ==> r15c13 ≠ 9 x-wing-in-horiz-sectors: n9{r13 r14}{c13 c14} ==> r11c14 ≠ 9 x-wing-in-horiz-sectors: n9{r13 r14}{c13 c14} ==> r11c13 ≠ 9 biv-chain[2]: r13n9{c13 c14} - r14n9{c14 c13} ==> vr10c13 ≠ 12478 ctr-to-verti-sector ==> r11c13 ≠ 8 ctr-to-verti-sector ==> r13c13 ≠ 8 naked-single ==> r13c13 = 9 naked-single ==> r14c13 = 7 naked-single ==> r14c14 = 9 naked-single ==> r13c14 = 8 naked-single ==> vr10c13 = 12379 naked-single ==> r15c13 = 3 naked-single ==> r12c13 = 1 naked-single ==> r12c14 = 3 naked-single ==> r11c13 = 2 ctr-to-verti-sector ==> r11c14 ≠ 6 ctr-to-verti-sector ==> r15c14 ≠ 6 ctr-to-verti-sector ==> r11c14 ≠ 1 verti-sector-to-ctr ==> vr10c14 ≠ 23789 naked-single ==> vr10c14 = 34589 naked-single ==> r15c14 = 5 naked-single ==> r11c14 = 4 ctr-to-horiz-sector ==> r11c9 ≠ 1 ctr-to-horiz-sector ==> r11c10 ≠ 1 ctr-to-horiz-sector ==> r11c11 ≠ 1 ctr-to-horiz-sector ==> r11c12 ≠ 1 biv-chain[2]: c10n2{r13 r12} - c10n1{r12 r13} ==> r13c10 ≠ 3 biv-chain[2]: c10n2{r13 r12} - c10n1{r12 r13} ==> r13c10 ≠ 4 biv-chain[2]: c10n2{r13 r12} - c10n1{r12 r13} ==> r13c10 ≠ 5 biv-chain[2]: c10n2{r13 r12} - c10n1{r12 r13} ==> r13c10 ≠ 6 biv-chain[2]: c10n2{r13 r12} - c10n1{r12 r13} ==> r13c10 ≠ 7 biv-chain[2]: c10n1{r12 r13} - c10n2{r13 r12} ==> r12c10 ≠ 3 biv-chain[2]: c10n1{r12 r13} - c10n2{r13 r12} ==> r12c10 ≠ 4 biv-chain[2]: c10n1{r12 r13} - c10n2{r13 r12} ==> r12c10 ≠ 5 biv-chain[2]: c10n1{r12 r13} - c10n2{r13 r12} ==> r12c10 ≠ 6 biv-chain[2]: c10n1{r12 r13} - c10n2{r13 r12} ==> r12c10 ≠ 7 biv-chain[2]: c10n1{r12 r13} - c10n2{r13 r12} ==> r12c10 ≠ 8 biv-chain[2]: vr17c11{n1234579 n1234678} - r24c11{n9 n8} ==> r18c11 ≠ 8 biv-chain[2]: vr17c11{n1234579 n1234678} - r24c11{n9 n8} ==> r19c11 ≠ 8 biv-chain[2]: vr17c11{n1234579 n1234678} - r24c11{n9 n8} ==> r20c11 ≠ 8 biv-chain[2]: vr17c11{n1234579 n1234678} - r24c11{n9 n8} ==> r21c11 ≠ 8 biv-chain[2]: vr17c11{n1234579 n1234678} - r24c11{n9 n8} ==> r23c11 ≠ 8 biv-chain[2]: vr17c11{n1234678 n1234579} - r24c11{n8 n9} ==> r18c11 ≠ 9 biv-chain[2]: vr17c11{n1234678 n1234579} - r24c11{n8 n9} ==> r19c11 ≠ 9 biv-chain[2]: vr17c11{n1234678 n1234579} - r24c11{n8 n9} ==> r20c11 ≠ 9 biv-chain[2]: vr17c11{n1234678 n1234579} - r24c11{n8 n9} ==> r21c11 ≠ 9 biv-chain[2]: vr17c11{n1234678 n1234579} - r24c11{n8 n9} ==> r23c11 ≠ 9 biv-chain[2]: hr3c4{n1456789 n2356789} - r3c8{n1 n3} ==> r3c5 ≠ 3 biv-chain[2]: hr3c4{n1456789 n2356789} - r3c8{n1 n3} ==> r3c7 ≠ 3 biv-chain[2]: hr3c4{n1456789 n2356789} - r3c8{n1 n3} ==> r3c9 ≠ 3 biv-chain[2]: hr3c4{n1456789 n2356789} - r3c8{n1 n3} ==> r3c11 ≠ 3 biv-chain[2]: hr3c4{n2356789 n1456789} - r3c8{n3 n1} ==> r3c5 ≠ 1 biv-chain[2]: hr3c4{n2356789 n1456789} - r3c8{n3 n1} ==> r3c7 ≠ 1 biv-chain[2]: hr3c4{n2356789 n1456789} - r3c8{n3 n1} ==> r3c9 ≠ 1 biv-chain[2]: hr3c4{n2356789 n1456789} - r3c8{n3 n1} ==> r3c11 ≠ 1 biv-chain[2]: hr8c9{n12347 n12356} - r8c14{n7 n5} ==> r8c11 ≠ 5 biv-chain[2]: hr8c9{n12347 n12356} - r8c14{n7 n5} ==> r8c12 ≠ 5 biv-chain[2]: hr8c9{n12347 n12356} - r8c14{n7 n5} ==> r8c13 ≠ 5 biv-chain[2]: r8c13{n7 n6} - hr8c9{n12347 n12356} ==> r8c12 ≠ 7 biv-chain[2]: r8c13{n7 n6} - hr8c9{n12347 n12356} ==> r8c14 ≠ 7 naked-single ==> r8c14 = 5 naked-single ==> hr8c9 = 12356 naked-single ==> r8c13 = 6 verti-sector-to-ctr ==> vr6c12 ≠ 24578 verti-sector-to-ctr ==> vr6c12 ≠ 24569 verti-sector-to-ctr ==> vr6c12 ≠ 23489 verti-sector-to-ctr ==> vr6c12 ≠ 14678 verti-sector-to-ctr ==> vr6c12 ≠ 14579 x-wing-in-verti-sectors: n7{c13 c14}{r4 r9} ==> r9c12 ≠ 7 biv-chain[2]: c13n7{r4 r9} - c14n7{r9 r4} ==> hr4c10 ≠ 5689 naked-single ==> hr4c10 = 4789 naked-single ==> r4c13 = 7 naked-single ==> r9c13 = 5 naked-single ==> r4c14 = 9 naked-single ==> r9c14 = 7 naked-single ==> hr9c11 = 579 naked-single ==> r9c12 = 9 naked-single ==> r10c12 = 8 naked-single ==> r10c11 = 9 ctr-to-verti-sector ==> r11c12 ≠ 7 cell-to-verti-ctr ==> vr3c12 ≠ 19 ctr-to-verti-sector ==> r5c12 ≠ 1 verti-sector-to-ctr ==> vr3c12 ≠ 46 naked-single ==> vr3c12 = 28 naked-single ==> r5c12 = 2 naked-single ==> r4c12 = 8 naked-single ==> r4c11 = 4 naked-single ==> r5c11 = 1 naked-single ==> r7c11 = 2 naked-single ==> r8c11 = 3 naked-single ==> r6c11 = 6 naked-single ==> r7c12 = 1 naked-single ==> r8c12 = 2 naked-single ==> r8c10 = 1 naked-single ==> r5c9 = 4 naked-single ==> vr7c10 = 14 naked-single ==> r9c10 = 4 naked-single ==> vr6c12 = 12689 naked-single ==> r11c12 = 6 naked-single ==> hr11c8 = 245679 naked-single ==> hr6c4 = 1245689 ctr-to-horiz-sector ==> r9c7 ≠ 2 ctr-to-horiz-sector ==> r9c9 ≠ 2 ctr-to-horiz-sector ==> r9c7 ≠ 1 ctr-to-horiz-sector ==> r9c9 ≠ 1 cell-to-horiz-ctr ==> hr2c8 ≠ 169 cell-to-horiz-ctr ==> hr2c8 ≠ 367 cell-to-horiz-ctr ==> hr2c8 ≠ 349 cell-to-verti-ctr ==> vr9c11 ≠ 1234689 naked-single ==> vr9c11 = 1235679 cell-to-verti-ctr ==> vr10c10 ≠ 12348 ctr-to-verti-sector ==> r14c10 ≠ 8 ctr-to-verti-sector ==> r15c10 ≠ 8 verti-sector-to-ctr ==> vr7c6 ≠ 48 ctr-to-verti-sector ==> r8c6 ≠ 8 hidden-single-in-magic-horiz-sector ==> r8c4 = 8 naked-single ==> vr6c4 = 12358 ctr-to-verti-sector ==> r9c6 ≠ 8 cell-to-horiz-ctr ==> hr10c3 ≠ 47 ctr-to-horiz-sector ==> r10c5 ≠ 4 ctr-to-horiz-sector ==> r10c5 ≠ 7 horiz-sector-to-ctr ==> hr7c1 ≠ 4568 horiz-sector-to-ctr ==> hr7c1 ≠ 3578 horiz-sector-to-ctr ==> hr7c1 ≠ 2678 horiz-sector-to-ctr ==> hr7c1 ≠ 2489 horiz-sector-to-ctr ==> hr7c1 ≠ 1589 naked-pairs-in-horiz-sector: r15{c10 c11}{n6 n7} ==> r15c12 ≠ 7 naked-pairs-in-horiz-sector: r15{c10 c11}{n6 n7} ==> r15c12 ≠ 6 naked-pairs-in-horiz-sector: r15{c10 c11}{n6 n7} ==> r15c9 ≠ 7 naked-pairs-in-horiz-sector: r15{c10 c11}{n6 n7} ==> r15c9 ≠ 6 naked-pairs-in-horiz-sector: r11{c10 c11}{n5 n7} ==> r11c9 ≠ 7 naked-pairs-in-horiz-sector: r11{c10 c11}{n5 n7} ==> r11c9 ≠ 5 naked-single ==> r11c9 = 9 naked-single ==> r15c9 = 8 naked-single ==> r16c9 = 6 naked-single ==> r15c12 = 9 cell-to-verti-ctr ==> vr15c8 ≠ 67 ctr-to-verti-sector ==> r17c8 ≠ 6 ctr-to-verti-sector ==> r17c8 ≠ 7 horiz-sector-to-ctr ==> hr16c10 ≠ 19 ctr-to-horiz-sector ==> r16c11 ≠ 1 ctr-to-horiz-sector ==> r16c12 ≠ 1 x-wing-in-verti-sectors: n1{c10 c11}{r12 r13} ==> r13c9 ≠ 1 x-wing-in-verti-sectors: n1{c10 c11}{r12 r13} ==> r13c8 ≠ 1 x-wing-in-verti-sectors: n1{c10 c11}{r12 r13} ==> r13c7 ≠ 1 x-wing-in-verti-sectors: n1{c10 c11}{r12 r13} ==> r13c5 ≠ 1 x-wing-in-verti-sectors: n1{c10 c11}{r12 r13} ==> r12c9 ≠ 1 x-wing-in-verti-sectors: n1{c10 c11}{r12 r13} ==> r12c8 ≠ 1 verti-sector-to-ctr ==> vr11c8 ≠ 159 verti-sector-to-ctr ==> vr11c8 ≠ 168 x-wing-in-verti-sectors: n1{c10 c11}{r12 r13} ==> r12c7 ≠ 1 x-wing-in-verti-sectors: n1{c10 c11}{r12 r13} ==> r12c5 ≠ 1 biv-chain[2]: r16c8{n9 n8} - vr15c8{n49 n58} ==> r17c8 ≠ 9 cell-to-horiz-ctr ==> hr17c5 ≠ 13679 biv-chain[2]: vr15c8{n49 n58} - r16c8{n9 n8} ==> r17c8 ≠ 8 cell-to-horiz-ctr ==> hr17c5 ≠ 12689 cell-to-horiz-ctr ==> hr17c5 ≠ 23678 biv-chain[2]: vr10c10{n12456 n12357} - r15c10{n6 n7} ==> r11c10 ≠ 7 naked-single ==> r11c10 = 5 naked-single ==> r11c11 = 7 naked-single ==> r15c11 = 6 naked-single ==> r15c10 = 7 naked-single ==> vr10c10 = 12357 naked-single ==> r14c10 = 3 naked-single ==> r14c11 = 5 cell-to-horiz-ctr ==> hr16c10 ≠ 46 ctr-to-horiz-sector ==> r16c12 ≠ 4 ctr-to-horiz-sector ==> r16c12 ≠ 6 biv-chain[2]: r16c11{n3 n2} - hr16c10{n37 n28} ==> r16c12 ≠ 3 biv-chain[2]: hr16c10{n37 n28} - r16c11{n3 n2} ==> r16c12 ≠ 2 biv-chain[2]: hr20c4{n1234569 n1234578} - r20c6{n9 n8} ==> r20c5 ≠ 8 biv-chain[2]: hr20c4{n1234569 n1234578} - r20c6{n9 n8} ==> r20c7 ≠ 8 biv-chain[2]: hr20c4{n1234569 n1234578} - r20c6{n9 n8} ==> r20c8 ≠ 8 biv-chain[2]: hr20c4{n1234569 n1234578} - r20c6{n9 n8} ==> r20c10 ≠ 8 biv-chain[2]: hr20c4{n1234578 n1234569} - r20c6{n8 n9} ==> r20c5 ≠ 9 biv-chain[2]: hr20c4{n1234578 n1234569} - r20c6{n8 n9} ==> r20c7 ≠ 9 biv-chain[2]: hr20c4{n1234578 n1234569} - r20c6{n8 n9} ==> r20c10 ≠ 9 biv-chain[2]: r20c10{n7 n6} - vr19c10{n78 n69} ==> r21c10 ≠ 7 biv-chain[2]: r20c10{n7 n6} - hr20c4{n1234578 n1234569} ==> r20c5 ≠ 7 biv-chain[2]: r20c10{n7 n6} - hr20c4{n1234578 n1234569} ==> r20c7 ≠ 7 biv-chain[2]: r20c10{n7 n6} - hr20c4{n1234578 n1234569} ==> r20c8 ≠ 7 biv-chain[2]: r20c10{n7 n6} - hr20c4{n1234578 n1234569} ==> r20c11 ≠ 7 biv-chain[2]: hr20c4{n1234578 n1234569} - r20c10{n7 n6} ==> r20c5 ≠ 6 biv-chain[2]: hr20c4{n1234578 n1234569} - r20c10{n7 n6} ==> r20c7 ≠ 6 biv-chain[2]: hr20c4{n1234578 n1234569} - r20c10{n7 n6} ==> r20c8 ≠ 6 biv-chain[2]: hr20c4{n1234578 n1234569} - r20c10{n7 n6} ==> r20c11 ≠ 6 biv-chain[2]: vr19c10{n78 n69} - r20c10{n7 n6} ==> r21c10 ≠ 6 biv-chain[2]: hr23c4{n1234579 n1234678} - r23c6{n9 n8} ==> r23c5 ≠ 8 biv-chain[2]: hr23c4{n1234579 n1234678} - r23c6{n9 n8} ==> r23c7 ≠ 8 biv-chain[2]: hr23c4{n1234678 n1234579} - r23c6{n8 n9} ==> r23c5 ≠ 9 biv-chain[2]: hr23c4{n1234678 n1234579} - r23c6{n8 n9} ==> r23c7 ≠ 9 biv-chain[2]: r21c4{n8 n9} - r21c6{n9 n8} ==> hr21c1 ≠ 245679 biv-chain[2]: r21c4{n9 n8} - r21c6{n8 n9} ==> hr21c1 ≠ 345678 biv-chain[2]: r22c9{n6 n8} - r24c9{n8 n9} ==> vr18c9 ≠ 123478 ctr-to-verti-sector ==> r19c9 ≠ 7 ctr-to-verti-sector ==> r21c9 ≠ 7 biv-chain[2]: vr18c9{n123568 n123469} - r24c9{n8 n9} ==> r19c9 ≠ 9 biv-chain[2]: vr18c9{n123568 n123469} - r24c9{n8 n9} ==> r21c9 ≠ 9 biv-chain[2]: r24c9{n8 n9} - vr18c9{n123568 n123469} ==> r19c9 ≠ 8 biv-chain[2]: r24c9{n8 n9} - vr18c9{n123568 n123469} ==> r21c9 ≠ 8 biv-chain[2]: r24c9{n8 n9} - vr18c9{n123568 n123469} ==> r22c9 ≠ 8 naked-single ==> r22c9 = 6 naked-single ==> r22c7 = 8 whip[2]: r18c6{n3 n1} - hr18c1{n23789 .} ==> r18c5 ≠ 3 whip[2]: r18c6{n1 n3} - hr18c1{n14789 .} ==> r18c5 ≠ 1 whip[2]: r18c6{n3 n1} - hr18c1{n23789 .} ==> r18c4 ≠ 3 whip[2]: r18c6{n1 n3} - hr18c1{n14789 .} ==> r18c4 ≠ 1 whip[2]: r18c6{n3 n1} - hr18c1{n23789 .} ==> r18c3 ≠ 3 whip[2]: r18c6{n1 n3} - hr18c1{n14789 .} ==> r18c3 ≠ 1 whip[2]: r18c6{n3 n1} - hr18c1{n23789 .} ==> r18c2 ≠ 3 whip[2]: r18c6{n1 n3} - hr18c1{n14789 .} ==> r18c2 ≠ 1 whip[2]: r4c8{n3 n1} - hr4c6{n369 .} ==> r4c9 ≠ 3 whip[2]: r4c8{n1 n3} - hr4c6{n189 .} ==> r4c9 ≠ 1 whip[2]: r4c8{n3 n1} - hr4c6{n369 .} ==> r4c7 ≠ 3 whip[2]: r4c8{n1 n3} - hr4c6{n189 .} ==> r4c7 ≠ 1 whip[2]: r16c4{n3 n1} - vr14c4{n34789 .} ==> r19c4 ≠ 3 whip[2]: r16c4{n3 n1} - vr14c4{n34789 .} ==> r17c4 ≠ 3 whip[2]: r16c4{n1 n3} - vr14c4{n16789 .} ==> r19c4 ≠ 1 whip[2]: r16c4{n1 n3} - vr14c4{n16789 .} ==> r17c4 ≠ 1 whip[2]: r16c4{n3 n1} - vr14c4{n34789 .} ==> r15c4 ≠ 3 whip[2]: r16c4{n1 n3} - vr14c4{n16789 .} ==> r15c4 ≠ 1 whip[2]: r6c9{n2 n1} - vr1c9{n234689 .} ==> r3c9 ≠ 2 whip[2]: r6c9{n2 n1} - vr1c9{n234689 .} ==> r2c9 ≠ 2 horiz-sector-to-ctr ==> hr2c8 ≠ 259 ctr-to-horiz-sector ==> r2c10 ≠ 9 ctr-to-horiz-sector ==> r2c9 ≠ 9 horiz-sector-to-ctr ==> hr2c8 ≠ 268 ctr-to-horiz-sector ==> r2c10 ≠ 6 ctr-to-horiz-sector ==> r2c9 ≠ 6 whip[2]: r6c9{n1 n2} - vr1c9{n134789 .} ==> r2c9 ≠ 1 horiz-sector-to-ctr ==> hr2c8 ≠ 178 whip[2]: hr24c1{n45 n27} - r24c2{n1 .} ==> r24c3 ≠ 7 whip[2]: hr24c1{n45 n18} - r24c3{n2 .} ==> r24c2 ≠ 1 whip[2]: hr20c1{n34 n25} - r20c2{n1 .} ==> r20c3 ≠ 5 whip[2]: hr19c6{n578 n479} - r19c9{n3 .} ==> r19c8 ≠ 4 whip[2]: hr19c6{n578 n389} - r19c9{n4 .} ==> r19c8 ≠ 3 whip[2]: hr19c6{n578 n569} - r19c8{n7 .} ==> r19c7 ≠ 6 whip[2]: hr19c6{n578 n479} - r19c9{n3 .} ==> r19c7 ≠ 4 whip[2]: hr19c6{n578 n389} - r19c9{n4 .} ==> r19c7 ≠ 3 whip[2]: hr19c1{n2345 n1238} - r19c4{n4 .} ==> r19c5 ≠ 8 whip[2]: hr19c1{n2345 n1238} - r19c4{n4 .} ==> r19c2 ≠ 8 whip[2]: hr18c9{n23456 n12359} - r18c14{n4 .} ==> r18c13 ≠ 9 whip[2]: hr18c9{n23456 n12359} - r18c14{n4 .} ==> r18c10 ≠ 9 whip[2]: r18c13{n8 n5} - r18c10{n5 .} ==> hr18c9 ≠ 12359 ctr-to-horiz-sector ==> r18c14 ≠ 9 whip[2]: hr17c1{n234 n126} - r17c4{n4 .} ==> r17c3 ≠ 6 whip[2]: hr17c1{n234 n135} - r17c4{n4 .} ==> r17c3 ≠ 5 whip[2]: hr17c1{n135 n234} - r17c4{n5 .} ==> r17c3 ≠ 4 whip[2]: hr17c1{n234 n126} - r17c4{n4 .} ==> r17c2 ≠ 6 whip[2]: hr17c1{n234 n135} - r17c4{n4 .} ==> r17c2 ≠ 5 whip[2]: hr17c1{n135 n234} - r17c4{n5 .} ==> r17c2 ≠ 4 whip[2]: hr10c3{n38 n56} - r10c4{n2 .} ==> r10c5 ≠ 5 whip[2]: hr10c3{n56 n38} - r10c4{n2 .} ==> r10c5 ≠ 3 whip[2]: hr10c3{n56 n29} - r10c4{n3 .} ==> r10c5 ≠ 2 whip[2]: hr7c6{n569 n479} - r7c9{n3 .} ==> r7c7 ≠ 7 whip[2]: hr7c1{n3569 n1679} - r7c4{n2 .} ==> r7c5 ≠ 1 whip[2]: hr7c1{n3569 n1679} - r7c4{n2 .} ==> r7c3 ≠ 1 whip[2]: hr7c1{n3569 n1679} - r7c4{n2 .} ==> r7c2 ≠ 1 whip[2]: hr6c1{n25 n34} - r6c2{n1 .} ==> r6c3 ≠ 4 whip[2]: hr6c1{n34 n25} - r6c3{n1 .} ==> r6c2 ≠ 2 whip[2]: hr4c1{n4789 n5689} - r4c3{n4 .} ==> r4c2 ≠ 6 whip[2]: vr18c7{n134568 n123489} - r19c7{n5 .} ==> r24c7 ≠ 9 whip[2]: hr24c4{n578 n389} - r24c7{n7 .} ==> r24c5 ≠ 3 whip[2]: vr14c4{n35689 n34789} - r17c4{n5 .} ==> r19c4 ≠ 4 whip[2]: hr19c1{n2345 n1247} - r19c4{n8 .} ==> r19c5 ≠ 7 whip[2]: hr19c1{n2345 n1247} - r19c4{n8 .} ==> r19c3 ≠ 7 whip[2]: hr19c1{n2345 n1247} - r19c4{n8 .} ==> r19c2 ≠ 7 whip[2]: vr14c4{n35689 n34789} - r17c4{n5 .} ==> r18c4 ≠ 4 whip[2]: vr16c10{n68 n59} - r18c10{n6 .} ==> r17c10 ≠ 5 whip[2]: vr14c4{n35689 n34789} - r17c4{n5 .} ==> r15c4 ≠ 4 whip[2]: vr1c5{n1456789 n2356789} - r6c5{n1 .} ==> r7c5 ≠ 2 whip[2]: vr1c5{n1456789 n2356789} - r6c5{n1 .} ==> r5c5 ≠ 2 whip[2]: vr1c5{n1456789 n2356789} - r5c5{n4 .} ==> r7c5 ≠ 3 whip[2]: vr1c10{n58 n67} - r2c10{n4 .} ==> r3c10 ≠ 7 whip[2]: vr1c10{n67 n49} - r2c10{n5 .} ==> r3c10 ≠ 4 whip[2]: vr1c5{n1456789 n2356789} - r6c5{n1 .} ==> r3c5 ≠ 2 g-whip[2]: r19c9{n5 n34} - hr19c6{n578 .} ==> r19c7 ≠ 5 cell-to-verti-ctr ==> vr18c7 ≠ 134568 whip[2]: r19c7{n7 n9} - vr18c7{n123678 .} ==> r21c7 ≠ 7 whip[2]: r19c7{n7 n9} - vr18c7{n123678 .} ==> r24c7 ≠ 7 whip[2]: hr24c4{n578 n479} - r24c7{n6 .} ==> r24c5 ≠ 4 whip[3]: r10c7{n2 n1} - c9n1{r10 r17} - hr17c5{n23489 .} ==> r17c7 ≠ 2 whip[3]: r2c11{n5 n8} - hr2c8{n457 n358} - r2c10{n4 .} ==> r2c9 ≠ 5 whip[2]: hr2c8{n358 n457} - r2c9{n8 .} ==> r2c10 ≠ 7 cell-to-verti-ctr ==> vr1c10 ≠ 67 ctr-to-verti-sector ==> r3c10 ≠ 6 whip[3]: r2c11{n8 n5} - r2c10{n5 n4} - hr2c8{n358 .} ==> r2c9 ≠ 8 whip[3]: r18c13{n7 n8} - r18c14{n8 n6} - r18c10{n6 .} ==> hr18c9 ≠ 12368 whip[3]: hr9c5{n34789 n45679} - r9c9{n3 n7} - r9c6{n7 .} ==> r9c7 ≠ 5 whip[3]: hr9c5{n45679 n34789} - r9c9{n5 n7} - r9c6{n7 .} ==> r9c7 ≠ 3 whip[3]: vr1c5{n2356789 n1456789} - r6c5{n2 n1} - r5c5{n1 .} ==> r7c5 ≠ 4 whip[3]: vr1c5{n2356789 n1456789} - r6c5{n2 n1} - r5c5{n1 .} ==> r3c5 ≠ 4 whip[3]: vr18c7{n124578 n123678} - r24c7{n5 n3} - r23c7{n3 .} ==> r21c7 ≠ 6 whip[3]: vr18c7{n123678 n124578} - r24c7{n6 n4} - r23c7{n4 .} ==> r20c7 ≠ 5 whip[3]: vr18c7{n123678 n124578} - r24c7{n6 n4} - r23c7{n4 .} ==> r21c7 ≠ 5 whip[4]: r17c6{n3 n1} - hr17c5{n34568 n13589} - r17c9{n7 n5} - r17c8{n5 .} ==> r17c7 ≠ 3 whip[4]: r17c6{n1 n3} - hr17c5{n14678 n13589} - r17c9{n7 n5} - r17c8{n5 .} ==> r17c7 ≠ 1 whip[4]: hr18c9{n23456 n12458} - r18c10{n6 n5} - r18c13{n5 n4} - r18c14{n4 .} ==> r18c12 ≠ 8 whip[5]: c11n8{r3 r2} - hr2c8{n457 n358} - r2c10{n4 n5} - r3c10{n5 n9} - vr1c10{n58 .} ==> r3c9 ≠ 8 whip[5]: r3c11{n8 n5} - r3c10{n5 n9} - r3c6{n9 n6} - r3c9{n6 n7} - r3c5{n7 .} ==> r3c7 ≠ 8 whip[5]: c11n8{r3 r2} - hr2c8{n457 n358} - r2c10{n4 n5} - r3c10{n5 n9} - vr1c10{n58 .} ==> r3c6 ≠ 8 whip[2]: vr1c6{n59 n68} - r3c6{n9 .} ==> r2c6 ≠ 6 whip[5]: r3c11{n5 n8} - r3c10{n8 n9} - r3c6{n9 n6} - r3c9{n6 n7} - r3c5{n7 .} ==> r3c7 ≠ 5 whip[5]: r23n3{c7 c5} - r20n3{c5 c8} - vr18c8{n46 n37} - r19c8{n8 n7} - r19c7{n7 .} ==> vr18c7 ≠ 124578 ctr-to-verti-sector ==> r23c7 ≠ 5 ctr-to-verti-sector ==> r24c7 ≠ 5 cell-to-horiz-ctr ==> hr24c4 ≠ 578 whip[2]: hr24c4{n479 n569} - r24c7{n3 .} ==> r24c5 ≠ 6 whip[3]: vr18c7{n123678 n123489} - r23c7{n6 n3} - r24c7{n3 .} ==> r21c7 ≠ 4 whip[3]: vr18c7{n123678 n123489} - r23c7{n6 n3} - r24c7{n3 .} ==> r20c7 ≠ 4 whip[5]: hr23c4{n1234678 n1234579} - c6n8{r23 r24} - hr24c4{n569 n389} - r24c7{n4 n3} - r23n3{c7 .} ==> r23c5 ≠ 5 g-whip[5]: r18c10{n6 n3589} - hr18c9{n12467 n23456} - r18c14{n8 n4} - r18c13{n4 n5} - r18c10{n5 .} ==> r18c12 ≠ 6 g-whip[5]: r18c10{n6 n3589} - hr18c9{n12467 n23456} - r18c14{n8 n4} - r18c13{n4 n5} - r18c10{n5 .} ==> r18c11 ≠ 6 g-whip[5]: r17c14{n8 n124679} - hr17c11{n358 n123678} - r17c13{n4 n7} - r17c14{n7 n6} - hr17c11{n178 .} ==> r17c12 ≠ 8 whip[6]: hr21c8{n235789 n245689} - r21c9{n3 n5} - r21c11{n5 n6} - vr17c11{n1234579 n1234678} - r24n9{c11 c9} - vr18c9{n123568 .} ==> r21c14 ≠ 4 whip[6]: hr21c8{n235789 n245689} - r21c9{n3 n5} - r21c11{n5 n6} - vr17c11{n1234579 n1234678} - r24n9{c11 c9} - vr18c9{n123568 .} ==> r21c13 ≠ 4 whip[6]: hr7c1{n3569 n3479} - r7c4{n5 n3} - r7c3{n3 n7} - r8c3{n7 n6} - r4c3{n6 n4} - r9n4{c3 .} ==> r7c2 ≠ 4 whip[7]: vr18c7{n123489 n123678} - r24c7{n4 n3} - r23n3{c7 c5} - r20n3{c5 c8} - vr18c8{n46 n37} - r19c8{n8 n7} - r19c7{n7 .} ==> r23c7 ≠ 6 whip[3]: r23c7{n3 n4} - vr18c7{n123678 n123489} - r24c7{n6 .} ==> r21c7 ≠ 3 whip[3]: r23c7{n3 n4} - vr18c7{n123678 n123489} - r24c7{n6 .} ==> r20c7 ≠ 3 naked-single ==> r20c7 = 2 naked-single ==> r21c7 = 1 cell-to-verti-ctr ==> vr18c8 ≠ 28 ctr-to-verti-sector ==> r19c8 ≠ 8 cell-to-horiz-ctr ==> hr19c6 ≠ 389 ctr-to-horiz-sector ==> r19c9 ≠ 3 horiz-sector-to-ctr ==> hr19c6 ≠ 578 biv-chain[2]: c11n2{r19 r18} - c11n1{r18 r19} ==> hr19c10 ≠ 3469 biv-chain[2]: c11n2{r19 r18} - c11n1{r18 r19} ==> hr19c10 ≠ 3478 biv-chain[2]: c11n2{r19 r18} - c11n1{r18 r19} ==> hr19c10 ≠ 3568 biv-chain[2]: c11n2{r19 r18} - c11n1{r18 r19} ==> hr19c10 ≠ 4567 biv-chain[2]: c11n2{r19 r18} - c11n1{r18 r19} ==> r19c11 ≠ 4 biv-chain[2]: c11n2{r19 r18} - c11n1{r18 r19} ==> r19c11 ≠ 5 biv-chain[2]: c11n2{r19 r18} - c11n1{r18 r19} ==> r19c11 ≠ 6 biv-chain[2]: c11n2{r19 r18} - c11n1{r18 r19} ==> r19c11 ≠ 7 biv-chain[2]: c11n1{r18 r19} - c11n2{r19 r18} ==> r18c11 ≠ 4 biv-chain[2]: c11n1{r18 r19} - c11n2{r19 r18} ==> r18c11 ≠ 5 biv-chain[2]: c11n1{r18 r19} - c11n2{r19 r18} ==> r18c11 ≠ 7 hidden-single-for-magic-digit-in-verti-sector ==> r21c11 = 7 naked-single ==> hr21c8 = 235789 naked-pairs-in-verti-sector: c14{r21 r23}{n8 n9} ==> r19c14 ≠ 9 naked-pairs-in-verti-sector: c14{r21 r23}{n8 n9} ==> r19c14 ≠ 8 cell-to-horiz-ctr ==> hr19c10 ≠ 2389 ctr-to-horiz-sector ==> r19c12 ≠ 3 naked-pairs-in-verti-sector: c14{r21 r23}{n8 n9} ==> r18c14 ≠ 8 naked-pairs-in-verti-sector: c14{r21 r23}{n8 n9} ==> r17c14 ≠ 9 cell-to-horiz-ctr ==> hr17c11 ≠ 259 naked-pairs-in-verti-sector: c14{r21 r23}{n8 n9} ==> r17c14 ≠ 8 cell-to-horiz-ctr ==> hr17c11 ≠ 358 naked-pairs-in-horiz-sector: r21{c10 c14}{n8 n9} ==> r21c13 ≠ 9 naked-pairs-in-horiz-sector: r21{c10 c14}{n8 n9} ==> r21c13 ≠ 8 naked-single ==> r21c13 = 5 naked-single ==> r21c9 = 3 biv-chain[2]: r19c8{n7 n6} - hr19c6{n479 n569} ==> r19c7 ≠ 7 naked-single ==> r19c7 = 9 naked-single ==> vr18c7 = 123489 cell-to-horiz-ctr ==> hr24c4 ≠ 569 ctr-to-horiz-sector ==> r24c5 ≠ 5 whip[2]: r19c11{n2 n1} - hr19c10{n2479 .} ==> r19c12 ≠ 2 whip[2]: r19c11{n1 n2} - hr19c10{n1489 .} ==> r19c12 ≠ 1 whip[3]: hr18c9{n23456 n12458} - r18c14{n6 n4} - r18c13{n4 .} ==> r18c10 ≠ 8 biv-chain[2]: r18c10{n6 n5} - vr16c10{n68 n59} ==> r17c10 ≠ 6 whip[4]: r18c10{n5 n6} - hr18c9{n12458 n23456} - r18c13{n7 n4} - r18c14{n4 .} ==> r18c12 ≠ 5 g-whip[4]: hr17c11{n457 n12356789} - r17c14{n4 n7} - hr17c11{n169 n367} - r17c13{n4 .} ==> r17c12 ≠ 6 g-whip[4]: hr17c11{n367 n12345789} - r17c14{n6 n7} - hr17c11{n169 n457} - r17c13{n8 .} ==> r17c12 ≠ 4 g-whip[4]: hr19c10{n2578 n124789} - r19c14{n6 n7} - hr19c10{n1489 n2479} - r19c12{n5 .} ==> r19c13 ≠ 4 whip[5]: r17c12{n7 n2} - vr14c12{n16789 n25789} - r18c12{n1 n7} - c13n7{r18 r19} - c14n7{r19 .} ==> hr17c11 ≠ 268 ctr-to-horiz-sector ==> r17c12 ≠ 2 whip[5]: r17c12{n7 n1} - vr14c12{n25789 n16789} - r18c12{n2 n7} - c13n7{r18 r19} - c14n7{r19 .} ==> hr17c11 ≠ 169 g-whip[5]: r17c14{n7 n46} - hr17c11{n178 n34567} - r17c13{n9 n4} - hr17c11{n367 n457} - r17c14{n6 .} ==> r17c12 ≠ 7 whip[2]: vr14c12{n45679 n16789} - r17c12{n5 .} ==> r18c12 ≠ 1 g-whip[4]: vr14c12{n16789 n23456789} - r17c12{n1 n3} - vr14c12{n25789 n35689} - r18c12{n2 .} ==> r19c12 ≠ 5 horiz-sector-to-ctr ==> hr19c10 ≠ 2578 horiz-sector-to-ctr ==> hr19c10 ≠ 2569 horiz-sector-to-ctr ==> hr19c10 ≠ 1579 g-whip[5]: hr18c9{n12458 n1234567} - r18c13{n8 n4} - r18c14{n4 n6} - hr18c9{n13457 n12467} - r18c10{n5 .} ==> r18c12 ≠ 7 cell-to-verti-ctr ==> vr14c12 ≠ 16789 ctr-to-verti-sector ==> r17c12 ≠ 1 cell-to-horiz-ctr ==> hr17c11 ≠ 178 ctr-to-horiz-sector ==> r17c13 ≠ 8 g-whip[4]: r17c12{n5 n3} - vr14c12{n25789 n3456789} - r18c12{n2 n4} - c13n4{r18 .} ==> hr17c11 ≠ 367 ctr-to-horiz-sector ==> r17c14 ≠ 6 whip[3]: c11n2{r18 r19} - hr19c10{n1678 n2479} - c14n6{r19 .} ==> hr18c9 ≠ 13457 whip[4]: c14n6{r19 r18} - r18c10{n6 n5} - hr18c9{n12467 n23456} - c11n1{r18 .} ==> hr19c10 ≠ 2479 ctr-to-horiz-sector ==> r19c11 ≠ 2 naked-single ==> r19c11 = 1 naked-single ==> r18c11 = 2 cell-to-verti-ctr ==> vr14c12 ≠ 25789 horiz-sector-to-ctr ==> hr18c9 ≠ 12467 ctr-to-horiz-sector ==> r18c13 ≠ 7 ctr-to-horiz-sector ==> r18c14 ≠ 7 horiz-sector-to-ctr ==> hr18c9 ≠ 12458 naked-single ==> hr18c9 = 23456 naked-single ==> r18c13 = 4 naked-single ==> r18c14 = 6 naked-single ==> r18c10 = 5 naked-single ==> r18c12 = 3 naked-single ==> r17c12 = 5 naked-single ==> vr14c12 = 35689 naked-single ==> r16c12 = 8 naked-single ==> r19c12 = 6 naked-single ==> hr19c10 = 1678 naked-single ==> r19c14 = 7 naked-single ==> r19c13 = 8 naked-single ==> r23c13 = 9 naked-single ==> r23c14 = 8 naked-single ==> r21c14 = 9 naked-single ==> r21c10 = 8 naked-single ==> r17c13 = 7 naked-single ==> r17c14 = 4 naked-single ==> vr19c10 = 78 naked-single ==> r20c10 = 7 naked-single ==> hr20c4 = 1234578 naked-single ==> r20c6 = 8 naked-single ==> r21c6 = 9 naked-single ==> r21c4 = 8 naked-single ==> r22c4 = 9 naked-single ==> hr16c10 = 28 naked-single ==> r16c11 = 2 naked-single ==> hr17c11 = 457 naked-single ==> vr16c10 = 59 naked-single ==> r17c10 = 9 ctr-to-horiz-sector ==> r17c7 ≠ 6 whip[4]: r17c8{n5 n4} - vr15c8{n58 n49} - r16n8{c8 c7} - r17c7{n8 .} ==> hr17c5 ≠ 23489 biv-chain[7]: r20n5{c5 c11} - c11n4{r20 r23} - r23c7{n4 n3} - r24c7{n3 n4} - hr24c4{n389 n479} - r24c6{n8 n9} - c5n9{r24 r18} ==> r18c5 ≠ 5 whip[8]: r3c11{n5 n8} - r3c10{n8 n9} - vr1c10{n58 n49} - r2c10{n8 n4} - hr2c8{n358 n457} - r2c9{n3 n7} - r3c9{n7 n6} - r3c6{n6 .} ==> r3c5 ≠ 5 whip[8]: r7c4{n5 n1} - r9c4{n1 n2} - c2n2{r9 r5} - c3n2{r5 r6} - c3n3{r6 r5} - c3n1{r5 r9} - c2n1{r9 r6} - hr6c1{n25 .} ==> hr7c1 ≠ 1679 ctr-to-horiz-sector ==> r7c4 ≠ 1 whip[4]: hr7c1{n3479 n3569} - r7c3{n2 n3} - r7c4{n3 n5} - r7c2{n5 .} ==> r7c5 ≠ 6 g-whip[5]: r7c2{n7 n12568} - hr7c1{n3479 n2579} - r7c3{n3 n2} - r7c2{n2 n5} - r7c4{n5 .} ==> r7c5 ≠ 7 whip[8]: hr7c1{n3569 n2579} - r7c3{n6 n7} - r8c3{n7 n6} - c2n6{r8 r6} - r7c2{n6 n5} - r8c2{n5 n7} - r4c2{n7 n4} - r4c3{n4 .} ==> r7c4 ≠ 2 whip[4]: r7c4{n5 n3} - hr7c1{n2579 n3569} - r7c2{n2 n6} - r7c3{n6 .} ==> r7c5 ≠ 5 naked-single ==> r7c5 = 9 hidden-single-in-magic-horiz-sector ==> r8c6 = 9 naked-single ==> vr7c6 = 39 naked-single ==> r9c6 = 3 naked-single ==> hr9c5 = 34789 naked-single ==> r9c9 = 7 naked-single ==> r14c9 = 4 naked-single ==> r9c7 = 8 naked-single ==> r16c7 = 9 naked-single ==> r16c8 = 8 naked-single ==> vr15c8 = 58 naked-single ==> r17c8 = 5 cell-to-horiz-ctr ==> hr17c5 ≠ 13589 biv-chain[2]: hr17c5{n23579 n14579} - r17c6{n3 n1} ==> r17c9 ≠ 1 hidden-single-in-magic-verti-sector ==> r10c9 = 1 naked-single ==> r10c7 = 2 cell-to-horiz-ctr ==> hr17c5 ≠ 14579 naked-single ==> hr17c5 = 23579 naked-single ==> r17c7 = 7 naked-single ==> r14c7 = 6 naked-single ==> r17c6 = 3 naked-single ==> r18c6 = 1 naked-single ==> r17c9 = 2 ctr-to-horiz-sector ==> r18c3 ≠ 2 ctr-to-horiz-sector ==> r18c5 ≠ 2 cell-to-verti-ctr ==> vr11c8 ≠ 456 x-wing-in-verti-sectors: n1{c3 c7}{r11 r15} ==> r15c5 ≠ 1 x-wing-in-verti-sectors: n1{c3 c7}{r11 r15} ==> r11c5 ≠ 1 x-wing-in-verti-sectors: n1{c3 c7}{r11 r15} ==> r11c4 ≠ 1 hidden-single-for-magic-digit-in-verti-sector ==> r9c4 = 1 x-wing-in-verti-sectors: n1{c2 c3}{r5 r6} ==> r5c7 ≠ 1 x-wing-in-verti-sectors: n1{c2 c3}{r5 r6} ==> r5c5 ≠ 1 biv-chain[2]: r5c5{n4 n3} - vr1c5{n1456789 n2356789} ==> r6c5 ≠ 4 naked-pairs-in-horiz-sector: r6{c5 c9}{n1 n2} ==> r6c7 ≠ 2 cell-to-verti-ctr ==> vr1c7 ≠ 235679 naked-pairs-in-horiz-sector: r6{c5 c9}{n1 n2} ==> r6c7 ≠ 1 naked-single ==> r6c7 = 4 cell-to-horiz-ctr ==> hr7c6 ≠ 479 ctr-to-horiz-sector ==> r7c9 ≠ 7 cell-to-verti-ctr ==> vr1c7 ≠ 145679 horiz-sector-to-ctr ==> hr3c4 ≠ 1456789 naked-single ==> hr3c4 = 2356789 naked-single ==> r3c8 = 3 naked-single ==> r4c8 = 1 naked-single ==> hr4c6 = 189 verti-sector-to-ctr ==> vr1c7 ≠ 134789 biv-chain[2]: c2n1{r6 r5} - c3n1{r5 r6} ==> hr6c1 ≠ 25 ctr-to-horiz-sector ==> r6c2 ≠ 5 ctr-to-horiz-sector ==> r6c3 ≠ 2 biv-chain[2]: c2n1{r6 r5} - c3n1{r5 r6} ==> hr6c1 ≠ 34 naked-single ==> hr6c1 = 16 whip[2]: hr2c4{n679 n589} - r2c5{n7 .} ==> r2c7 ≠ 5 whip[2]: hr2c4{n679 n589} - r2c5{n7 .} ==> r2c6 ≠ 5 biv-chain[2]: r2c6{n9 n8} - vr1c6{n59 n68} ==> r3c6 ≠ 9 whip[2]: vr11c8{n357 n249} - r14c8{n7 .} ==> r12c8 ≠ 9 g-whip[2]: vr11c8{n348 n234567} - r14c8{n9 .} ==> r13c8 ≠ 7 g-whip[2]: vr11c8{n357 n1234568} - r14c8{n9 .} ==> r12c8 ≠ 8 g-whip[2]: vr11c8{n348 n234567} - r14c8{n9 .} ==> r12c8 ≠ 7 x-wing-in-horiz-sectors: n7{r12 r13}{c5 c6} ==> r15c6 ≠ 7 x-wing-in-horiz-sectors: n7{r12 r13}{c5 c6} ==> r15c5 ≠ 7 x-wing-in-horiz-sectors: n7{r12 r13}{c5 c6} ==> r14c6 ≠ 7 x-wing-in-horiz-sectors: n7{r12 r13}{c5 c6} ==> r14c5 ≠ 7 hidden-single-in-magic-horiz-sector ==> r14c8 = 7 ctr-to-verti-sector ==> r13c8 ≠ 4 ctr-to-verti-sector ==> r12c8 ≠ 4 x-wing-in-horiz-sectors: n7{r12 r13}{c5 c6} ==> r11c6 ≠ 7 x-wing-in-horiz-sectors: n7{r12 r13}{c5 c6} ==> r11c5 ≠ 7 biv-chain[2]: r12n7{c6 c5} - r13n7{c5 c6} ==> vr10c6 ≠ 45689 ctr-to-verti-sector ==> r11c6 ≠ 4 ctr-to-verti-sector ==> r12c6 ≠ 4 ctr-to-verti-sector ==> r13c6 ≠ 4 ctr-to-verti-sector ==> r15c6 ≠ 4 x-wing-in-horiz-sectors: n4{r12 r13}{c5 c7} ==> r15c7 ≠ 4 x-wing-in-horiz-sectors: n4{r12 r13}{c5 c7} ==> r15c5 ≠ 4 x-wing-in-horiz-sectors: n4{r12 r13}{c5 c7} ==> r11c7 ≠ 4 x-wing-in-horiz-sectors: n4{r12 r13}{c5 c7} ==> r11c5 ≠ 4 biv-chain[2]: r13n4{c5 c7} - r12n4{c7 c5} ==> vr9c5 ≠ 1356789 naked-single ==> vr9c5 = 2346789 naked-single ==> r16c5 = 3 naked-single ==> r16c4 = 1 naked-single ==> vr14c4 = 16789 naked-single ==> r17c4 = 6 naked-single ==> hr17c1 = 126 naked-single ==> r17c2 = 1 naked-single ==> r17c3 = 2 cell-to-horiz-ctr ==> hr24c1 ≠ 27 ctr-to-horiz-sector ==> r24c2 ≠ 7 cell-to-horiz-ctr ==> hr20c1 ≠ 25 ctr-to-horiz-sector ==> r20c2 ≠ 5 cell-to-horiz-ctr ==> hr19c1 ≠ 1346 cell-to-horiz-ctr ==> hr19c1 ≠ 1256 ctr-to-horiz-sector ==> r19c2 ≠ 6 ctr-to-horiz-sector ==> r19c3 ≠ 6 ctr-to-horiz-sector ==> r19c5 ≠ 6 cell-to-horiz-ctr ==> hr19c1 ≠ 2345 ctr-to-horiz-sector ==> r19c5 ≠ 5 ctr-to-horiz-sector ==> r19c3 ≠ 5 ctr-to-horiz-sector ==> r19c2 ≠ 5 biv-chain[2]: hr19c1{n1247 n1238} - r19c2{n4 n3} ==> r19c3 ≠ 3 biv-chain[2]: hr19c1{n1247 n1238} - r19c2{n4 n3} ==> r19c5 ≠ 3 biv-chain[2]: r19c2{n4 n3} - hr19c1{n1247 n1238} ==> r19c3 ≠ 4 naked-single ==> r19c3 = 1 cell-to-horiz-ctr ==> hr24c1 ≠ 18 ctr-to-horiz-sector ==> r24c2 ≠ 8 horiz-sector-to-ctr ==> hr20c1 ≠ 16 naked-single ==> hr20c1 = 34 verti-sector-to-ctr ==> vr17c5 ≠ 1345689 verti-sector-to-ctr ==> vr17c5 ≠ 1245789 verti-sector-to-ctr ==> vr17c5 ≠ 1236789 naked-single ==> vr17c5 = 2345679 hidden-single-in-magic-horiz-sector ==> r22c2 = 8 naked-pairs-in-verti-sector: c2{r19 r20}{n3 n4} ==> r24c2 ≠ 4 naked-pairs-in-verti-sector: c2{r19 r20}{n3 n4} ==> r24c2 ≠ 3 naked-pairs-in-verti-sector: c2{r19 r20}{n3 n4} ==> r21c2 ≠ 4 naked-pairs-in-verti-sector: c2{r19 r20}{n3 n4} ==> r21c2 ≠ 3 naked-pairs-in-verti-sector: c2{r19 r20}{n3 n4} ==> r18c2 ≠ 4 biv-chain[2]: r19c2{n4 n3} - hr19c1{n1247 n1238} ==> r19c5 ≠ 4 naked-single ==> r19c5 = 2 horiz-sector-to-ctr ==> hr21c1 ≠ 126789 biv-chain[2]: r24c2{n6 n5} - hr24c1{n36 n45} ==> r24c3 ≠ 6 biv-chain[2]: hr24c1{n36 n45} - r24c2{n6 n5} ==> r24c3 ≠ 5 naked-pairs-in-verti-sector: c3{r20 r24}{n3 n4} ==> r21c3 ≠ 4 naked-pairs-in-verti-sector: c3{r20 r24}{n3 n4} ==> r21c3 ≠ 3 naked-pairs-in-verti-sector: c3{r20 r24}{n3 n4} ==> r18c3 ≠ 4 whip[2]: hr18c1{n15689 n14789} - r18c3{n6 .} ==> r18c2 ≠ 7 hidden-single-in-magic-verti-sector ==> r21c2 = 7 naked-single ==> hr21c1 = 135789 naked-single ==> r21c3 = 5 naked-single ==> r21c5 = 3 hidden-single-for-magic-digit-in-horiz-sector ==> r23c7 = 3 naked-single ==> r24c7 = 4 naked-single ==> hr24c4 = 479 naked-single ==> r24c6 = 9 naked-single ==> r24c5 = 7 naked-single ==> r22c5 = 6 naked-single ==> r23c5 = 4 naked-single ==> r20c5 = 5 naked-single ==> r20c11 = 4 naked-single ==> r20c8 = 3 naked-single ==> r18c5 = 9 naked-single ==> r22c3 = 7 naked-single ==> r18c3 = 6 naked-single ==> r18c2 = 5 naked-single ==> r24c2 = 6 naked-single ==> r23c6 = 8 naked-single ==> hr23c4 = 1234678 naked-single ==> r23c11 = 6 naked-single ==> vr17c11 = 1234678 naked-single ==> r24c11 = 8 naked-single ==> r24c9 = 9 naked-single ==> vr18c9 = 123469 naked-single ==> r19c9 = 4 naked-single ==> hr19c6 = 479 naked-single ==> r19c8 = 7 naked-single ==> hr24c1 = 36 naked-single ==> r24c3 = 3 naked-single ==> r20c3 = 4 naked-single ==> r20c2 = 3 naked-single ==> r19c2 = 4 naked-single ==> hr19c1 = 1247 naked-single ==> r19c4 = 7 naked-single ==> r18c4 = 8 naked-single ==> r15c4 = 9 naked-single ==> hr18c1 = 15689 naked-single ==> vr18c8 = 37 ctr-to-horiz-sector ==> r15c6 ≠ 8 ctr-to-horiz-sector ==> r15c5 ≠ 8 ctr-to-horiz-sector ==> r15c2 ≠ 8 hidden-single-in-magic-verti-sector ==> r11c2 = 8 naked-single ==> hr11c1 = 123468 naked-pairs-in-verti-sector: c5{r11 r15}{n2 n6} ==> r13c5 ≠ 6 naked-pairs-in-verti-sector: c5{r11 r15}{n2 n6} ==> r13c5 ≠ 2 naked-pairs-in-verti-sector: c5{r11 r15}{n2 n6} ==> r12c5 ≠ 6 naked-pairs-in-verti-sector: c5{r11 r15}{n2 n6} ==> r12c5 ≠ 2 naked-pairs-in-verti-sector: c5{r11 r15}{n2 n6} ==> r10c5 ≠ 6 cell-to-horiz-ctr ==> hr10c3 ≠ 56 ctr-to-horiz-sector ==> r10c4 ≠ 5 naked-pairs-in-verti-sector: c4{r10 r11}{n2 n3} ==> r7c4 ≠ 3 naked-single ==> r7c4 = 5 ctr-to-horiz-sector ==> r7c3 ≠ 4 naked-pairs-in-verti-sector: c5{r10 r14}{n8 n9} ==> r12c5 ≠ 9 naked-pairs-in-verti-sector: c5{r10 r14}{n8 n9} ==> r12c5 ≠ 8 biv-chain[2]: c6n8{r12 r14} - c6n9{r14 r12} ==> r12c6 ≠ 2 biv-chain[2]: c6n8{r12 r14} - c6n9{r14 r12} ==> r12c6 ≠ 3 biv-chain[2]: c6n8{r12 r14} - c6n9{r14 r12} ==> r12c6 ≠ 5 biv-chain[2]: c6n8{r12 r14} - c6n9{r14 r12} ==> r12c6 ≠ 6 biv-chain[2]: c6n8{r12 r14} - c6n9{r14 r12} ==> r12c6 ≠ 7 hidden-single-for-magic-digit-in-horiz-sector ==> r12c5 = 7 naked-single ==> r13c5 = 4 hidden-single-in-magic-verti-sector ==> r12c7 = 4 hidden-single-in-magic-horiz-sector ==> r13c6 = 7 hidden-single-in-magic-horiz-sector ==> r13c8 = 6 naked-single ==> vr11c8 = 267 naked-single ==> r12c8 = 2 naked-single ==> r12c10 = 1 naked-single ==> r13c10 = 2 naked-single ==> r12c11 = 3 naked-single ==> r13c11 = 1 naked-single ==> r12c9 = 5 naked-single ==> r13c9 = 3 naked-single ==> r13c7 = 5 naked-single ==> hr12c4 = 1234579 naked-single ==> r12c6 = 9 naked-single ==> r14c6 = 8 naked-single ==> r14c5 = 9 naked-single ==> r10c5 = 8 naked-single ==> hr10c3 = 38 naked-single ==> r10c4 = 3 naked-single ==> r11c4 = 2 naked-single ==> r11c5 = 6 naked-single ==> r15c5 = 2 naked-single ==> r11c6 = 3 naked-single ==> r11c7 = 1 naked-single ==> r15c7 = 3 naked-single ==> r11c3 = 4 naked-single ==> r15c3 = 1 naked-single ==> r13c3 = 3 naked-single ==> hr13c1 = 37 naked-single ==> r13c2 = 7 naked-single ==> r15c2 = 6 naked-single ==> r15c6 = 5 naked-single ==> hr15c1 = 123569 naked-single ==> vr10c6 = 35789 whip[2]: hr7c1{n2579 n3569} - r7c2{n2 .} ==> r7c3 ≠ 6 whip[4]: r5n1{c2 c3} - c3n3{r5 r7} - c3n2{r7 r9} - r9c2{n2 .} ==> r5c2 ≠ 4 whip[4]: c2n5{r8 r4} - hr4c1{n4789 n5689} - r4c3{n4 n6} - r8c3{n6 .} ==> r8c2 ≠ 7 biv-chain[5]: c3n3{r5 r7} - hr7c1{n2579 n3569} - c2n7{r7 r4} - c2n4{r4 r9} - r9c3{n4 n2} ==> r5c3 ≠ 2 whip[5]: c3n3{r5 r7} - hr7c1{n2579 n3569} - c3n2{r7 r9} - c2n2{r9 r5} - r5c7{n2 .} ==> r5c5 ≠ 3 naked-single ==> r5c5 = 4 naked-single ==> vr1c5 = 1456789 naked-single ==> r6c5 = 1 naked-single ==> r6c9 = 2 biv-chain[2]: c2n4{r4 r9} - c3n4{r9 r4} ==> hr4c1 ≠ 5689 naked-single ==> hr4c1 = 4789 hidden-single-in-magic-verti-sector ==> r8c2 = 5 hidden-single-for-magic-digit-in-verti-sector ==> r2c5 = 5 naked-single ==> hr2c4 = 589 naked-pairs-in-verti-sector: c7{r2 r4}{n8 n9} ==> r7c7 ≠ 8 naked-pairs-in-verti-sector: c7{r2 r4}{n8 n9} ==> r3c7 ≠ 9 biv-chain[2]: r2c7{n9 n8} - r4c7{n8 n9} ==> vr1c7 ≠ 245678 naked-single ==> vr1c7 = 234689 biv-chain[2]: r7c7{n6 n3} - hr7c6{n569 n389} ==> r7c9 ≠ 6 biv-chain[2]: hr7c6{n569 n389} - r7c7{n6 n3} ==> r7c9 ≠ 3 biv-chain[2]: r2c9{n7 n3} - vr1c9{n245678 n234689} ==> r3c9 ≠ 7 hidden-single-for-magic-digit-in-horiz-sector ==> r3c5 = 7 naked-single ==> r8c5 = 6 naked-single ==> r8c3 = 7 naked-single ==> r4c3 = 4 naked-single ==> r9c3 = 2 naked-single ==> r9c2 = 4 naked-single ==> r7c3 = 3 naked-single ==> r5c3 = 1 naked-single ==> r6c3 = 6 naked-single ==> r6c2 = 1 naked-single ==> r5c2 = 2 naked-single ==> r5c7 = 3 naked-single ==> r7c7 = 6 naked-single ==> r3c7 = 2 naked-single ==> r4c2 = 7 naked-single ==> r7c2 = 6 naked-single ==> hr7c6 = 569 naked-single ==> r7c9 = 5 naked-single ==> vr1c9 = 245678 naked-single ==> r4c9 = 8 naked-single ==> r4c7 = 9 naked-single ==> r2c7 = 8 naked-single ==> r2c6 = 9 naked-single ==> r3c9 = 6 naked-single ==> r3c6 = 5 naked-single ==> r3c11 = 8 naked-single ==> r3c10 = 9 naked-single ==> r2c11 = 5 naked-single ==> r2c9 = 7 naked-single ==> hr2c8 = 457 naked-single ==> r2c10 = 4 PUZZLE SOLVED. rating-type = gW+S, MOST COMPLEX RULE TRIED = W[8] ------------------------ -95-598-745-98---------- -89-7523698-21---------- -7498-918-4879---------- -215473-471236---------- -16-1548296-14---------- -6359-695-2143---------- -57869---13265---------- -421-38974-957---------- ---38-271-98------------ -842631-957624---------- -97-7942513-13---------- -73-4756321-98---------- -42-9867435-79---------- -619253-876935---------- ---13-986-28------------ -126-37529-574---------- -56891---52346---------- -4172-974-1687---------- -34-5823174-32---------- -758391-387259---------- -8796-896-3125---------- -98-4837216-98---------- -63-794-968-13---------- *********************************************************************************************** *** KakuRules 2.0.s based on CSP-Rules 2.0.s, config = gW+S *** using CLIPS 6.31-r738 *********************************************************************************************** (The 10 ending "-" at the end of each line in the solution puzzle are due to the fact that KakuRules requires squares puzzles - a limitation I should correct some day.)

You proposed this puzzle as a challenge, but the challenge is now for you: write a fully detailed, simpler, pure logic solution (i.e. one in which no chain of length greater than 8 will be required).

by **Mathimagics** » Sun Jan 19, 2020 10:19 am

Hi Denis,

I only meant a challenge for P&P solvers. It is not intended to be challenge for software - actually, with grids of this size and cell density, for P&P solving I only generate puzzles that are solvable by my "linear solver", without recursion.

Your challenge is interesting, but I am not sure exactly what you mean by a "chain" here?

Cheers
MM

Website · by **denis_berthier** » Sun Jan 19, 2020 10:37 pm

I meant the universal types of chains used in my resolution path: bivalue-chains, whips, g-whips, or any other type of chain people prefer. I don't know what your linear solver does, but it sounds much like T&E(1). If so, this should allow the mentioned chains to solve these puzzles.

But the challenge is not about how you solve it (hand or computer), it's about the patterns you use. If you don't like "my" chains, you can still use different ones.

by **creint** » Mon Jan 20, 2020 4:47 pm

Solved with my solver using only singles and calculations in 1.8 seconds.

Website · by **denis_berthier** » Mon Jan 20, 2020 10:53 pm

creint wrote:Solved with my solver using only singles and calculations in 1.8 seconds.

What do you mean by "calculations"? T&E ?

by **Mathimagics** » Tue Jan 21, 2020 7:53 am

denis_berthier wrote:I don't know what your linear solver does, but it sounds much like T&E(1).

Indeed, my simple Kakuro solver (SKS) uses only simple candidate eliminations based on testing "Can cell (R, C) = D?". The test looks only at the intersecting runs, H & V, to see if both runs can still be completed with (R, C) = D given current domains.

The solution path produced (if the puzzle is solved) is simply a list of individual candidate eliminations + naked singles.

An elementary Kakuro puzzle "triage" classification might be:

solvable by SKS

solvable by "advanced techniques" (eg your whips/chains)

T&E required

by **creint** » Wed Jan 22, 2020 4:38 pm

denis_berthier wrote:
creint wrote:Solved with my solver using only singles and calculations in 1.8 seconds.

What do you mean by "calculations"? T&E ?

Digit cannot be placed in a cell because sum value would be different than expected. All within one constraint. Some solvers can check combinations in multiple constraints.

Website · by **denis_berthier** » Wed Jan 22, 2020 10:36 pm

creint wrote:Digit cannot be placed in a cell because sum value would be different than expected. All within one constraint. Some solvers can check combinations in multiple constraints.

So, your "calculations" include an unrestricted number of combination checks, right?

by **creint** » Thu Jan 23, 2020 4:57 pm

For every constraint for every cell in constraint for every digit in cell: if there is no combination containing this digit in this cell then exclude. Only restricted to a single calculation constraint.

Website · by **denis_berthier** » Thu Jan 23, 2020 11:05 pm

creint wrote:For every constraint for every cell in constraint for every digit in cell: if there is no combination containing this digit in this cell then exclude. Only restricted to a single calculation constraint.

I don't understand how you can solve this puzzle that way. It can't be solved by only such local constraints. Some form or another of chain is required.

by **creint** » Fri Jan 24, 2020 4:46 pm

denis_berthier wrote:
creint wrote:For every constraint for every cell in constraint for every digit in cell: if there is no combination containing this digit in this cell then exclude. Only restricted to a single calculation constraint.

I don't understand how you can solve this puzzle that way. It can't be solved by only such local constraints. Some form or another of chain is required.

Sum 17 in 2 cells then only 8 and 9 are possible in a cell. 8+9 or 9+8
Sum 14 in 2 cell then only 5689 are possible in a cell. 5+9, 6+8, 8+6, 9+5

Sum 14 in 2 cells digits 569, 56 then only 9+5 is possible.
Sum 28 in 4 cells digits 4567, 467, 9, 45678 then -6, _, _, -4567
-6 because 6, 47, 9, 4578 has no combination of sum 28. Same for the other exclusions.

Website · by **denis_berthier** » Fri Jan 24, 2020 10:19 pm

Hi creint
I see what you mean, but those are only local exclusions for the easiest cells, based only on local constraints. This type of reasoning is not enough for this puzzle. You must have harder steps for some eliminations, involving several sectors.

by **creint** » Sat Jan 25, 2020 6:21 pm

No, my solver used only singles and calculations, how do you want to debug this?
I could give a screenshot of each step, or pencilmarks like these.

First step of calculations

Hidden Text: Show

Code: Select all: 5689 569 56789 5689 56789 123456789 456789 1234568 9 89 89 9 123456789 5689 123456789 13 123456789 456789 1234568 12 1 456789 45679 5689 456789 123456789 13 123456789 4568 46789 45679 456789 12457 123457 5 123457 3457 123457 12345678 7 1234568 1234678 1234567 1345678 12456 123456 123456789 345789 123456789 89 123456789 79 1234568 1234 134 12456789 12345679 123456789 123456789 3456789 89 3456789 1234 1234 1234 134 56789 5679 56789 56789 5789 1234 123456 1234567 1234567 134567 124 124 124 345789 123456789 79 123456789 1234 456789 45679 456789 23456789 23456789 1234567 7 1234567 89 89 46789 1234567 123456789 123456789 23456789 123456789 123456789 12345678 123456789 123456789 123456789 123456789 79 7 123456789 23456789 123456789 123456789 123456789 12345678 123456789 13 13 46789 123467 1234567 234567 1234567 1234567 1234567 1234567 1234567 89 89 4 1245 3456789 3456789 3456789 3456789 3456789 345678 3456789 79 79 46789 1234567 123456789 123456789 23456789 123456789 356789 35678 356789 356789 356789 356789 13 13 689 689 689 12346789 12346789 13456 123456 123456 13 123456789 456789 123456789 5689 123456789 12345789 23456789 13456789 12345678 123456789 123456789 13 5689 123456789 123456789 12345789 23456789 1345678 12345678 12345678 12345678 3456789 346789 3456789 123456789 123456789 12345789 23456789 13456 123456 123456789 89 123456789 12346789 123456789 6789 123456789 1234 234 13456789 12345678 89 123456789 89 123456789 123456789 6789 123456789 12 12345789 23456789 6789 678 89 6789 689 9 689 1235 12 1235 235 89 8 123456789 89 123456789 79 123456789 123456 123456789 89 89 1345678 12345678 3456789 89 3456789 689 6 689 13 3

After singles, second step of calculations

Hidden Text: Show

Code: Select all: 9 5 56789 5689 56789 123456789 456789 1234568 9 8 8 9 2456789 5689 2456789 13 2456789 456789 24568 2 1 45679 4567 9 456789 6789 13 6789 4568 46789 4567 5679 1247 12347 5 12347 7 12347 12346 7 12346 12346 1346 6 12456 12356 1234567 5 1234567 8 1234567 9 123456 1 4 1245679 1234567 12345678 123456789 345678 9 345678 1234 1234 134 34 5679 567 56789 56789 5789 1234 123456 1234567 134567 567 124 124 124 3457 345678 9 345678 1234 5789 4567 5679 2345678 23456789 12 7 12 89 89 678 1234 12345678 12345678 2345678 12345678 123456789 12345678 123456789 123456789 2 2456 9 7 123456789 23456789 123456789 123456789 123456789 12345678 123456789 1 13 678 234 1234567 234567 1234567 1234567 1234567 1234567 1234567 9 89 4 2 3456789 3456789 3456789 3456789 3456789 345678 3456789 7 79 678 1234 56789 123456789 23456789 123456789 356789 35678 356789 356789 3 56789 13 13 689 689 689 12346789 12346789 13456 123456 456 13 123456789 457 123456789 5689 123456789 2345789 2456789 45678 24567 56789 2456789 13 5689 1234567 123456789 2345789 2456789 1345678 1234567 5678 12345678 3456789 346789 345 1234567 123456789 2345789 2456789 13456 12346 12345 89 12345 1234 12345 67 12345 3 2 134567 1234567 89 1234567 89 1234567 345 6789 34567 12 345789 456789 678 67 89 6789 68 9 6 1235 12 235 25 9 8 123456 89 123456 7 12345 1 123456 89 89 345678 123456 3456789 89 3456789 89 6 89 1 3

Website · by **denis_berthier** » Sun Jan 26, 2020 1:23 am

creint wrote:Sum 17 in 2 cells then only 8 and 9 are possible in a cell. 8+9 or 9+8
Sum 14 in 2 cell then only 5689 are possible in a cell. 5+9, 6+8, 8+6, 9+5

Sum 14 in 2 cells digits 569, 56 then only 9+5 is possible.
Sum 28 in 4 cells digits 4567, 467, 9, 45678 then -6, _, _, -4567
-6 because 6, 47, 9, 4578 has no combination of sum 28. Same for the other exclusions.

After your last post, I understand better what you meant above.
Each elimination combines knowledge from several sectors.

Even for your own purposes, I think you could make your output more explicit.

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