The New Sudoku Players' Forum

[Smythe Dakota]

This may be a beginners' question but I am wondering if anyone uses the "completing chutes" method to solve puzzles more quickly.

For example, in the sample puzzle that greets viewers at the top of this website:

Code: Select all

+-------+-------+-------+
|   6   | 1   4 |   5   |
|     8 | 3   5 | 6     |
| 2     |       |     1 |
+-------+-------+-------+
| 8     | 4   7 |     6 |
|     6 |       | 3     |
| 7     | 9   1 |     4 |
+-------+-------+-------+
| 5     |       |     2 |
|     7 | 2   6 | 9     |
|   4   | 5   8 |   7   |
+-------+-------+-------+



-- one can quickly observe that, in the middle band (rows 4-5-6) there is an upper 7 and a lower 7, thus there must be a middle 7 in the remaining (right-hand) box, i.e. r5c7, r5c8, or r5c9. In this case, of these three cells, only r5c9 is eligible to receive a 7, so right away we have placed a 7.

By similar reasoning (again in the middle band) we can quickly place a 6 in r6c5, and a 4 in r5c1.

Having placed this 4, we can now look at the left stack (columns 1-2-3) and observe that, since there is a left 4 and a center 4, there must be a right 4 in the remaining (top) box, i.e. r1c3, r2c3, or r3c3. Again, in this case only r3c3 is eligible to receive a 4, so there we are.

This technique works especially well at the outset of solving a puzzle. First check the three (horizontal) bands, then the three (vertical) stacks. Then check the three bands again, since some cells may have now been filled in from checking the stacks.

"Completing chutes" is nothing really new or earth-shaking, it's just a special case of the basic technique of asking "where can a 4 go in this row (or column or box)?". But, visually, it's quicker to see, and seems to get things off to a fast start.

Again, please excuse me if this is a beginners' observation. By now I can usually solve Monday's, Tuesday's, and Wednesday's puzzles on the train going in to work, while (for me) Thursday's and Friday's usually require extra time and at least one "reductio ad absurdum" (translation, trial and error) chain. And I haven't studied all those advanced techniques, like the flying swordfish, bare-naked duos, triple-XXX-rated airplane wings, etc. (Maybe, inadvertently, I have just described one of those.) I'm just thinking of this method as a speed-up technique.

Bill Smythe

[jimbob]

Bill,

I always start every puzzle in this way.

[ronk]

And I haven't studied all those advanced techniques, like the flying swordfish, bare-naked duos, triple-XXX-rated airplane wings, etc. (Maybe, inadvertently, I have just described one of those.)

IMO you described "hidden singles".

[coloin]

Your puzzle - and the 7 can be inserted at r5c9

Code: Select all

+-------+-------+-------+
|   6   | 1   4 |   5   |
|     8 | 3   5 | 6     |
| 2     |       |     1 |
+-------+-------+-------+
| 8     | 4   7 |     6 |
|     6 |       | 3     |
| 7     | 9   1 |     4 |
+-------+-------+-------+
| 5     |       |     2 |
|     7 | 2   6 | 9     |
|   4   | 5   8 |   7   |
+-------+-------+-------+

These are the important clues.

Code: Select all

+---+---+---+
|...|...|...|
|...|...|...|
|...|...|...|
+---+---+---+
|...|..7|...|
|...|...|3..|
|7..|...|...|
+---+---+---+
|...|...|...|
|...|...|...|
|...|...|.7.|
+---+---+---+


Of all the unavoidable sets in this grid... - and there are many [see below] - these 4 clues are in every set that the r5c9 clue is in - hence it can be inserted with certainty.

This is the basis of sudoku solving !

Here are [some!] of the "unavoidable sets" - a valid puzzle has to have one clue in each of these sets.

The 4 given clues are described by position - 46,57,61 & 98.
The inserted clue is 59

As an example one set is highlighted

C:\suxxprog>unavoid pappa.txt
963174258
178325649
254689731
821437596
496852317
735961824
589713462
317246985
642598173

Found 209 unavoidable sets (36 of size 4 or 6).

The maximum # of disjoint unavoidable sets (max clique number -- MCN) is 12.
One such maximal collection is:
{14,15,74,75}
{16,18,26,28}
{35,36,95,96}
{53,55,63,65}
{56,58,66,68}
{11,12,51,52,91,92}
{17,19,67,69,77,79}
{21,22,61,62,81,82}
{24,25,44,45,84,85}
{27,29,47,49,87,89}
{31,32,41,42,71,72}
{37,38,39,97,98,99}



C:\suxxprog>unav27 pappa.txt
11 13 14 15 16 17 21 22 24 25 26 28 31 32 33 36 37 38
11 13 14 15 16 17 21 23 24 25 27 29 33 34 35 36 37 39
11 13 14 16 21 23 24 25 31 33 35 36
11 13 14 16 21 22 24 26 27 28 29 32 33 34 36 37 38 39
11 13 14 19 21 23 24 29
11 12 14 16 18 21 23 24 25 28 31 32 33 34 35 36 38
11 12 13 14 16 18 21 29 32 33 34 36 38 39
11 12 14 15 19 21 22 23 24 25 26 28 29 31 32 33 34 35 36 38
11 12 14 15 19 21 22 25 26 29 31 32 34 35 36 39
11 12 14 15 19 21 22 23 24 27 28 29 33 34 35 37 38
11 12 13 17 18 19 22 23 26 27 28 29 31 32 33 36 37 38
11 12 13 17 18 19 21 23 24 25 27 29 31 32 34 35 38 39
11 12 16 17 18 23 24 25 27 28 31 32 33 34 35 36 38
11 12 13 16 17 18 22 27 31 32 33 36 37 38
11 12 13 16 17 18 23 24 25 27 31 32 33 34 35 36 38
11 15 17 19 21 29 31 35 37 39
11 12 15 17 19 22 23 24 26 27 28 29 31 32 33 34 35 36 37 38
11 13 15 16 17 18 19 22 23 24 25 26 27 29 31 32 33 34 35 36 37 38
11 13 14 15 16 17 19 22 24 25 26 27 28 31 32 33 34 35 36 37 38 39
11 12 13 14 15 16 17 18 19 31 32 33 34 35 36 37 38 39
15 17 23 24 25 27 28 33 34 35 37 38
15 17 21 22 25 26 29 31 32 36 37 39
21 22 23 24 25 26 27 28 29 31 32 33 34 35 36 37 38 39
12 14 15 19 22 27 34 35 37 39
12 14 15 19 21 22 23 24 25 26 28 29 31 32 33 34 35 36 38 39
13 14 15 17 19 23 24 25 27 34 35 37 39
13 14 19 22 23 24 26 27 29 32 34 36 37 39
13 14 19 21 23 24 25 31 35 39
12 13 14 18 19 23 26 28 29 32 33 34 36 38 39
13 16 18 21 22 23 24 25 26 27 29 31 32 33 34 35 36 37 38 39
41 43 46 47 61 63 66 67
42 43 46 48 49 52 53 56 58 59
42 44 45 48 49 62 64 65 68 69
71 74 76 78 81 86 88 89 94 96 98 99
71 72 73 74 75 76 77 78 79 81 82 83 84 85 86 87 88 89
71 72 73 74 75 76 78 79 81 82 83 84 86 87 88 89 95 97
71 72 73 74 76 77 78 79 81 82 83 84 86 87 88 89 92 97
71 72 73 75 76 77 79 81 82 83 84 85 86 87 88 89 91 93 94 96 98
71 72 73 74 75 76 77 81 82 83 85 87 88 89 94 96 98 99
71 72 73 74 75 76 77 81 82 83 85 86 87 88 91 94 96 98
71 72 73 75 76 79 81 82 84 85 86 89 91 92 93 94 95 96
71 72 73 75 76 79 81 82 83 84 86 87 88 89 91 93 94 95 96 97 98
71 72 73 74 75 76 81 82 83 84 85 89 92 93 94 95 96 99
71 72 73 74 75 76 81 82 83 87 88 89 94 95 96 97 98 99
71 72 73 74 75 76 81 82 83 84 85 86 91 92 93 94 95 96
71 72 73 74 75 76 81 82 83 86 87 88 91 94 95 96 97 98
71 72 73 76 77 79 81 84 85 86 87 89 91 92 93 94 95 96
71 72 73 76 77 79 81 82 83 84 86 87 88 89 91 92 93 94 96 97 98
71 72 73 74 76 77 81 83 84 85 87 89 92 93 94 95 96 99
71 72 73 74 76 77 81 82 83 87 88 89 92 94 96 97 98 99
71 72 73 74 76 77 81 83 84 85 86 87 91 92 93 94 95 96
71 72 73 74 76 77 81 82 83 86 87 88 91 92 94 96 97 98
71 72 73 75 76 77 79 81 84 86 89 91 92 93 94 95 96 97
71 72 73 74 75 76 77 81 83 84 89 92 93 94 95 96 97 99
71 72 73 74 75 76 77 81 83 84 86 91 92 93 94 95 96 97
72 73 74 75 76 77 79 82 83 84 85 87 88 89 94 96 98 99
72 73 74 75 76 79 82 83 84 87 88 89 94 95 96 97 98 99
72 73 74 76 77 79 82 83 84 87 88 89 92 94 96 97 98 99
74 76 78 79 83 84 86 88 93 96 98 99
74 76 78 79 84 86 88 89 94 96 98 99
74 76 78 79 81 83 84 86 91 93 98 99
74 76 78 79 81 84 86 89 91 94 98 99
72 73 75 76 77 78 79 81 82 83 85 86 87 88 91 93 98 99
72 73 75 76 78 79 81 82 83 86 87 88 91 93 95 97 98 99
72 73 76 77 78 79 81 82 83 86 87 88 91 92 93 97 98 99
72 73 75 76 77 79 82 83 85 87 88 93 96 98 99
72 73 75 76 79 82 85 92 93 95 96 99
72 73 75 76 79 82 83 87 88 93 95 96 97 98 99
72 73 76 77 79 85 87 92 93 95 96 99
72 73 76 77 79 82 83 87 88 92 93 96 97 98 99
72 73 75 76 77 79 92 93 95 96 97 99
81 83 84 86 88 89 91 93 94 96 98 99
71 74 78 79 81 83 84 89 91 93 98 99
71 72 73 75 77 78 79 82 83 84 85 87 88 89 91 93 94 98
71 72 73 75 77 78 79 81 82 83 85 87 88 89 91 93 98 99
71 72 73 75 78 79 82 84 85 86 88 89 91 92 93 94 95 96
71 72 73 75 78 79 82 83 84 87 88 89 91 93 94 95 97 98
71 72 73 75 78 79 81 82 85 86 88 89 91 92 93 95 96 99
71 72 73 75 78 79 81 82 83 87 88 89 91 93 95 97 98 99
71 72 73 74 75 78 82 83 84 85 86 88 91 92 93 94 95 96
71 72 73 77 78 79 84 85 86 87 88 89 91 92 93 94 95 96
71 72 73 77 78 79 82 83 84 87 88 89 91 92 93 94 97 98
71 72 73 77 78 79 81 85 86 87 88 89 91 92 93 95 96 99
71 72 73 77 78 79 81 82 83 87 88 89 91 92 93 97 98 99
71 72 73 74 77 78 83 84 85 86 87 88 91 92 93 94 95 96
71 72 73 75 77 78 79 84 86 88 89 91 92 93 94 95 96 97
71 72 73 75 77 78 79 81 86 88 89 91 92 93 95 96 97 99
71 72 73 74 75 77 78 83 84 86 88 91 92 93 94 95 96 97
11 13 21 22 23 31 33 41 42 51 52 72 73 81 82 83 92 93
11 13 61 63 71 73 81 83
11 12 13 31 32 33 62 63 71 73 91 92 93
11 12 13 22 23 31 32 33 61 62 71 72 73 81 83 91 92 93
11 12 31 32 42 43 52 53 61 62 63 71 73 81 82 83 91 93
11 12 22 23 31 32 42 43 52 53 71 72 73 82 83 91 93
12 13 22 23 32 33 42 43 52 53 62 63 72 73 82 83 92 93
12 13 22 23 52 53 61 62 72 73 81 83
12 13 32 33 51 53 62 63 91 92
32 33 42 43 61 62 63 81 82 83 92 93
21 22 32 33 41 42 43 61 63 71 72 82 83 92 93
21 22 41 42 43 51 52 53 61 63 71 72 73 82 83 91 92 93
22 23 42 43 51 52 53 72 73 82 83 91 92 93
15 16 24 25 26 34 35 36 44 46 54 55 64 65 66 74 75 76 84 85 94 96
15 16 24 26 44 46 54 56 65 66 74 75 76 84 85 86 94 96
15 16 45 46 65 66 75 76 85 86
34 35 54 55 64 65 94 95
24 26 44 45 54 56 65 66 75 76 84 85 86 94 96
14 15 16 24 26 34 35 44 45 46 54 56 64 65 66 75 76 84 86 94 95 96
14 15 16 24 26 34 35 36 44 45 46 54 55 56 64 66 75 76 84 86 94 95
14 15 16 24 26 34 36 44 45 46 54 56 64 66 75 76 84 86 94 96
14 16 24 26 34 35 44 46 54 56 64 65 66 74 76 84 86 94 95 96
14 16 24 26 34 35 36 44 46 54 55 56 64 66 74 76 84 86 94 95
14 16 24 26 34 36 44 46 54 56 64 66 74 76 84 86 94 96
14 16 44 45 46 65 66 74 75 76 85 86
14 15 16 24 25 26 34 35 45 46 54 56 64 65 66 74 76 84 85 86 94 95 96
14 15 16 24 25 34 36 45 46 64 66 74 76 84 85 86
27 29 48 49 67 68 69 77 78 79 87 88
28 29 48 49 68 69 78 79
17 18 27 28 29 47 48 49 68 69 77 79
17 18 47 48 67 68 87 88
17 18 19 28 29 47 48 67 68 69 87 89
17 19 47 48 49 67 68 78 79 87 88 89
16 17 25 28 31 33 42 43 44 51 56 68 69 75 77 79 82 84 85 93 97
14 16 17 21 25 28 31 33 39 42 44 51 56 68 69 75 79 82 84 92 93 97
14 17 21 25 31 39 58 59 74 75 79 83 84 93 97 98
42 43 54 56 58 66 67 82 84 88 93 96 97
14 17 21 25 31 39 54 58 66 67 68 75 79 84 88 96 97
14 17 23 25 31 35 39 41 42 43 75 79 82 84 93 97
14 17 19 21 25 31 39 67 68 72 75 79 82 84 88
14 17 19 21 23 25 31 39 41 42 54 58 66 67 72 75 79 82 84 88 93 96 97
14 17 19 35 39 42 43 54 56 67 68 72 75 79 82 84 88 93 96 97
14 19 21 23 25 31 35 39 41 42 54 56 72 75 82 84 93 96
11 14 17 21 25 29 43 48 52 58 73 75 79 82 84 87
11 14 21 25 31 36 42 43 48 52 56 58 64 68 73 75 82 84 93 95
11 17 21 29 31 39 43 48 52 58 73 79 82 87 93 97
11 14 17 31 36 39 64 66 73 75 79 93 95 97
56 58 66 68
21 25 29 31 36 39 42 43 48 52 56 58 64 68 82 84 87 93 95 97
14 17 25 29 36 39 64 66 75 79 84 87 95 97
14 17 21 25 31 39 42 43 75 79 82 84 93 97
13 14 21 24 38 39 43 45 51 57 58 62 66 69 75 76 77 81 85 92 99
13 14 18 21 26 32 38 39 43 45 47 55 58 62 66 71 75 76 82 89 94 97
12 14 21 24 27 34 38 57 58 65 66 75 78 82 86 91 97
12 14 21 27 34 38 39 76 78 81 82 86 91 97 99
12 13 14 21 24 45 49 53 57 58 62 65 66 75 76 78 81 82 86 91 97 99
12 13 14 21 24 27 34 39 45 49 53 57 62 65 81 82 97 99
12 13 14 21 24 27 34 38 43 45 57 58 62 66 75 76 78 82 86 91 97
13 14 21 24 43 45 49 53 57 58 62 65 66 75 78 81 82 86 91 97 99
38 39 43 49 53 58 76 78 81 86 91 99
24 27 34 38 39 43 49 53 57 58
13 14 21 24 43 46 61 66 81 83
13 14 21 22 24 43 45 61 62 66 75 76
13 14 15 21 24 43 45 46 62 66 74 76 81 82
13 14 15 21 22 24 43 45 46 61 62 66 74 76
37 39 57 59 97 99
37 38 39 97 98 99
37 38 39 57 58 59
37 38 57 58 97 98
38 39 58 59 98 99
57 58 59 97 98 99
43 46 61 62 66 81 82 83
21 22 61 62 81 82
13 15 22 24 43 45 62 66 74 75 76 82 83
13 15 22 24 43 45 46 62 66 74 76 82 83
13 15 21 22 24 45 46 74 76 81 82 83
13 14 19 21 24 35 38 41 43 45 54 57 62 67 72 76 81 88 96 99
13 19 21 23 24 35 38 43 45 54 57 62 66 67 72 75 76 81 82 88 96 99
13 14 21 24 43 45 62 66 75 76 81 82
38 39 57 58 97 99
12 14 16 21 27 28 33 34 39 43 44 49 51 53 58 65 69 75 78 82 85 86 91 97
11 14 21 28 33 36 39 43 44 51 52 58 64 66 69 73 75 77 82 85 87
11 14 21 28 33 36 39 43 44 51 52 58 64 66 69 73 75 82 85 92 95
11 14 21 28 33 36 39 43 44 51 52 58 64 66 69 73 75 82 87 95 97
11 16 33 36 51 52 73 77 82 87 92 97
11 16 33 36 51 52 73 75 77 92 95 97
82 85 87 92 95 97
75 77 85 87 95 97
75 77 82 85 92 97
43 48 52 58 73 75 77 82 85 87
43 48 52 58 73 75 82 85 92 95
14 16 21 28 33 39 43 44 51 58 66 69
14 18 21 23 26 32 39 41 47 55 58 63 66 67 71 75 82 89 94 97
14 18 23 26 32 35 39 41 43 47 54 55 58 63 66 71 72 75 82 89 94 96 97
14 18 21 26 32 39 43 47 55 58 63 66 71 75 82 89 94 97
12 14 21 22 27 34 39 46 49 53 58 59 61 65 66 75 78 82 83 86 91 97
12 14 15 21 27 34 39 43 46 49 53 58 59 61 65 66 74 78 82 83 86 91 97 98
12 15 22 27 34 37 39 43 46 49 53 58 61 65 74 78 83 86 91 97 98
12 15 21 22 27 34 37 39 43 49 53 58 65 66 74 75 78 82 86 91 97
12 14 21 27 34 39 43 49 53 58 72 78 82 88 91 97
12 14 19 21 27 34 35 39 43 49 53 54 58 65 67 82 86 88 91 96 97
65 66 72 75 78 82 86 88
14 19 35 39 54 58 65 66 67 75 78 86 88 96 97
12 19 23 27 34 35 41 49 53 54 65 66 67 72 75 82 86 91 96
12 14 19 23 27 34 35 39 41 49 53 54 58 66 67 72 75 78 91 96 97
11 14 21 27 29 34 36 39 43 49 53 58 64 65 75 78 86 87 95 97
11 12 14 21 27 34 39 43 48 49 52 53 58 73 75 78 91 95 97
11 12 14 21 29 34 36 39 64 65 82 86 87 91 95 97
11 12 43 48 52 58 65 66 73 75 82 86 91 95
12 14 21 27 34 39 43 49 53 58 65 66 75 78 82 86 91 97
12 14 21 27 29 34 39 48 49 52 53 73 78 82 87 91 97
27 29 43 48 49 52 53 58 65 66 73 75 78 82 86 87
15 19 21 22 35 37 41 43 46 54 59 61 67 72 74 82 83 88 96 98
15 19 21 22 23 35 37 43 46 54 59 61 66 67 72 74 83 88 96 98
14 15 19 35 37 54 58 59 66 67 72 75 82 88 96 97 98
14 19 21 22 35 37 39 41 43 46 54 59 61 67 72 74 75 82 83 88 96 98
14 19 22 23 35 37 39 41 46 54 59 61 67 72 74 75 83 88 96 98
14 19 21 22 23 35 37 39 43 46 54 59 61 66 67 72 74 75 83 88 96 98
11 15 21 22 29 36 37 43 46 48 52 59 61 64 66 73 74 75 82 87 95 97 98
37 39 58 59 97 98
21 22 43 46 61 66 82 83
14 15 74 75
11 14 21 29 36 39 64 66
43 48 52 58 73 75 82 87 95 97
21 23 41 43
14 19 35 39 54 58 66 67 72 75 82 88 96 97
13 16 17 24 28 31 33 38 42 44 51 56 57 62 68 69 76 77 79 81 84 92 93 99
13 17 31 38 42 44 45 56 57 62 68 76 79 81 84 85 93 99
16 17 24 25 28 31 33 42 44 45 51 56 68 69 77 79 92 93
24 25 44 45 84 85
13 16 17 25 28 31 33 38 42 45 51 56 57 62 68 69 76 77 79 81 85 92 93 99
13 16 24 25 28 33 38 51 57 62 69 76 77 81 84 85 92 99
17 18 24 26 31 32 42 47 63 68 71 76 79 93 94 99
17 18 24 26 31 32 42 47 63 68 71 76 81 84 93 94
12 17 25 27 31 34 42 45 53 56 62 65 84 86 91 93
12 13 17 24 25 27 31 34 38 42 45 49 53 56 57 76 78 79 81 84 93 99
12 13 17 24 25 27 31 34 38 42 45 49 53 56 57 76 78 81 84 86 91 93 99
12 13 17 24 25 27 31 34 38 53 56 57 62 65 68 84 86 91 93
12 13 24 27 34 38 42 49 53 57 62 68 78 79
13 17 24 25 31 38 45 49 56 57 65 68 76 78 79 81 84 93 99
13 17 24 25 31 38 45 49 56 57 65 68 76 78 81 84 86 91 93 99
42 45 49 62 65 68 78 79
13 15 17 22 24 25 31 37 42 45 46 56 59 61 62 68 76 79 81 84 93 98 99
13 15 17 22 24 31 37 45 46 61 62 74 76 79 81 84 93 99
15 17 22 24 25 31 37 38 42 45 46 56 57 61 62 68 74 76
24 25 37 38 42 45 56 57 59 62 68 74 76 79 83 84 93 98
13 15 17 22 25 31 38 42 46 56 57 59 61 68 81 83 98 99
13 15 22 24 25 42 45 46 56 59 61 62 68 76 79 81 83 84 93 98 99
13 17 23 24 25 31 35 38 41 42 45 56 57 62 68 72 76 81 84 93 96
13 17 23 24 31 35 38 41 42 45 54 57 62 67 68 72 76 79 93 96 99
13 17 23 24 31 35 38 41 45 54 56 57 62 67 68 72 76 81 84 93 96
13 17 19 31 38 41 42 67 68 72 79 81 88 93 99
13 17 19 31 38 54 56 67 68 81 84 88 93 96 99
23 25 31 35 41 42 72 76 79 93 96 99
23 24 25 31 35 41 45 54 56 57 62 67 72 76 81 84 93 96
17 19 24 25 41 42 45 56 57 62 67 68 72 76 79 81 84 88
17 19 54 56 67 68 76 79 84 88 96 99
13 19 23 25 31 35 38 41 42 72 79 81 88 93 99
13 19 23 25 31 35 38 54 56 81 84 88 93 96 99
13 17 19 23 24 25 35 38 41 42 45 56 57 62 68 72 76 79 81 84 88
13 17 19 23 24 25 35 38 42 45 54 56 57 62 68 76 79 84 88 96 99
13 17 19 23 24 35 38 41 45 54 56 57 62 67 68 72 76 79 81 84 88
13 17 24 25 31 38 42 45 56 57 62 68 76 79 81 84 93 99
16 17 25 28 31 32 42 44 51 56 68 69 71 77 79 84 85 89 92 94
16 17 25 28 42 44 47 51 55 56 63 68 69 71 77 85 89 92 93 94
16 17 18 25 28 31 32 33 42 44 47 51 56 63 68 69 77 79 84 85
31 32 33 71 79 84 89 92 93 94
31 33 44 47 51 55 63 69 71 77 79 84 85 89 93 94
17 18 32 33 42 47 63 68 92 93
12 17 25 27 31 33 34 42 49 51 53 56 65 68 78 79 84 86
12 17 25 27 28 31 33 34 42 49 51 53 56 65 68 69 77 79 84 86
12 17 25 27 28 31 34 42 49 53 56 65 68 69 77 79 84 86 91 93
12 16 17 25 28 31 34 42 44 49 51 56 65 68 69 77 79 84 85 86 91 92
12 16 17 25 28 31 34 42 44 49 51 56 65 68 77 78 79 84 85 86 91 92
12 16 17 25 27 33 34 42 44 49 53 56 65 69 84 85 86 92 93
12 16 31 33 34 44 49 65 69 85 86 91 92 93
16 17 25 28 42 44 51 53 56 68 69 77 79 84 85 91 92 93
31 33 51 53 91 93
15 17 22 25 28 31 33 37 42 46 56 59 61 68 69 77 79 92 93
15 16 17 25 28 31 33 42 44 46 51 56 59 61 68 74 77 79 83 85 92 93 98
16 17 22 25 28 31 33 37 42 44 51 56 74 77 83 84 85 92 98
16 17 25 28 31 33 42 44 51 56 59 61 68 74 77 79 83 84 85 92 93 98
22 28 33 37 68 69 74 77 79 83 84 92 93 98
16 17 25 28 54 56 67 68 72 77 84 85 88 92 96
16 17 19 25 28 31 33 42 44 51 56 67 68 69 84 85 92 93
23 25 31 35 41 42 54 56 67 68 69 72 77 79 84 88 93 96
31 33 42 44 51 54 56 67 68 69 72 77 79 84 88 92 93 96
17 19 67 69 77 79
17 19 31 33 42 44 51 54 56 67 68 72 79 84 88 92 93 96
16 19 23 25 28 31 33 35 42 44 51 56 68 69 72 79 84 85 88 92 93 96
16 19 23 25 28 31 35 41 42 67 69 72 77 85 88 92 93 96
16 19 23 25 28 31 35 41 42 44 51 56 68 69 72 79 84 85 88 93 96
16 19 23 28 31 33 35 41 42 44 51 56 68 69 72 79 84 85 88 93 96
16 17 19 23 25 28 31 35 41 42 72 77 79 85 88 92 93 96
16 17 19 23 28 33 35 72 77 79 85 88 92 96
16 17 25 28 31 33 42 44 51 56 68 69 77 79 84 85 92 93
15 17 18 22 25 31 32 37 42 46 47 56 59 63 68 71 74 79 93 94 98
15 17 18 22 25 31 32 37 42 46 47 56 59 63 68 71 79 83 89 93 98
15 17 22 25 31 37 42 46 56 59 61 63 68 71 74 79 93 94 98
15 17 22 25 31 37 42 46 56 59 61 63 68 71 79 83 89 93 98
15 18 22 26 32 37 46 47 55 59 74 79 84 89 94 98
15 18 22 26 32 37 46 47 55 59 83 84 89 93 94 98
61 63 71 74 79 83 84 89
61 63 71 74 83 84 93 94
31 32 41 42 71 72
23 25 31 35 41 47 63 67 71 79 84 89 93 94
23 26 31 32 35 41 42 47 54 55 63 67 94 96
17 18 41 42 47 63 68 71 72 79 84 89 93 94
17 18 19 23 26 32 35 42 47 54 55 56 63 68 72 79 84 88 93 96
17 18 19 23 26 32 35 42 47 54 55 63 68 72 79 84 88 93 94 96
17 19 23 26 32 35 42 47 54 55 56 63 67 68 72 79 84 88 93 96
17 19 23 26 32 35 42 47 54 55 63 67 68 72 79 84 88 93 94 96
17 19 41 42 47 63 67 68 71 72 79 84 88 89 93 94
17 19 31 32 42 47 63 67 68 71 79 84 88 89 93 94
17 19 23 25 26 31 32 35 54 55 67 68 71 72 79 84 88 93 94 96
17 19 23 25 26 31 35 41 42 54 55 67 68 72 79 84 88 93 94 96
17 19 23 25 31 32 35 54 56 67 68 71 72 79 84 88 93 96
25 26 55 56
17 18 31 32 42 47 63 68 71 79 84 89 93 94
12 15 17 31 34 37 42 46 49 53 56 59 61 65 68 74 78 79 83 84 86 91 93 98
12 17 22 27 31 37 42 46 56 59 61 68 74 79 83 84 93 98
15 17 25 27 34 37 46 49 53 59 61 65 74 78 83 86 91 98
22 25 27 31 34 37 42 46 49 53 56 59 61 65 68 74 78 79 83 84 86 91 93 98
12 15 17 22 25 27
12 15 22 25 31 34 42 49 53 56 65 68 78 79 84 86 91 93
11 17 25 29 31 34 36 73 79 84 86 87 93 95
11 12 17 25 27 29 31 34 52 53 64 65 73 79 84 87 91 93 95
11 12 17 25 27 29 31 36 42 48 49 53 56 65 68 73 78 86 87 91 93 95
11 12 17 25 27 29 31 36 42 49 53 56 65 68 73 78 79 86 87 91 93 95
11 12 27 29 42 48 49 52 53 56 64 65 68 73 78 84 86 87 91 95
11 12 27 29 42 49 52 53 56 64 65 68 73 78 79 84 86 87 91 95
12 17 25 27 31 34 42 49 53 56 65 68 78 79 84 86 91 93
12 17 25 27 31 34 36 42 48 52 53 56 64 65 68 91 93
34 36 42 48 52 56 64 68 84 86[size]
15 17 22 25 31 37 42 46 54 56 59 61 68 74 79 84 88 96 98
[size=7]
15 17 22 23 25 31 37 42 46 54 56 59 61 68 72 74 79 93 96 98
15 17 19 22 23 25 31 37 42 46 56 59 61 67 68 72 79 83 88 93 98
54 56 83 84 88 93 96 98
22 23 54 56 72 74 83 84 93 96
23 25 31 35 41 42 46 54 56 59 61 68 72 74 79 93 96 98
17 19 41 46 54 59 61 67 68 74 79 84 88 96 98
17 19 22 23 67 68 72 74 79 83 84 88
17 19 23 25 31 35 41 42 67 68 72 79 83 88 93 98
15 19 22 23 25 31 35 37 42 46 56 59 61 67 68 72 79 83 88 93 98
15 17 19 22 23 25 31 35 37 41 42 46 54 59 61 67 68 72 74 79 93 96 98
15 17 22 25 31 37 42 46 56 59 61 68 74 79 83 84 93 98
11 17 25 29 31 36 42 48 52 56 64 68 73 79 84 87 93 95
17 19 23 25 31 35 41 42 54 56 67 68 72 79 84 88 93 96
12 13 16 24 27 28 33 38 44 45 49 51 53 57 62 65 69 76 78 81 85 86 91 92 99
13 16 24 27 33 34 38 44 45 51 57 62 69 76 77 78 81 85 92 99
15 16 22 28 33 37 45 46 57 59 61 62 76 77 81 83 85 92 98 99
15 16 45 46 51 57 62 69 76 77 81 85 92 99
22 28 33 37 44 45 46 57 59 61 62 74 76 77 81 83 85 92 98 99
44 45 46 51 57 62 69 74 76 77 81 85 92 99
13 15 16 24 28 33 38 44 45 46
13 16 24 28 33 38 44 46 74 76
13 15 22 24 37 38 44 45 51 57 62 69 74 77 81 83 85 92 98 99
13 15 22 24 28 33 37 38 44 45 74 77 83 85 92 98
13 15 16 22 24 37 38 51 57 62 69 74 76 77 81 83 85 92 98 99
13 15 16 22 24 28 33 37 38 74 76 77 83 85 92 98
16 19 24 28 35 38 44 45 67 69 72 77 85 88 92 96
13 16 23 24 33 35 41 45 51 54 57 62 69 76 77 81 85 92 99
13 16 19 33 35 38 51 57 62 67 69 72 76 77 81 85 88 92 96 99
13 16 19 23 24 28 41 44 45 54 57 62 67 69 72 76 77 81 85 88 92 96 99
11 13 16 24 28 33 36 38 44 45 73 76 77 81 85 87
11 16 33 36 51 57 62 69 73 77 81 87 92 99
11 13 16 24 29 33 36 38 45 48 51 52 57 62 64 69 73 76 77 92 95 99
13 16 24 28 33 38 44 45 51 57 62 69 76 77 81 85 92 99
13 16 24 28 29 33 38 44 45 48 52 57 62 64 76 77 85 87 95 99
13 15 18 22 24 32 37 45 46 47 55 59 62 63 71 74 76 81 89 94 98 99
13 15 18 22 26 32 37 38 46 47 55 59 61 62 71 74 81 83 89 94 98
23 26 32 35 41 45 54 55 57 62 63 67 71 72 76 81 89 94 96 99
13 18 23 24 32 35 38 41 45 47 54 57 63 67 71 72
13 19 23 26 32 35 38 54 55 62 63 71 76 81 88 89 94 96 99
11 13 18 24 29 32 36 38 45 48 52 55 57 63 64 73 76 81 87 94 95 99
11 13 18 24 26 29 32 36 47 48 52 55 63 64 73 76 81 87 89 94 95 99
11 13 18 24 26 29 32 36 45 47 48 52 57 62 63 64 73 76 81 87 89 95 99
11 18 24 29 32 36 38 45 48 52 57 62 63 64 71 73 76 81 87 95 99
11 18 24 29 32 36 38 45 48 52 55 57 63 64 71 73 76 81 87 94 95 99
11 18 24 26 29 32 36 45 47 48 52 57 62 63 64 71 73 76 81 87 89 95 99
45 47 55 57
24 26 71 76 81 89 94 99
13 18 32 38 62 63
12 13 15 22 27 34 37 38 45 46 49 53 57 61 62 65 74 78 83 86 91 98 99
12 13 15 22 27 34 37 38 45 46 49 53 57 61 62 65 74 76 78 81 83 86 91 99
12 13 15 22 24 27 34 38 45 46 49 53 57 61 62 65 74 78 83 86 91 98 99
12 15 22 27 34 37 38 46 49 53 57 59 61 65 74 76 78 81 83 86 91 99
12 15 22 24 27 34 38 46 49 53 57 59 61 65 74 78 83 86 91 98 99
12 15 22 24 27 34 38 46 49 53 57 59 61 65 74 76 78 81 83 86 91 99
12 13 34 35 53 54 57 62 65 67
12 13 19 23 27 41 49 53 57 62 67 72 78 86 88 91 96
24 27 34 35 38 45 49 54 57 65 67 76 78 81 86 91 99
13 19 23 24 27 34 38 41 45 49 62 65 67 72 76 78 81 86 88 91 96 99
12 19 23 24 27 35 38 41 45 49 53 54 57 72 76 78 81 86 88 91 96 99
12 13 19 23 24 34 35 38 41 45 53 54 62 65 72 76 81 88 96 99
11 13 34 36 64 65 73 76 81 86 91 95
11 13 27 29 48 49 73 76 78 81 86 87
11 12 13 24 27 34 36 38 45 49 53 57 62 64 65 73 76 78 91 95 99
11 12 13 24 27 29 34 38 45 48 49 53 57 62 65 73 78 81 87 91 99
11 12 24 27 34 36 38 45 49 52 53 57 62 64 65 73 76 78 91 95 99
11 12 24 27 29 34 38 45 48 49 52 53 57 62 65 73 78 81 87 91 99
12 13 24 27 34 38 45 49 53 57 62 65 76 78 81 86 91 99
24 27 29 36 38 45 48 49 52 57 62 64 76 78 86 87 95 99
24 29 34 36 38 45 48 52 57 62 64 65 81 86 87 91 95 99
15 19 35 37 38 41 46 54 57 59 61 62 67 72 74 76 81 88 96 99
15 19 22 23 35 37 38 57 59 61 62 72 76 81 83 88 96 99
13 15 22 24 41 45 46 54 57 62 67 72 74 81 83 88 96 98
13 15 22 23 24 45 46 72 74 76 83 88 96 98
13 15 19 22 23 24 35 37 41 45 54 57 59 61 62 67 81 83
13 15 19 23 24 35 37 41 45 46 54 59 61 67 74 76
13 19 23 24 35 37 38 41 45 46 54 59 61 67 74 76 98 99
37 38 57 59 98 99
13 15 22 24 45 46 61 62 74 76 81 83
11 13 19 23 24 29 35 38 41 45 48 52 54 57 72 73 76 81 87 88 96 99
11 13 24 29 36 38 45 48 52 57 62 64 73 76 81 87 95 99
11 13 24 29 36 38 45 48 54 57 64 67 73 76 81 87 95 99
11 19 23 29 41 48 52 57 62 67 72 73 87 88
11 13 19 23 24 29 36 38 41 45 48 62 64 67 72 73 76 81 87 88 95 99
52 54 57 62 64 67
13 19 23 24 35 38 41 45 52 54 62 64 72 76 81 88 96 99
13 19 23 24 35 38 41 45 54 57 62 67 72 76 81 88 96 99
15 16 18 22 26 32 33 37 44 46 47 55 59 83 85 89 92 94 98
15 16 32 33 44 46 61 63 71 74 83 85 92 94
15 18 22 26 28 33 37 46 47 51 55 59 63 69 71 77 83 89 92 98
32 33 44 47 51 55 63 69 71 77 85 89 92 94
16 18 26 28
12 15 16 33 34 44 46 49 51 53 59 61 65 91 92
12 16 22 27 33 34 37 44 46 51 53 74 78 83 86 91 92 98
22 28 33 37 44 46 49 65 69 74 77 83 85 86 92 98
12 15 22 28 33 34 37 44 49 53 59 61 65 69 74 77 83 85 91 92 98
11 16 27 29 33 34 36 44 48 49 51 53 64 65 69 73 78 85 86 87 91 95
11 16 27 28 29 33 34 36 44 48 49 51 53 65 69 73 78 85 86 87 91 95
11 12 16 33 34 44 49 51 52 53 65 69 85 86
11 12 16 33 36 73 77 85 87 91 92 95
11 12 51 52 91 92
12 16 33 34 44 49 51 53 65 69 85 86 91 92
12 16 33 34 36 52 53 64 65 73 77 85 86 87 92 95
27 28 77 78
27 29 34 36 48 49 51 52 53 64 65 73 78 86 87 91 92 95
11 15 16 22 29 36 37 44 46 48 52 59 61 64 74 77 85 87 95 98
11 15 22 28 29 33 36 37 46 48 52 59 61 64 73 74 77 92 95 98
15 16 22 28 33 37 44 46 74 77 83 85 92 98
15 16 44 46 73 74 77 83 85 87
22 28 33 37 73 77 83 87 92 98
51 59 61 69
11 16 19 23 29 33 35 51 52 73 77 85 87 92 95 96
11 16 33 35 36 41 48 51 52 54 64 67 72 73 77 85 87 88 92 96
11 16 33 36 51 52 73 77 85 87 92 95
11 16 28 29 33 35 36 41 48 51 52 54 64 67 69 72 73 77 85 88 92 96
11 16 19 23 28 33 36 41 44 48 52 54 64 67 69 72 73 87 88
41 44 51 54
28 29 44 48 64 69
16 19 23 28 33 35 67 69 72 77 85 88 92 96
12 15 18 22 26 27 32 34 37 46 47 49 53 59 61 65 83 86 89 91 94 98
12 18 26 27 32 34 47 49 74 78 86 89 94 98
15 18 22 26 32 37 46 47 55 59 61 63 71 78 83 89 91 98
71 74 78 91 94 98
12 15 18 22 26 27 32 34 37 46 47 49 55 59 61 63 71 74 78 83 86 89
12 15 22 27 34 37 46 49 53 59 61 65 71 74 83 86 91 94
12 18 23 26 27 32 35 41 47 49 53 54 55 65 67 71 72 78 86 89 91 94 96
23 26 32 34 35 41 47 53 54 63 65 67 71 72 94 96
12 19 23 26 27 32 34 41 47 49 63 67 71 72 78 86 88 91 94 96
11 18 26 27 29 32 36 47 48 49 52 55 63 64 71 73 94 95
11 12 18 26 27 32 34 36 47 48 52 55 63 64 71 73 78 86 87 91 94 95
11 12 18 26 27 32 34 36 47 48 52 55 63 64 65 71 73 78 86 87 91 95
11 12 18 26 29 32 34 36 48 49 52 53 71 73 78 86 89 91 94
11 12 34 36 47 48 49 52 53 55 63 64 73 78 86 87 89 91 94 95
11 12 34 36 47 48 49 52 53 64 65 73 78 86 87 89 91 95
11 12 27 29 34 36 48 49 52 53 55 63 64 73 78 86 87 91 94 95
12 18 26 27 32 34 47 49 71 78 86 89 91 94
12 18 26 27 29 32 34 71 78 86 87 89 91 94
53 55 63 65
27 29 47 49 87 89
15 18 22 23 26 32 35 37 46 47 54 55 59 72 74 83 88 89 96 98
15 18 23 26 32 35 37 46 47 54 55 59 61 63 71 72 74 83 88 89 96 98
22 23 26 32 35 54 55 72 74 83 88 94 96 98
23 26 32 35 54 55 61 63 71 72 74 83 88 94 96 98
15 19 22 26 32 35 37 41 46 54 55 59 61 67 94 96
15 18 19 35 37 41 46 47 54 59 61 63 67 71 74 83 89 94 96 98
15 18 19 22 23 35 37 41 46 47 54 59 63 67 71 72 74 83 89 94 96 98
15 18 19 22 23 35 37 41 46 54 59 61 63 67 71 72 74 83 89 94 96 98
15 18 22 26 32 37 46 47 55 59 61 63 71 74 83 89 94 98
11 18 26 29 32 35 36 47 48 52 54 55 63 64 71 73 87 89 94 96
11 18 26 29 32 36 47 48 52 55 63 64 71 73 87 89 94 95
11 18 26 29 32 36 41 48 52 54 64 67 71 72 87 88 89
11 18 26 29 32 36 41 47 48 52 54 64 67 71 72 87 89
11 18 23 26 29 32 35 47 48 54 55 71 72 73 87 89 94 96
11 19 23 26 29 32 35 36 41 48 52 54 55 63 64 67 87 88 94 96
11 19 23 26 29 32 36 41 48 52 55 63 64 67 87 88 94 95
11 19 23 29 41 47 63 67 71 73
11 18 19 23 26 29 32 35 36 41 47 48 52 54 55 63 64 67 87 89 94 96
11 18 19 23 26 29 32 36 41 47 48 52 55 63 64 67 87 89 94 95
11 18 19 23 29 41 47 48 52 54 64 67 72 73 87 89
23 26 32 35 41 47 54 55 63 67 71 72 94 96
23 26 32 35 52 55 63 64 72 73 94 95 96
23 26 32 35 52 54 55 63 64 72 73 94 96
41 47 52 54 63 64 67 71 72 73
18 19 88 89
22 23 41 46 49 53 54 59 72 74 83 86 91 96
15 19 35 37 41 49 53 59 61 67 83 86 88 91 96 98
12 15 22 23 27 34 37 41 46 53 54 61 65 72 74 78 91 96 98
12 15 19 22 27 34 35 37 41 46 49 61 65 67 74 78 86 88 96 98
12 15 19 23 27 34 35 37 53 54 59 61 65 67 72 78 83 88 91 98
12 19 22 23 27 34 35 46 49 54 59 65 67 72 74 78 83 86 88
12 15 22 27 34 37 46 49 53 59 61 65 74 78 83 86 91 98
11 12 27 29 34 36 48 49 52 53 64 65 73 78 86 87 91 95
11 12 19 23 27 29 34 35 36 41 48 49 52 53 54 65 67 72 73 78 86 87 88 91 95
12 19 23 27 34 35 41 49 53 54 65 67 72 78 86 88 91 96
12 19 23 27 34 36 41 49 53 54 64 65 67 72 78 86 88 91 95 96
11 15 19 22 29 35 37 52 54 59 61 64 67
11 15 22 29 36 37 46 48 52 59 61 64 73 74 83 87 95 98
11 19 23 29 41 46 52 54 61 64 72 73 74 83 88 96 98
11 19 23 29 41 48 52 54 64 67 72 73 87 88
11 19 22 23 29 41 46 52 54 61 64 72 74 83 88 96 98
11 15 19 23 29 36 37 41 46 48 52 54 59 72 73 74 83 87 88 95 96 98
11 15 19 22 23 29 36 37 41 46 48 52 54 59 72 74 83 87 88 95 96 98
35 36 95 96
15 19 35 36 37 46 48 54 59 64 67 73 74 83 87 95 98
15 19 22 23 35 37 41 46 54 59 61 67 72 74 83 88 96 98

[tso]

Could you point to a post that introduces and/or explains "unavoidable sets"?

[coloin]

The unavoidable sets have been around in many posts, notably when Red Ed tried to count them up !
http://forum.enjoysudoku.com/viewtopic.php?t=2747&start=0
this post gets a bit heavy when you start to consider minimal and non minimal unavoidables !

I think the main point about unavoidable sets is that there are so many of them and they are fairly difficult to see.

Take the above grid and a few of the sets as an example - one clue from each set is needed for a unique solution ! Every set has to be "hit" by a given clue - otherwise you get more than one grid solution.

Code: Select all

+---+---+---+
|963|174|258|
|178|325|649|
|254|689|731|
+---+---+---+
|821|437|596|
|496|852|317|
|735|961|824|
+---+---+---+
|589|713|462|
|317|246|985|
|642|598|173|
+---+---+---+

Found 209 unavoidable sets (36 of size 4 or 6).

The maximum # of disjoint unavoidable sets (max clique number -- MCN) is 12.
One such maximal collection is:
{14,15,74,75}
{16,18,26,28}
{35,36,95,96}
{53,55,63,65}
{56,58,66,68}
{11,12,51,52,91,92}
{17,19,67,69,77,79}
{21,22,61,62,81,82}
{24,25,44,45,84,85}
{27,29,47,49,87,89}
{31,32,41,42,71,72}
{37,38,39,97,98,99}


The ones of size 4 are the easiest to spot - here is {14,15,74,75}

Code: Select all

+---+---+---+
|963|..4|258|
|178|325|649|
|254|689|731|
+---+---+---+
|821|437|596|
|496|852|317|
|735|961|824|
+---+---+---+
|589|..3|462|
|317|246|985|
|642|598|173|
+---+---+---+ this has two grid solutions

And one not easy to spot is {41 43 46 47 61 63 66 67 }

Code: Select all

+---+---+---+
|963|174|258|
|178|325|649|
|254|689|731|
+---+---+---+
|.2.|43.|.96|
|496|852|317|
|.3.|96.|.24|
+---+---+---+
|589|713|462|
|317|246|985|
|642|598|173|
+---+---+---+ this has two grid solutions

And one impossible to spot is {12 15 18 22 26 27 32 34 37 46 47 49 55 59 61 63 71 74 78 83 86 89 }

Code: Select all

+---+---+---+
|9.3|1.4|2.8|
|1.8|32.|.49|
|2.4|.89|.31|
+---+---+---+
|821|43.|.9.|
|496|8.2|31.|
|.3.|961|824|
+---+---+---+
|.89|.13|4.2|
|31.|24.|98.|
|642|598|173|
+---+---+---+ this has two grid solutions

Each one of those sets in that very long list needs at least one clue !

[Moschopulus]

Could you point to a post that introduces and/or explains "unavoidable sets"?

An unavoidable set in a completed Sudoku grid is a set of cells (and their digits) such that some permutation of the cells results in another valid completed grid. An equivalent definition is this: a subset of a Sudoku grid is unavoidable if and only if deleting all the clues in the subset from the Sudoku grid results in a Sudoku puzzle that is not uniquely solvable. An unavoidable set is said to be minimal if no proper subset is itself unavoidable.

You can find more explanation here and included in the download of checker is a program "unavoid" which outputs all unavoidable sets up to size 12, and some of size 13 and 14. coloin is using this above. Type "unavoid grid -p" to get all the unavoidable sets.

coloin is also using a program of dukuso's which finds some unavoidable sets (but not all) of larger sizes.

[Smythe Dakota]

.... IMO you described "hidden singles".

Please excuse my not being up on terminology, but if a hidden single is simply "there is only one place in this row where a 4 can appear", then of course completing the chutes is nothing more than a special case of a hidden single. But it is a hidden single that can be spotted more quickly than most, so it's a useful speed-up technique in solving a puzzle, especially at the outset of the solving process.

Bill Smythe

[Heuresement]

Bill, I think that you will find that most sudoku puzzlers apply such techniques continually throughout the solving process. After a while, it becomes easier to recognise such patterns as they reappear in many puzzles.

Please note that although these have been called hidden singles, they are not really hidden at this stage. They only become hidden if you start considering the individual cell candidates for a row, column or box.

Please also note that these techniques can be applied throughout the solving process, as the more cells that are filled in, the more such singles become apparent.

Myself, I generally exhaust the puzzle of all such singles before I apply other techniques, eg doubles etc. and will only use pencilmarks etc if I get completely stuck.

Michael

[Smythe Dakota]

Hidden singles aren't always that easy to spot. In the newspaper puzzle today, I stared it for a long time before I found a hidden single (or whatever you want to call it) in a column that had only 4 cells filled in. One doesn't usually find such things in columns (or rows or boxes) with fewer than 5 filled in, so one is often inclined not to look.

When I did finally find it, it was only because I was still trying to complete chutes. There was an obscure reason why a 2 couldn't appear in a certain mini-column of a certain box, therefore it had to appear in that same mini-column of another box in the same stack. This, in turn, forced the appearance of another 2 in a different mini-column of a different box (in the same stack), and there was only one eligible cell in that mini-column. Lo and behold, that turned out to be the only eligible cell in the entire column, so I was led to the simple hidden-single logic by trying to apply a (somewhat contorted) chute completion.

Yes, I also use chute completion at later stages of the solving process, perhaps later than I should, as there are diminishing returns. Oh, well.

Bill Smythe