Overview: - How do conspiratorial miscounting circles work? - What combinations are (im)possible - Refine the pairing circles - Finish How do conspiratorial miscounting circles work ---------------------------------------------- If n occurs n times, and also n+n times, then it occurs 3n times, so only 1, 2, or 3 can be bad. Because neither set is a valid set of counting circles, One of these must be true: 1&2 are overcounted, 1&3 are, 2&3 are, 1, 2, and 3 are. Specific to this 13+13 scenario: If 1,2&3 are overcounted, there are 8 correctly counted cells that can't use 1, 2, or 3. Therefore each set only overcounts one value. If 1&2 are overcounted, the correctly counted cells must split 8+9 = {35,8}+{36,45,9} If 1&3 are overcounted, the correctly counted cells must split 6+8 = {24,6}+8 If 2&3 are overcounted, the correctly counted cells must split 5+6 = {14,5}+6 For completeness: Overcounting 1&2: 13/13: [1 4 0 0 0 0 0 8 0] [2 2 0 4 5 0 0 0 0] 13/13: [1 4 0 0 0 0 0 8 0] [2 2 3 0 0 6 0 0 0] 13/13: [2 2 0 0 0 0 0 0 9] [1 4 0 0 0 0 0 8 0] 13/13: [2 2 0 0 0 0 0 0 9] [1 4 3 0 5 0 0 0 0] Overcounting 1&3: 13/13: [2 0 3 0 0 0 0 8 0] [1 0 6 0 0 6 0 0 0] 13/13: [2 0 3 0 0 0 0 8 0] [1 2 6 4 0 0 0 0 0] Overcounting 2&3: 13/13: [0 4 3 0 0 6 0 0 0] [0 2 6 0 5 0 0 0 0] 13/13: [0 4 3 0 0 6 0 0 0] [1 2 6 4 0 0 0 0 0] Which combinations are (im)possible ----------------------------------- box 4 lacks green, and boxes 2 and 8 lack blue, so 9 is impossible further, if there's an 8, it's green, but boxes 3/4/5/6/8 fail. Therefore its' one of these: 13/13: [0 4 3 0 0 6 0 0 0] [0 2 6 0 5 0 0 0 0] 13/13: [0 4 3 0 0 6 0 0 0] [1 2 6 4 0 0 0 0 0] and there are 9 3s (and 6 2s). 7 of the 9 3s are placed, 4 green, 3 blue, only boxes 3 and 6 are ambiguous The final 3s have the same colour, and must yielding a 6/3 split, so are both green The colour with 3 3s - blue - has 6 6s and 4 2s, but empty blue occupies 7 rows/boxes/columns However, if r8c8 is 6, that knocks out 2 rows and boxes, so must be 2 this pushes a bunch of 2s and 6s round the grid, but not resolvnging box 1,3,5 yet. Can green contain 5 5s? No, so it's 1 1, 2 2s, and 4 4s - pencil mark all and clean the 2s Green [124] makes box 5s blue [26] a 6, so blue needs 1 more 6, and 2 2s. so r1c2's the 6 Row 5 and column 1 cover 5 green circles, so r5c1 isn't 4, and r4c4 and r6c8 are 4s. => r5c2 is a 4 r5c1 being 2 prevents there being a 2nd 2, so is 1. => All circles complete! => All 2s complete => All 6s complete Refine the pairing circles -------------------------- A 1 can't be on a pairing circle unless 2 also is, and box 1's 2 can't pair with a 3 This uniquely palces all 1s. 4s can be pencilled into one of two cells in boxes 1,2,8, and 9, only box 3 has 3 options. => box 9's 4 is on a [45] white kropke => box 6's 5 is in r5c8 => box 3's 6 connects to a 7 Row 5's kropke must contain an 8 => box 6's pairing circle is [79] above 8 => box 4's 8 can only be in r4c2 => box 7's 8 can only be in r7c3 => box 8's pairing circle must contain its 8, making a [48] pair - which resolves 4 above 8 => box 9's 8 goes in r9c8, and its [45] pair resolves 5 above 4. => box 3's 9 is in r2c8 => box 5's kropke forces the 8 into r5c6 => colum 6's kropke resolves 4 above 5 => box 1's [45] pair resolves 4 above 5 => box 3 reolves 4 above 5 above 8 Finish ------ By this stage you're well into the speedrun. Proof of unique circle contents, once 8s and 9s are dismissed: https://sigh.github.io/Interactive-Sudoku-Solver/?q=.Or.And.ContainExact%7E2_2_2_2_3_3_3_6_6_6_6_6_6%7ER1C2%7ER2C2%7ER1C7%7ER2C6%7ER4C8%7ER5C4%7ER6C5%7ER7C6%7ER8C8%7ER9C9%7ER9C6%7ER8C4%7ER8C3.ContainExact%7E2_2_3_3_3_3_3_3_5_5_5_5_5%7ER3C1%7ER1C8%7ER4C4%7ER5C3%7ER5C2%7ER5C1%7ER4C7%7ER5C9%7ER6C8%7ER7C9%7ER7C1%7ER9C1%7ER9C2.End.And.ContainExact%7E2_2_3_3_3_3_3_3_5_5_5_5_5%7ER1C2%7ER2C2%7ER1C7%7ER2C6%7ER4C8%7ER5C4%7ER6C5%7ER7C6%7ER8C8%7ER9C9%7ER9C6%7ER8C4%7ER8C3.ContainExact%7E2_2_2_2_3_3_3_6_6_6_6_6_6%7ER3C1%7ER1C8%7ER4C4%7ER5C1%7ER5C2%7ER5C3%7ER6C8%7ER5C9%7ER7C9%7ER7C1%7ER9C1%7ER9C2%7ER4C7.End.And.ContainExact%7E1_2_2_3_3_3_3_3_3_4_4_4_4%7ER1C2%7ER2C2%7ER1C7%7ER2C6%7ER4C8%7ER5C4%7ER6C5%7ER7C6%7ER8C8%7ER9C9%7ER9C6%7ER8C4%7ER8C3.ContainExact%7E2_2_2_2_3_3_3_6_6_6_6_6_6%7ER3C1%7ER1C8%7ER5C9%7ER6C8%7ER4C7%7ER7C9%7ER4C4%7ER5C1%7ER5C2%7ER5C3%7ER7C1%7ER9C1%7ER9C2.End.And.ContainExact%7E2_2_2_2_3_3_3_6_6_6_6_6_6%7ER1C2%7ER2C2%7ER1C7%7ER2C6%7ER4C8%7ER5C4%7ER6C5%7ER7C6%7ER8C8%7ER9C9%7ER9C6%7ER8C4%7ER8C3.ContainExact%7E1_2_2_3_3_3_3_3_3_4_4_4_4%7ER3C1%7ER1C8%7ER4C7%7ER5C9%7ER6C8%7ER7C9%7ER4C4%7ER5C1%7ER5C2%7ER5C3%7ER7C1%7ER9C1%7ER9C2.End.End Proof of rest of puzzle given the above digits: https://sigh.github.io/Interactive-Sudoku-Solver/?q=.%7ER1C2_6%7ER2C2_2%7ER3C1_3%7ER2C6_3%7ER1C7_2%7ER1C8_3%7ER4C8_6%7ER4C7_3%7ER4C4_4%7ER5C4_6%7ER6C5_3%7ER7C6_6%7ER6C8_4%7ER7C9_3%7ER8C8_2%7ER9C9_6%7ER9C6_2%7ER9C2_3%7ER8C3_6%7ER8C4_3%7ER5C3_3%7ER6C1_6%7ER5C2_4%7ER2C5_6%7ER3C5_2%7ER3C7_6%7ER5C1_1%7ER7C1_2%7ER9C1_4%7ER6C4_2%7ER5C9_2.Or.Renban%7ER2C2%7ER3C2.Renban%7ER2C2%7ER3C3.Renban%7ER2C2%7ER2C3.End.Or.Renban%7ER2C3%7ER2C2.Renban%7ER2C3%7ER3C2.Renban%7ER2C3%7ER3C3.End.Or.Renban%7ER3C2%7ER2C2.Renban%7ER3C2%7ER2C3.Renban%7ER3C2%7ER3C3.End.Or.Renban%7ER3C3%7ER3C2.Renban%7ER3C3%7ER2C2.Renban%7ER3C3%7ER2C3.End.Or.Renban%7ER5C8%7ER5C7.Renban%7ER5C8%7ER6C7.Renban%7ER5C8%7ER6C8.End.Or.Renban%7ER6C8%7ER6C7.Renban%7ER6C8%7ER5C7.Renban%7ER6C8%7ER5C8.End.Or.Renban%7ER6C7%7ER5C7.Renban%7ER6C7%7ER5C8.Renban%7ER6C7%7ER6C8.End.Or.Renban%7ER7C2%7ER7C3.Renban%7ER7C2%7ER8C3.Renban%7ER7C2%7ER8C2.End.Or.Renban%7ER7C3%7ER7C2.Renban%7ER7C3%7ER8C2.Renban%7ER7C3%7ER8C3.End.Or.Renban%7ER8C3%7ER8C2.Renban%7ER8C3%7ER7C2.Renban%7ER8C3%7ER7C3.End.Or.Renban%7ER8C2%7ER7C2.Renban%7ER8C2%7ER7C3.Renban%7ER8C2%7ER8C3.End.Or.Renban%7ER5C7%7ER5C8.Renban%7ER5C7%7ER6C8.Renban%7ER5C7%7ER6C7.End.Or.Renban%7ER7C4%7ER7C5.Renban%7ER7C4%7ER8C5.Renban%7ER7C4%7ER8C4.End.Or.Renban%7ER7C5%7ER7C4.Renban%7ER7C5%7ER8C4.Renban%7ER7C5%7ER8C5.End.Or.Renban%7ER8C5%7ER8C4.Renban%7ER8C5%7ER7C4.Renban%7ER8C5%7ER7C5.End.Or.Renban%7ER8C4%7ER7C4.Renban%7ER8C4%7ER7C5.Renban%7ER8C4%7ER8C5.End.WhiteDot%7ER1C1%7ER2C1.WhiteDot%7ER3C6%7ER4C6.WhiteDot%7ER5C5%7ER5C6.WhiteDot%7ER3C7%7ER3C8.WhiteDot%7ER7C7%7ER8C7