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SSS ORC UK 21D 24 25 http://www.rcbroughton.co.uk/sudoku/forum/viewtopic.php?f=13&t=1489 |
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Author: | Andrew [ Thu May 16, 2019 5:24 am ] |
Post subject: | Re: SSS ORC UK 21D 24 25 |
Thanks HATMAN for this latest batch of puzzles! I'm also British but also a Canadian citizen so have both UK and Canadian passports. HATMAN is probably right that 24 is the easiest of these three; 21D definitely seems harder after an easy start (but see my post for that puzzle), I haven't yet tried 25. 24 felt about the same level as other easier Mean ORC and SSS ORC puzzles. The final killer cage was definitely a key part of this puzzle, leading to the solution. Here is my walkthrough for SSS ORC UK 24: Sums are for heights of skyscrapers visible from the edges. Only those higher than the previous one(s) count toward the sums. On the odd-numbered rows and columns, which are normal ones containing 1-9, each sum must include 9, the height of the highest skyscraper, which can be seen from each edge. However this doesn’t necessarily apply for the even-numbered rows and columns where repeated numbers are allowed; for these rows and columns the height of the highest skyscraper may be less than 9. Since this an ORC puzzle, I’ve stated placements in odd rows/columns. x not 0, y not 7. It has been specified that all totals in the even-numbered rows and columns must contain at least one of 7,8,9; also that they contain at least one of 1,2,3 although that was said not to be particularly helpful. The four killer cages have the same total. Skyscrapers starting with 8 or 9 must total 8 (possible for even rows/columns), 9 or 17. Since no totals are given as 17 and xy cannot be 8, 9 or 17, rows/columns starting/finishing with 8,9 must be ones with unspecified totals. 1a. Upper total in C1 = 11 must contain 9 -> R12C1 = [29] (cannot be [219], NC), placed for C1, 2 placed for R1, no 8 in R3C1 (NC) 1b. Unspecified lower totals only in C38, right-hand total in R9 = 14 must contain 9 -> R9C89 = [95], placed for R9, 5 placed for C9, all four killer cages must total 14 1c. Other unspecified lower total in C3 -> R9C3 = 8, placed for R9 and C3 1d. Left-hand total in R9 = 21 must include 8,9 with 14(3) cage at R9C1 -> R9C12 = [42], placed for R9, 4 placed for C1 1e. R34C1 = 14(2) cage = [68], placed for C1, 6 placed for R3 1f. Unspecified upper totals only in C57 -> R1C57 = {89}, locked for R1 1g. Unspecified right-hand totals only in R26 -> R26C9 = {89}, locked for C9 Clean-ups: R1C1 = 2 -> no 1,3 in R1C2, no 1,2,3 in R2C2 R2C1 = 9 -> no 8 in R2C2, no 8,9 in R3C2 R3C1 = 6 -> no 5,6,7 in R24C2, no 5,7 in R3C2 R4C1 = 8 -> no 9 in R4C2, no 7 in R5C1, no 7,8,9 in R5C2 R9C1 = 4 -> no 3,5 in R8C1, no 3,4,5 in R8C2 R9C2 = 2 -> no 1 in R8C12, no 1,2,3 in R8C3 R9C3 = 8 -> no 7,8,9 in R8C24, no 7,9 in R8C3, no 7 in R9C4 R9C8 = 9 -> no 8,9 in R8C7, no 8 in R8C8 R9C9 = 5 -> no 4,5,6 in R8C8, no 4,6 in R8C9 R1C5 = {89} -> no 8,9 in R2C456 R1C7 = {89} -> no 8,9 in R2C78 R2C9 = {89} -> no 8,9 in R3C8 R6C9 = {89} -> no 8,9 in R57C8 R8C3 = {456} -> no 5 in R7C234 Naked pair {89} in R1C57 -> no 7 in R12C6 2a. R8C1 = 7, placed for C1, no 6 in R8C2 2b. R8C2 = 2 -> no 1,3 in R7C1 2c. R7C1 = 5, placed for R7 and C1 Clean-ups: R7C1 = 5 -> no 4,5,6 in R6C2, no 4,6 in R7C2 R7C2 = 2 -> no 1,3 in R7C2, no 1,2,3 in R7C3 R8C1 = 7 -> no 7,8 in R7C2 R5C1 = {13} -> no 2 in R456C2 R6C1 = {13} -> no 2 in R7C2 Naked pair {13} in R56C1 -> no 1,3,4 in R5C2 3a. R7C2 = 9, placed for R7 3b. Left-hand total for R5 = 15 must contain 9 -> R5C12 = [15], placed for R5, 1 placed for C1 3c. Right-hand total for R5 = 30 must contain 9 and 8 for R5 so can only be [98742/9876] (because of NC, cannot be [98643] because no space for 7 in R5) -> R5C3 = 9, R5C5 = 8. R5C7 = 7, all placed for R5, 9 placed for C3, 8 placed for C5, 7 placed for C7, R5C9 = {26} 3d. R1C5 = 9, placed for R1 and C5 3e. R1C7 = 8, placed for C7 Clean-ups: R1C7 = 8 -> no 7 in R12C8 R5C1 = 1 -> no 1 in R46C2 R5C2 = 5 -> no 4 in R4C2, no 4,5,6 in R46C3 R5C3 = 9 -> no 8 in R4C2, no 8,9 in R4C4 + R6C24 R4C2 = 3 -> no 2,4 in R3C2, no 2,3,4 in R3C3, no 2 in R4C3 R5C5 = 8 -> no 7 in R46C45, no 7,8,9 in R46C6 R5C7 = 7 -> no 6 in R46C67 + R5C68, no 6,7,8 in R46C8 R3C2 = {13} -> no 2 in R2C3 R5C6 = {234} -> no 3 in R46C57 R5C8 = {234} -> no 3 in R4C9 7 in R7 only in R9C56 -> no 6,7 in R8C5, no 6 in R9C56 4a. R6C3 = 2 (hidden single in C3) -> no 2,3 in R5C4 4b. R5C4 = {46} -> no 5 in R4C45 4b. 14(3) cage at R4C3 = [716] (cannot be 7{34}/[761], NC), 7 placed for C3, 6 placed for C5, no 1 in R3C3, no 6 in R5C4 4c. R3C3 = 5, place for R3 and C3 4d. Naked pair {46} in R78C3, locked for C3 4e. R5C4 = 4, placed for R5 4f. R5C9 = 6 (hidden single in R5), locked for C9 Clean-ups: R3C3 = 5 -> no 4 in R2C2 + R3C4, no 4,5,6 in R2C4 R4C3 = 7 -> no 7,8 in R3C4 R4C4 = 1 -> no 2 in R3C4, no 1,2 in R3C5 R4C5 = 6 -> no 7 in R3C56, no 5 in R4C6 R5C4 = 4 -> no 3,5 in R6C4, no 4,5 in R6C5 R5C9 = 6 -> no 5 in R46C8, no 7 in R4C9 R6C3 = 2 -> no 3 in R6C2, no 1 in R6C4, no 1,2,3 in R7C4 R6C5 = {12} -> no 2 in R5C6, no 1,2 in R7C56 5a. R5C6 = 3, placed for R5 -> R5C8 = 2, no 2 in R6C5 5b. R6C2 = 7 -> no 6 in R7C3 5c. R7C3 = 4, placed for R7 and C3 5d. R6C5 = 1, placed for C5 5e. R8C3 = 6 -> no 6,7 in R7C4, no 6 in R9C4 5f. R7C4 = 8, placed for R7, no 7 in R7C5 5g. R7C5 = 3, placed for R7 and C5, no 2,4 in R8C5 5h. R3C5 = 4, placed for R3 and C5 5i. R89C5 = [57], placed for C5, 7 placed for R9 5j. R9C7 = 6 (hidden single in R9), placed for C7 Clean-ups: R2C5 = 2 -> no 1,3 in R123C46 R3C5 = 4 -> no 4,5 in R2C6, no 3,4 in R4C6 R5C6 = 3 -> no 2 in R4C67, no 2,4 in R6C67 R4C6 = 1 -> no 2 in R3C6, no 1,2 in R3C7 R5C8 = 2 -> no 1 in R46C7, no 1,3 in R46C8, no 1,2 in R4C9 R4C9 = 4 -> no 3 in R3C89 R6C5 = 1 -> no 2 in R6C4 R7C3 = 4 -> no 4 in R6C4, no 3,4,5 in R8C4 R7C5 = 3 -> no 3 in R6C6, no 2 in R8C4, no 2,3,4 in R8C6 R8C5 = 5 -> no 6 in R7C6 + R8C46 R9C5 = 7 -> no 7,8 in R8C6 R9C7 = 6 -> no 5 in R8C67, no 7 in R8C8 6a. R3C4 = 9, placed for R3 6b. R3C7 = 3, placed for R3 and C7 -> R3C2 = 1, placed for R3, no 1 in R2C3, no 4 in R4C7 6c. R2C3 = 3, placed for C3 -> R1C3 = 1, placed for R1 6d. R3C6 = 8 -> no 9 in R4C7 6e. R4C7 = 5, placed for C7 6f. R4C9 = 4, placed for C9 Clean-ups: R2C3 = 3 -> no 4 in R1C24, no 2 in R2C4 R2C4 = 7 -> no 6 in R1C4 R3C7 = 3 -> no 2 in R2C6 + R3C8, no 2,4 in R2C7 + R4C8, no 2,3,4 in R2C8 R2C6 = 6 -> no 5 in R1C6 R3C8 = 7 -> no 6 in R2C8, no 8 in R2C9 7a. R278C7 = [124], 2 placed for R7 7b. R3C89 = [72], 2 placed for C9 7c. R7C689 = [761], 1 placed for C9 Clean-ups: R3C9 = 2 -> no 1 in R2C8 R7C7 = 2 -> no 1 in R68C6, no 2 in R6C8, no 1,2,3 in R8C8 R7C8 = 6 -> no 7 in R8C9 R8C7 = 4 -> no 3 in R9C6 8a. R18C9 = [73], 7 placed for R1 8b. R26C9 = [98] Clean-up: R6C9 = 8 -> no 9 in R6C8 and the rest is naked singles. Solution: 2 6 1 5 9 4 8 3 7 9 9 3 7 2 6 1 5 9 6 1 5 9 4 8 3 7 2 8 3 7 1 6 1 5 9 4 1 5 9 4 8 3 7 2 6 3 7 2 6 1 5 9 4 8 5 9 4 8 3 7 2 6 1 7 2 6 1 5 9 4 9 3 4 2 8 3 7 1 6 9 5 |
Author: | Andrew [ Tue May 21, 2019 1:07 am ] |
Post subject: | Re: SSS ORC UK 21D 24 25 |
Thanks HATMAN for adding the note about 20. I was struggling with this puzzle after an easy start, then checked my worksheet diagram again and found that I hadn't included the 20, so I checked with HATMAN what it is. After that I reworked my solving path from the start of step 5. Here is my walkthrough for SSS ORC UK 21D: Sums are for heights of skyscrapers visible from the edges. Only those higher than the previous one(s) count toward the sums. On the odd-numbered rows and columns, which are normal ones containing 1-9, each sum must include 9, the height of the highest skyscraper, which can be seen from each edge. However this doesn’t necessarily apply for the even-numbered rows and columns where repeated numbers are allowed; for these rows and columns the height of the highest skyscraper may be less than 9. Since this an ORC puzzle, I’ve stated placements in odd rows/columns. x not 0, y not 7, w is unknown. There is a total on the diagonal. It has been specified that all totals in the even-numbered rows and columns must contain at least one of 7,8,9; also that they contain at least one of 1,2,3 although that was said not to be particularly helpful. Three killer cages have the same total, the other killer cage has twice that total. Because the killer cages cannot be FNC or NC while R5C9 + R6C8 also cannot be AK, the three equal ones must total 4, 5, 6, 7 or 8 (cannot be 2 because three of the four killer cages cannot be [11]) and the other one must total 8, 10, 12, 14 or 16 1. Lower total in C8 = 35 can be only made up as [9.8.7.6.5/9.8.7.641/9.8.75.42] (cannot be [9.8.7.63.2/9.8.753.2.1/9.86.5.4.3/9.86.5.42.1/97.6.5.4.31/8.7.6.5.4.3.2] which require more than 9 cells to provide gaps between consecutive digits) -> R1C8 = 9, placed for R1, R3C8 = 8, placed for R3, R5C8 = 7, placed for R5, R9C8 = {125}, max R6C8 = 5, R7C8 = {126}, max R8C8 = 5, no 8 in R1C7, no 8 in R6C9 [Note. Since these are long permutations, I worked them out by looking at groups of the 2, 3 or 4 missing digits which total 10.] 2a. Skyscrapers starting with 8 or 9 must total 8 (possible for even rows/columns), 9 or 17. Since no totals are given as 17 and xy or 2w cannot be 8, 9 or 17, rows/columns starting/finishing with 8,9 must be ones with unspecified totals. 2b. Unspecified upper totals only in C467, R1C8 = 9, no 8 in R1C7 -> R1C4 = 8, placed for R1 2c. Unspecified left-hand totals only in R24 -> R24C1 = {89}, locked for C1 2d. Unspecified lower totals only in C26 -> R9C26 = {89}, locked for R9 2e. Unspecified right-hand totals only in R68 -> R68C9 = [98], placed for C9 Clean-ups after steps 1 and 2: R1C4 = 8 -> no 7 in R1C35, no 7,8,9 in R2C35, no 7,9 in R2C4 R1C8 = 9 -> no 8,9 in R2C7, no 8 in R2C8 R3C8 = 8 -> no 7 in R2C79 + R34C9, no 7,9 in R24C8 + R3C7, no 7,8,9 in R4C7 R4C1 = {89} -> no 8,9 in R5C2 R5C8 = 7 -> no 6 in R4C79 + R5C9, no 6,8 in R4C8 + R5C7, no 6,7,8 in R6C7 R8C9 = 8 -> no 7 in R79C9 R9C2 = {89} -> no 8,9 in R8C3 R9C6 = {89} -> no 8,9 in R8C57 Naked pair {89} in R24C1 -> no 7 in R3C1, no 7,9 in R3C2 3. Lower total in C8 = 35 can only be made up as [9.8.7.6.5/9.8.7.641] (cannot be [9.8.7.5.42] which would make R7C89 = {12}, not allowed by NC) -> R7C8 = 6, placed for R7, R9C8 = {15} 3a. R89C7 = [41/.5] -> no 4,5 in R9C9 Clean-up: R7C8 = 6 -> no 5 in R6C78 + R7C9 + R8C8, no 5,7 in R7C7, no 5,6,7 in R8C7 4a. R1C9 = 7 (hidden single in C9), placed for R1 4b. R7C7 = 8 (hidden single in C7), placed for R7 and D\, no 7,9 in R6C6, no 9 in R6C7 4c. R5C7 = 9 (hidden single in C7), placed for R5, no 8 in R5C6 4d. R9C7 = 7 (hidden single in C7), placed for R6, no 8 in R9C6 4e. R9C26 = [89], no 7 in R8C3 4f. 8 in R5 only in R5C34 -> no 7,8,9 in R46C3 4g. R5C3 = 8 (hidden single in C3), placed for R5, no 7,9 in R4C4 4h. R37C3 = {79} (hidden pair in C3) 4i. R2C2 + R3C3 = {79} (hidden pair on D\), no 8 in R2C1 4j. R24C1 = [98] Clean-ups: R1C9 = 7 -> no 6 in R2C89 R4C1 = 8 -> no 7,9 in R4C2 R5C3 = 8 -> no 8 in R4C2, no 7,8,9 in R6C24 R5C7 = 9 -> no 8,9 in R4C6 R7C7 = 8 -> no 7,9 in R7C6, no 7,8,9 in R8C6 R9C2 = 8 -> no 7 in R8C1, no 7,9 in R8C2 R9C7 = 7 -> no 6 in R8C6 R3C3 = {79} -> no 8 in R2C4 R7C3 = {79} -> no 8 in R8C24 Naked pair {79} in R2C2 + R3C3 -> no 6 in R2C3 + R3C2 5. Lower total on diagonal = 20 must contain 8,9 -> R8C8 + R9C9 must total 3 (cannot be [21], NC) -> R9C9 = 3, placed for R9, C9 and D\, no 2,4 in R8C8 5a. R8C8 = 1, placed for D\, no 1,2 in R7C9 5b. R7C9 = 4, placed for R7 and C9 5c. Lower total in C8 = 35 (step 3) = [9.8.7.6.5] (only remaining permutation) -> R9C8 = 5, placed for R9 5d. R3C9 = 6 (hidden single in C9), placed for R3, no 5 in R24C9 5e. R5C9 = 5 (hidden single in C9), placed for R5 Clean-ups: R3C9 = 6 -> no 5 in R24C8 R5C9 = 5 -> no 4 in R46C8 R7C9 = 4 -> no 3 in R6C8 R8C8 = 1 -> no 2 in R8C7 R9C8 = 5 -> no 4 in R8C7 R8C7 = {13} -> no 2 in R78C6 6a. 7 in C1 only in R67C1 -> no 6 in R6C12, no 7 in R7C2 6b. 6 in C7 only in R12C7 -> no 5,6 in R1C6, no 5 in R12C7, no 5,7 in R2C6 6c. 5 in C7 only in R34C7 -> no 4,5 in R3C6, no 4 in R34C7 7. 7,9 in R3 only in R3C3456, R3C6 = {79} -> R3C456 must contain one of 7,9 -> no 8 in R4C5 7a. R6C5 = 8 (hidden single in C5) -> no 7,9 in R7C45 7b. 9 in C5 only in R34C5 -> no 9 in R3C46 7c. 7 in R7 only in R7C13 -> no 6 in R8C2 [Now it’s time to use the killer cages; don’t think I can make any further progress without using them. See note at the start about possible killer cage totals before any eliminations.] 8a. R4C4 + R5C5 total 6, 7, 8 or 10, R5C9 + R6C8 total 6 or 7 so they must be two of the cages with the same total 8b. R23C6 is the only killer cage which can have twice the total so must total 12 (cannot total 14 because 6,8 only in R2C6 and no 7 in R2C6) -> R23C6 = [93], 3 placed for R3, no 9 in R3C5, no 2 in R3C7 8c. The three equal killer cages must total 6 8d. R1C67 = 6 = {24}, locked for R1 8e. R4C4 + R5C5 = 6 = {24}, locked for D\ 8f. R5C9 + R6C8 = 6 = [51] 8g. R2C7 = 6 (hidden single in C7) -> no 5 in R3C7 8h. R3C7 = 1, placed for R3 and C7 8i. R8C7 = 3, placed for C7 8j. R4C7 = 5 (hidden single in C7) 8k. R3C3 = 9 (hidden single in R3), placed for C3 and D\ 8l. R2C2 = 7 -> no 6 in R1C123 8m. R1C1 = 5, placed for R1, C1 and D\ 8n. R1C5 = 6 (hidden single in R1), placed for C5 8o. R6C6 = 6 -> no 5 in R7C56 8p. R7C2 = 9 (hidden single in R7) 8q. R7C3 = 7, placed for R7 8r. R6C1 = 7 (hidden single in C1) 8s. R7C4 = 5 (hidden single in R7) Clean-ups: R1C5 = 6 -> no 5,6 in R2C4, no 5 in R2C5 R3C6 = 3 -> no 2,3,4 in R24C5, no 2,4 in R3C5 +R4C6 R3C7 = 1 -> no 1,2 in R24C8, no 1 in R4C6 R4C7 = 5 -> no 6 in R4C6, no 4,6 in R5C6 R4C8 = 3 -> no 2 in R4C9 R6C1 = 7 -> no 6 in R5C12 R6C8 = 1 -> no 2 in R6C7 R6C7 = 4 -> no 3 in R57C6 R7C3 = 7 -> no 6 in R6C34 + R8C3, no 6,7 in R8C4 R7C4 = 5 -> no 4,5 in R68C3, no 4 in R68C4 Naked pair {13} in R1C23 -> no 1,2,3,4 in R2C3 Naked pair {24} in R4C4 + R5C5, no 1,5 in R4C5, no 1,2,3,4 in R5C4 9a. R2C3 = 5, placed for C3, no 4,5 in R3C24 9b. R2C5 = 1, placed for C5 9c. R3C1245 = [4275], 4 placed for C1, 5 placed for C5 9d. R4C9 = 1, placed for C9 -> R2C9 = 2, no 3 in R2C8 9e. R2C8 = 4 -> no 4 in R1C7 9f. R1C67 = [42] 9g. R3C4 = 7 -> no 6 in R4C3, no 7 in R3C5 9h. R7C6 = 1, placed for R7, no 2 in R7C5 9i. R7C5 = 3, placed for R7 and C5 9j. R7C1 = 2, placed for C1, no 1,3 in R8C1 9k. R8C1 = 6, R9C1 = 1, placed for R9 and C1 9l. R5C1 = 3, placed for R5, no 2,4 in R5C2 9m. R5C2 = 1, placed for R5 9n. R5C6 = 2, placed for R5 9o. R5C5 = 4, placed for C5 and D\ 9p. R9C5 = 2, placed for R9 and C5 9q. R9C3 = 6 (hidden single in C3), placed for R9 9r. R4C3 = 4 (hidden single in C3) Clean-ups: R2C3 = 5 -> no 4 in R2C4 R2C5 = 1 -> no 2 in R2C4 R3C1 = 4 -> no 3,4,5 in R4C2 R3C2 = 2 -> no 1 in R4C2 R5C1 = 3 -> no 2 in R4C2, no 2,3,4 in R6C2 R5C2 = 1 -> no 1,2 in R6C3 R5C5 = 4 -> no 3,5 in R4C6 + R6C4 R7C1 = 2 -> no 1 in R6C2, no 1,2,3 in R8C2 R7C5 = 3 -> no 2 in R6C4, no 2,3 in R8C4, no 3,4 in R8C6 R8C1 = 6 -> no 5 in R8C2 R9C3 = 6 -> no 5 in R8C4 R9C5 = 2 -> no 1 in R8C46 10a. R6C3 = 3, placed for C3 10b. R1C3 = 1, placed for R1 and C3, no 1 in R2C4 Solution: 5 3 1 8 6 4 2 9 7 9 7 5 3 1 9 6 4 2 4 2 9 7 5 3 1 8 6 8 6 4 2 9 7 5 3 1 3 1 8 6 4 2 9 7 5 7 5 3 1 8 6 4 1 9 2 9 7 5 3 1 8 6 4 6 4 2 9 7 5 3 1 8 1 8 6 4 2 9 7 5 3 Interesting to note that the digits of the solution total 405. I think that's the first time I've seen that for an ORC puzzle. |
Author: | Andrew [ Thu May 23, 2019 4:35 am ] |
Post subject: | Re: SSS ORC UK 21D 24 25 |
I sometimes find that Skyscraper sum puzzles require more care than usual in avoiding making invalid assumptions. This was one such puzzle. Step 2 and the start of step 3 took some thought; after that it was at about the same level as the other two in this batch. Here is my walkthrough for SSS ORC UK 25: Sums are for heights of skyscrapers visible from the edges. Only those higher than the previous one(s) count toward the sums. On the odd-numbered rows and columns, which are normal ones containing 1-9, each sum must include 9, the height of the highest skyscraper, which can be seen from each edge. However this doesn’t necessarily apply for the even-numbered rows and columns where repeated numbers are allowed; for these rows and columns the height of the highest skyscraper may be less than 9. Since this an ORC puzzle, I’ve stated placements in odd rows/columns. x not 0, y not 7, w is unknown. It has been specified that all totals in the even-numbered rows and columns must contain at least one of 7,8,9; also that they contain at least one of 1,2,3 although that was said not to be particularly helpful. The six killer cages have the same total. 1a. Skyscrapers starting with 8 or 9 must total 8 (possible for even rows/columns), 9 or 17. Since no totals are given as 17 and xy cannot be 8, 9 or 17, rows/columns starting/finishing with 8,9 must be ones with unspecified totals. 1b. Unspecified left-hand totals only in R13 -> R13C1 = {89}, locked for C1 1c. Unspecified right-hand totals only in R57 -> R57C9 = {89}, locked for C9 1d. Unspecified upper totals only in C169, no 8,9 in R1C9 -> R1C16 = {89}, locked for R1 1e. Unspecified lower totals only in C48 -> R9C48 = {89}, locked for R9 Clean-ups: R1C6 = {89} -> no 8,9 in R2C57 R3C1 = {89} -> no 8,9 in R234C2 R5C9 = {89} -> no 8,9 in R456C8 R7C9 = {89} -> no 8,9 in R78C8 R9C4 = {89} -> no 8,9 in R8C35 R9C8 = {89} -> no 8,9 in R8C7 Naked pair {89} in R13C1 -> no 7 in R2C12 Naked pair {89} in R57C9 -> no 7 in R6C89 2a. Upper total in C5 = 15 must contain 9 = [159/249/6.9] with possibly additional lower numbers hidden after the 4, 5 or 6 (cannot be [1239], NC) -> R1C5 = {126} 2b. Lower total 30 in C5 = 30 must contain 8,9 with 7 either included in the total or hidden after the 9 = [9.8.7.6/9.8.7 with two smaller numbers] (cannot be 97.8 with three smaller numbers totalling 13 because these could only be 6.52 or 64.3 which would require 8 cells for this total) -> R3C5 = 9, placed for R3 and C5, R5C5 = 8, placed for R5 and C5, R7C5 = 7, placed for R7 and C5 2c. R3C1 = 8, placed for R3 and C1 -> R1C1 = 9, placed for R1 2d. R5C9 = 9, placed for R5 and C9 2e. R7C9 = 8, placed for R7 2f. R123C5 = [159/249/6.9], no 6 in R2C5 Clean-ups: R1C6 = 8 -> no 7 in R12C7, no 7,9 in R2C6 R3C1 = 8 -> no 7 in R3C2 + R4C12 R3C5 = 9 -> no 8,9 in R2C4 + R4C46, no 8 in R2C6 R5C5 = 8 -> no 7 in R45C46, no 7,8,9 in R6C46 R7C5 = 7 -> no 6 in R6C456 + R7C46 + R8C5, no 6,7,8 in R8C46 R7C9 = 8 -> no 7 in R8C89 3a. Killer cage at R1C5 totals at least 15 -> all six killer cages must total at least 15 3b. Killer cage at R7C5 must total at least 15 with visible numbers totalling 13 to make lower total of 30 for C5 -> R9C5 = 6, placed for R9 and C5 3c. R123C5 = [159/249] -> killer cage at R1C5 totals 15 -> all killer cages must total 15 3d. R79C5 = [76] -> R8C5 = 2 (cage total), 2 placed for C5 3e. R123C5 = [159], 1,5 placed for C5, 1 placed for R1 Clean-ups: R1C5 = 1 -> no 2 in R1C4, no 1,2 in R2C46 R2C5 = 5 -> no 4,5,6 in R1C4 + R3C46, no 4,6 in R2C46 R8C5 = 2 -> no 1,2,3 in R7C46 + R9C6, no 1,3 in R8C46 R9C5 = 6 -> no 5 in R8C46, no 5,7 in R9C6 R2C6 = {35} -> no 4 in R123C7 R4C5 = {34} -> no 3 in R3C46, no 3,4 in R5C46 R6C5 = {34} -> no 4 in R7C46 Naked pair {34} in R46C5 -> no 2,5 in R5C46 Naked pair {59} in R7C46 -> no 4 in R6C5 4a. R6C5 = 3, placed for C5 4b. R9C6 = 4, placed for R9 4c. Naked pair {16} in R5C46, locked for R5 4d. Naked pair {59} in R7C46, locked for R7 Clean-ups: R4C5 = 4 -> no 3,5 in R4C46 R6C5 = 3 -> no 2,4 in R6C46 R9C6 = 4 -> no 3,4,5 in R8C7, no 3,5 in R9C7 5a. R46C7 = {89} (hidden pair in C7) 5b. Killer cage at R4C5 = 15, R4C5 = 4 -> R4C67 = 11 = [29], 9 placed for C7, no 1 in R5C6 5c. R6C7 = 8, no 9 in R7C6 5d. R5C46 = [16] 5e. R7C46 = [95] Clean-ups: R4C6 = 2 -> no 1 in R3C6, no 1,2,3 in R3C7, no 2,3 in R5C7 R5C4 = 1 -> no 1,2 in R46C3, no 2 in R4C4 + R5C3 R5C6 = 6 -> no 5,7 in R5C7, no 5 in R6C6 R6C7 = 8 -> no 7 in R5C8 R7C4 = 9 -> no 8,9 in R6C3 R7C6 = 5 -> no 4,6 in R7C7, no 4 in R8C6, no 6 in R8C7 6a. R5C7 = 4, placed for R5, no 3,5 in R5C8 6b. R5C8 = 2, placed for R5 Clean-ups: R5C7 = 4 -> no 3,4,5 in R46C8, no 3 in R6C6 R5C8 = 2 -> no 1 in R46C8, no 1,2,3 in R46C9 Naked triple {357} in R5C123 -> no 4,6 in R46C2 7a. R6C6 = 1 -> no 1,2 in R7C7 7b. R7C7 = 3, placed for R7 and C7, no 2 in R6C8 7c. R6C8 = 6, no 5 in R6C9 7d. Right-hand total in R6 = 2w, max R6C789 = [864] = 18 (R6C789 = [866] would only contribute 14 toward skyscraper sum) -> total must also contain 9 -> R6C2 = 9 Clean-up: R7C7 = 3 -> no 2,4 in R7C8, no 2 in R8C67, no 2,3,4 in R8C8 8a. 6 in C7 only in R123C7 -> no 5 in R2C67 8b. 5 in C7 only in R13C7 -> no 6 in R2C7 8c. R2C6 = 3 -> no 2 in R12C7 + R3C6 8d. R3C6 = 7, placed for R3, no 6 in R3C7 8e. R3C7 = 5, placed for R3 and C7 8f. R1C7 = 6, placed for R1 8g. R2C7 = 1, placed for C7 8h. R8C7 = 7, placed for C7, no 6 in R7C8, no 8 in R9C8 8i. R9C7 = 2, placed for R9 8j. R9C8 = 9, placed for R9 8k. R7C8 = 1, placed for R7 Clean-ups: R1C7 = 6 -> no 5,7 in R1C8, no 5,6,7 in R2C8 R2C7 = 1 -> no 2 in R12C8, no 1,2 in R3C8 R3C7 = 5 -> no 4 in R2C8, no 4,6 in R3C8 R7C8 = 1 -> no 1,2 in R8C9 R8C7 = 7 -> no 6 in R8C8 R9C4 = 8 -> no 7 in R89C3, no 9 in R8C4 R9C7 = 2 -> no 1 in R8C8 R8C8 = 5 -> no 4,6 in R8C9, no 5 in R9C9 9a. R3C8 = 3, placed for R3, no 2,4 in R3C9, no 2 in R4C8 9b. R4C8 = {67} -> no 6 in R3C9 9c. R3C9 = 1, placed for R3 and C9 9d. R3C4 = 2, placed for R3, no 3 in R2C4 9e. R2C4 = {57} -> no 6 in R3C3 9f. R3C3 = 4, placed for R3 and C3 Clean-ups: R3C2 = 6 -> no 5,6 in R24C1, no 5 in R24C2, no 5,6,7 in R24C3 R3C3 = 4 -> no 3,4 in R2C2, no 3 in R2C3 + R4C23, no 5 in R2C4, no 4 in R4C4 R2C4 = 7 -> no 7 in R1C3, no 8 in R2C3 R3C4 = 2 -> no 1,2 in R2C3, no 1 in R4C4 R3C8 = 3 -> no 2,3,4 in R2C9, no 4 in R4C9 R4C4 = 6 -> no 5,7 in R5C3 10a. R2C3 = 9, placed for C3 10b. R4C3 = 8, no 7 in R5C2 10c. R5C3 = 3, placed for R5 and C3 10d. R5C2 = 5, placed for R5, no 5,6 in R6C3 10e. R5C1 = 7, placed for C1 10f. R6C3 = 7, no 6 in R7C23 10g. R7C3 = 2, placed for R7 and C3 10h. R1C3 = 5, placed for R1 and C3 10i. R7C2 = 4, placed for R7 10j. R7C1 = 6, placed for C1 10k. R9C3 = 1, placed for R9 and C3 10l. R8C3 = 6, no 5,7 in R9C2 10m. R9C2 = 3, placed for R9 10n. R9C1 = 5, placed for C1 10o. R9C9 = 7, placed for C9 10p. Naked pair {56} in R24C9, locked for C9 10q. R6C9 = 4, placed for C9 10r. R8C9 = 3, placed for C9 10s. R1C9 = 2, placed for R1, no 3 in R1C8 10t. R1C8 = 4, placed for R1, no 5 in R2C9 10u. R2C9 = 6, placed for C9 Clean-ups: R4C9 = 5 -> no 6 in R4C8 R5C2 = 5 -> no 4 in R46C1 R5C3 = 3 -> no 2 in R4C2, no 3 in R6C4 R4C2 = 1 -> no 2 in R4C1 R7C2 = 4 -> no 3 in R6C1, no 3,4 in R8C1 R7C3 = 2 -> no 1 in R6C4, no 2 in R8C4 R9C2 = 3 -> no 2 in R8C1 10a. R2468C1 = [4321], no 3 in R1C2 10b. R1C24 = [73] 10c. Killer cage at R2C1 = 15, R2C13 = [49] = 13 -> R2C2 = 2 10d. Killer cage at R2C7 = 15, R2C79 = [16] = 7 -> R2C8 = 8 10e. Killer cage at R8C1 = 15, R8C13 = [16] = 7 -> R8C2 = 8 Solution: 9 7 5 3 1 8 6 4 2 4 2 9 7 5 3 1 8 6 8 6 4 2 9 7 5 3 1 3 1 8 6 4 2 9 7 5 7 5 3 1 8 6 4 2 9 2 9 7 5 3 1 8 6 4 6 4 2 9 7 5 3 1 8 1 8 6 4 2 9 7 5 3 5 3 1 8 6 4 2 9 7 I commented that the solution for 21D was unusual for an ORC puzzle in that the digits totalled 405. This one is even more unusual; as well as totalling 405, all the even rows and columns contain 1-9. After I’d found that the totals for the killer cages were 15, I’d thought it quite likely that at least one of them might be [555]. |
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