## Hidden triples

Advanced methods and approaches for solving Sudoku puzzles

### Hidden triples

Does anyone have a technique for spotting hidden triplets? Any help will be greatly appreciated.Thanks, Diane
dkendres

Posts: 3
Joined: 15 May 2018

### Re: Hidden triples

dkendres wrote:Does anyone have a technique for spotting hidden triplets? Any help will be greatly appreciated.Thanks, Diane

Hi Diane. Hidden subsets are easiest to spot directly by using the clues and solved cells, especially if solving without pencil marks. In the specific case of hidden triples, look for a house (box, row, col) in which the surrounding clues/solved cells (and perhaps other eliminations) restrict candidates of three different digits into three same cells within that house. Those cells form a hidden triple and you can eliminate any other candidates from them. If you're using pencil marks, then it's usually easier to look for naked subsets and find the hidden ones as their complements (the complementary naked and hidden subsets have the same eliminations so it doesn't matter which one you use). A couple of examples:

Code: Select all
`.-----------.-------------.-------------.| 7   2  3  | 9    46  56 | 8   45  1   || 5   9  4  | 8    7   1  | 2   3   6   || 6   1  8  | 35   34  2  | 9   7   45  |:-----------+-------------+-------------:| 8   3  12 | 25   19  45 | 7   6   459 || 49  6  12 | 235  8   7  | 13  45  459 || 49  5  7  | 6    19  34 | 13  8   2   |:-----------+-------------+-------------:| 2   8  5  | 1    36  36 | 4   9   7   || 3   7  6  | 4    2   9  | 5   1   8   || 1   4  9  | 7    5   8  | 6   2   3   |'-----------'-------------'-------------'`

Row 5: Hidden Triple (123) / Naked Triple (459) => -5 r5c4

Code: Select all
`.------------------.-------------------.-----------------.| 1468  1468    9  | 5     278    278  | 3    467   47   || 5     7       3  | 6     4      1    | 8    9     2    || 2     468     48 | 9     78     3    | 5    467   1    |:------------------+-------------------+-----------------:| 1468  12468   7  | 3     12689  289  | 249  1458  459  || 9     12348   5  | 2478  1278   278  | 24   1348  6    || 1468  123468  48 | 248   12689  5    | 7    1348  49   |:------------------+-------------------+-----------------:| 3     45      2  | 1     79     6    | 49   457   8    || 78    9       6  | 78    5      4    | 1    2     3    || 478   458     1  | 278   3      2789 | 6    457   4579 |'------------------'-------------------'-----------------'`

Column 8: Hidden Triple (138) / Naked Quad (4567)
Box 6: Hidden Triple (138) / Naked Quad (2459)
=> -45 r4c8, -4 r56c8

Code: Select all
`.-----------------.------------------.-----------.| 5     26   3    | 46    7     246  | 8   9   1 || 12    4    7    | 19    8     129  | 5   3   6 || 168   9    18   | 3     156   156  | 4   2   7 |:-----------------+------------------+-----------:| 1468  7    1458 | 2     1356  1456 | 36  48  9 || 3     68   2    | 4679  69    4679 | 1   48  5 || 1469  16   1459 | 1456  1356  8    | 36  7   2 |:-----------------+------------------+-----------:| 1289  128  6    | 1789  19    3    | 79  5   4 || 7     3    19   | 159   4     159  | 2   6   8 || 489   5    489  | 6789  2     679  | 79  1   3 |'-----------------'------------------'-----------'`

Box 8: Hidden Triple (678) / Naked Triple (159) => -19 r7c4, -9 r9c46

Code: Select all
`.-----------------.-------------------.--------------.| 6    45    2    | 3      158   1458 | 9   148  7   || 1    8     7    | 26     9     246  | 3   24   5   || 359  345   59   | 128    158   7    | 12  6    148 |:-----------------+-------------------+--------------:| 2    15    158  | 4      7     58   | 6   9    3   || 58   9     3    | 168    1568  1568 | 4   7    2   || 7    6     4    | 9      2     3    | 15  158  18  |:-----------------+-------------------+--------------:| 89   2     689  | 5      3     68   | 7   14   14  || 4    157   1568 | 12678  168   1268 | 25  3    9   || 35   1357  15   | 127    4     9    | 8   25   6   |'-----------------'-------------------'--------------'`

Box 7: Hidden Triple (689) / Naked Quad (1357) => -15 r8c3

Code: Select all
`.------------------.-------------.------------------.| 56     35    1   | 2    8   9  | 4    356    7    || 24569  3459  359 | 7    1   35 | 356  2356   8    || 257    357   8   | 4    6   35 | 9    1235   1235 |:------------------+-------------+------------------:| 459    1     6   | 358  79  2  | 357  34578  345  || 8      2     7   | 6    35  4  | 1    35     9    || 459    3459  359 | 58   79  1  | 2    45678  456  |:------------------+-------------+------------------:| 579    579   4   | 1    35  6  | 8    239    23   || 1      6     59  | 35   2   8  | 37   3479   34   || 3      8     2   | 9    4   7  | 56   156    156  |'------------------'-------------'------------------'`

Column 8: Hidden Triple (478) / Naked Sextuplet (123569) => -35 r4c8, -56 r6c8, -39 r8c8

Btw, these examples were generated by Hodoku. If you're using that program, you can easily generate examples of various patterns in the Learning or Practicing modes. Highly recommended.
-SpAce-: Show
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`   *             |    |               |    |    *        *        |=()=|    /  _  \    |=()=|               *            *    |    |   |-=( )=-|   |    |      *     *                     \  ¯  /                   *    `

"If one is to understand the great mystery, one must study all its aspects, not just the dogmatic narrow view of the Jedi."

SpAce

Posts: 2016
Joined: 22 May 2017

### Re: Hidden triples

There is no simple answer to this question. Basically looking for hidden sets is done without pencilmarks, they are only confusing - but cells with known pairs and locked candidates should be marked or remembered.

Some hidden triples or quads are easily spot, some are well hidden.

A good starting point is an empty minirow, where from the rest of the line or box several candidates are blocked.

E.g. look at r456c2, which cannot be 1258 from the box. 2 is already in column 2, so look, if 158 is blocked too for one of the 4 free cells. That's the case in row 9, and only r237c2 are possible for 258, building a hidden triple in column 2.
Code: Select all
` +-------+-------+-------+    +-------+-------+-------+ | . 7 2 | 8 . 4 | 1 . 3 |    | . 7 2 | 8 . 4 | 1 . 3 | | . . 4 | 2 . . | 5 7 . |    | . . 4 | 2 . . | 5 7 . | | . . . | . . . | 4 . . |    | . . . | . . . | 4 . . | +-------+-------+-------+    +-------+-------+-------+ | 2 . . | . 6 . | 8 . . |    |*2 X . | . 6 . | 8 . . | | 1 . 5 | 9 8 2 | 6 . 7 |    | 1 X*5 | 9 8 2 | 6 . 7 | | . . 8 | . 1 . | 2 . . |    | . X*8 | . 1 . | 2 . . | +-------+-------+-------+    +-------+-------+-------+ | . . 3 | . . 8 | 7 . . |    | . . 3 | . . 8 | 7 . . | | . 2 . | . . 9 | 3 . . |    | .*2 . | . . 9 | 3 . . | | 8 . . | 1 . 3 | 9 5 . |    |*8 X . |*1 . 3 | 9*5 . | +-------+-------+-------+    +-------+-------+-------+`

Similar here: 169 blocked for (cannot be in) r4c789, and in r4c5 from column 5
Code: Select all
` +-------+-------+-------+    +-------+-------+-------+ | 7 . 3 | 8 6 5 | 4 1 . |    | 7 . 3 | 8*6 5 | 4 1 . | | . 1 . | 4 3 2 | . 7 . |    | . 1 . | 4 3 2 | . 7 . | | . . . | 9 1 7 | 8 . . |    | . . . | 9*1 7 | 8 . . | +-------+-------+-------+    +-------+-------+-------+ | . 7 2 | . . . | . . . |    | . 7 2 | . X . | X X X | | 5 8 . | . 2 . | . 9 1 |    | 5 8 . | . 2 . | .*9*1 | | . . . | . . . | 2 6 . |    | . . . | . . . | 2*6 . | +-------+-------+-------+    +-------+-------+-------+ | . . 9 | 2 5 8 | . . . |    | . . 9 | 2 5 8 | . . . | | . . . | . . . | . 5 . |    | . . . | . . . | . 5 . | | . 5 7 | 3 9 . | . . . |    | . 5 7 | 3*9 . | . . . | +-------+-------+-------+    +-------+-------+-------+`

Here 359 both are blocked in row 9 and column 5 for box 8, leaving only 3 cells for them in box 8
Code: Select all
` +-------+-------+-------+    +-------+-------+-------+ | 1 3 4 | . 9 . | 8 2 6 |    | 1 3 4 | .*9 . | 8 2 6 | | 6 7 . | 4 . . | . 9 1 |    | 6 7 . | 4 . . | . 9 1 | | 9 . . | . . . | . . . |    | 9 . . | . . . | . . . | +-------+-------+-------+    +-------+-------+-------+ | 3 . 6 | 2 7 4 | . 8 9 |    | 3 . 6 | 2 7 4 | . 8 9 | | . 4 9 | 8 3 1 | 7 6 . |    | . 4 9 | 8*3 1 | 7 6 . | | 7 8 . | 6 5 9 | . . . |    | 7 8 . | 6*5 9 | . . . | +-------+-------+-------+    +-------+-------+-------+ | . . . | . 4 . | . . 8 |    | . . . | . 4 . | . . 8 | | 4 6 . | . . 8 | . 3 . |    | 4 6 . | . X 8 | . 3 . | | 8 9 3 | . . . | . 5 4 |    | 8*9*3 | X X X | .*5 4 | +-------+-------+-------+    +-------+-------+-------+`

If you look at the last sample of SpAce, you easily find a quad this way:
r123c8 are blocked for 4789, and you can find the same numbers in row 5 and 9, leaving 4 cells for them in column 8.
If you note, that 7 is locked in r7c12 (can only be there), also 478 cannot be in r8c8, and you have a triple 478 in r468c8 (which does not help much here).
Code: Select all
` +-------+-------+-------+    +-------+-------+-------+ | . . 1 | 2 8 9 | 4 . 7 |    | . . 1 | 2 8 9 |*4 X*7 | | . . . | 7 1 . | . . 8 |    | . . . | 7 1 . | . X*8 | | . . 8 | 4 6 . | 9 . . |    | . . 8 | 4 6 . |*9 X . | +-------+-------+-------+    +-------+-------+-------+ | . 1 6 | . . 2 | . . . |    | . 1 6 | . . 2 | . . . | | 8 2 7 | 6 . 4 | 1 . 9 |    |*8 2*7 | 6 .*4 | 1 X*9 | | . . . | . . 1 | 2 . . |    | . . . | . . 1 | 2 . . | +-------+-------+-------+    +-------+-------+-------+ | . . 4 | 1 . 6 | 8 . . |    | ^ ^*4 | 1 . 6 |*8 x . | | 1 6 . | . 2 8 | . . . |    | 1 6 . | . 2 8 | . . . | | 3 8 2 | 9 4 7 | . . . |    | 3*8 2 |*9*4*7 | . X . | +-------+-------+-------+    +-------+-------+-------+`

Now 2 samples, which are hard to spot.

You have to note, that 9 is locked in r89c8, thus r89c7 is blocked for 1239.
Out of them 129 are blocked for r2c7.
And with the 2(7) in r6c46 also for r6c7.
Code: Select all
` +-------+-------+-------+    +-------+-------+-------+ | 3 6 7 | 4 9 . | . 8 . |    | 3 6 7 | 4 9 ⁵ | . 8 . | | 9 2 4 | 1 8 . | . . 6 |    |*9*2 4 |*1 8 ⁵ | X . 6 | | 1 8 5 | . . . | . 4 . |    | 1 8 5 | . . . | . 4 . | +-------+-------+-------+    +-------+-------+-------+ | 5 3 6 | 9 1 4 | 7 2 8 |    | 5 3 6 | 9 1 4 | 7 2 8 | | 4 7 2 | 3 6 8 | . 5 . |    | 4 7 2 | 3 6 8 | . 5 . | | 8 1 9 | . 5 . | . . 4 |    | 8*1*9 | ² 5 ² | X . 4 | +-------+-------+-------+    +-------+-------+-------+ | 2 . 8 | . . . | 4 1 . |    | 2 . 8 | . . . | 4 1 . | | 6 4 1 | . . . | . . 3 |    | 6 4 1 | . . . | X ⁹ 3 | | 7 . 3 | . 4 1 | . . 2 |    | 7 . 3 | . 4 1 | X ⁹ 2 | +-------+-------+-------+    +-------+-------+-------+`

Here we have 19 in r45c1 and a hidden pair 58 in r56c4.
Also 7 is locked in r13c8.
Dropping 19 and 58 there could be a possible hidden triple with digits 34567 in row 5.
And 347 can be found in columns 29 and with the locked 7 in c8 (can't be in r5c297).
Code: Select all
` +-------+-------+-------+    +-------+-------+-------+ | 4 . . | 1 . 2 | 3 . . |    | 4 . . | 1 . 2 | 3 ⁷ . | | 3 . . | 4 6 . | 8 1 . |    | 3 . . | 4 6 . | 8 1 . | | 6 1 . | . . 8 | 5 . 4 |    | 6 1 . | q . 8 | 5 ⁷*4 | +-------+-------+-------+    +-------+-------+-------+ | . 5 . | . . . | . . 8 |    | p 5 . | q . . | . . 8 | | . . . | . 2 . | . . . |    | p X . | r 2 . | . X X | | 7 . . | . . . | . 3 . |    | 7 . . | r . . | .*3 . | +-------+-------+-------+    +-------+-------+-------+ | 8 7 6 | 2 . . | 9 4 1 |    | 8*7 6 | 2 . . | 9*4 1 | | 2 4 1 | 7 8 9 | 6 5 3 |    | 2*4 1 | 7 8 9 | 6 5*3 | | 5 3 9 | 6 4 1 | 2 8 7 |    | 5*3 9 | 6 4 1 | 2 8*7 | +-------+-------+-------+    +-------+-------+-------+`
eleven

Posts: 2223
Joined: 10 February 2008

### Re: Hidden triples

Hope, that a possible "locked" (in, for, to) confusion is now clarified. (thanks for the pm feedback).

As SpAce noticed, you can practice finding them using hodoku (look for "sudoku hodoku download", e.g. at sourceforge.net, to get the free java program).
Go to Mode/Practising and check Subsets/Hidden Triple.
Then to Options/Difficulty and select Medium.
With these settings "File/New random puzzle" always will generate a puzzle, which is easy, when you have found the hidden triple.
You can jump forward to the grid containing the triple by pressing the "Solve up to" button.
If you can't find the hidden triple, press "Next Hint".

Ah, and don't forget to uncheck Options/Show all candidates,
eleven

Posts: 2223
Joined: 10 February 2008