## Is there a puzzle with no (3D-)trivalue cell?

Everything about Sudoku that doesn't fit in one of the other sections

### Is there a puzzle with no (3D-)trivalue cell?

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Puzzles are known that have no (3D-)bivalue cell - [no bivalue/bilocal] - at the start and after basic moves. I don't have an example at hand but I clearly remember seeing several and I think I could find a few if I needed.

But my first question is about trivalue: does anyone know a puzzle that has no (3D-)bivalue cell AND no (3D-)trivalue cell - [no trivalue/trilocal] - also at the start and after basic moves?

In case an example exists, has anyone studied what's the simplest (say SER-wise) puzzle with this property?

Edit: corrected an error inside the first [], that made the question ambiguous.
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Last edited by denis_berthier on Mon Jan 11, 2021 8:34 am, edited 1 time in total.
denis_berthier
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Location: Paris

### Re: Is there a puzzle with no (3D-)trivalue cell?

I am not sure if there are puzzles without a bivalue/bilocal set of 2 similar pm values in a box,row or column ... Edit.. is this what you mean ?

Golden Nugget SE 11.9 has ? 6 [ 24/25, 41/63, 42/53, 51/62, 54/66, 81/92] !!! [wow]
Code: Select all
`+---+---+---+|...|...|.39||...|..1|..5||..3|.5.|8..|+---+---+---+|..8|.9.|..6||.7.|..2|...||1..|4..|...|+---+---+---+|..9|.8.|.5.||.2.|...|6..||4..|7..|...|+---+---+---+  Golden Nugget+----------------------+----------------------+----------------------+| 25678  14568  124567 | 268    2467   4678   | 1247   3      9      | | 26789  4689   2467   | 23689  23467  1      | 247    2467   5      | | 2679   1469   3      | 269    5      4679   | 8      12467  1247   | +----------------------+----------------------+----------------------+| 235    345    8      | 135    9      357    | 123457 1247   6      | | 3569   7      456    | 13568  136    2      | 13459  1489   1348   | | 1      3569   256    | 4      367    35678  | 23579  2789   2378   | +----------------------+----------------------+----------------------+| 367    136    9      | 1236   8      346    | 12347  5      12347  | | 3578   2      157    | 1359   134    3459   | 6      14789  13478  | | 4      13568  156    | 7      1236   3569   | 1239   1289   1238   | +----------------------+----------------------+----------------------+`

Here is a 20 clue puzzle I found with ED=10.4/10.4/9.7...
Code: Select all
`+---+---+---+|.1.|..2|..3||42.|...|...||..5|.6.|...|+---+---+---+|...|...|5..||.7.|..4|..2||..8|...|.9.|+---+---+---+|...|.8.|9..||...|.9.|.6.||3..|..1|..7|+---+---+---+  20C  +-------------------------+-------------------------+-------------------------+| 6789    1       679     | 45789   457     2       | 4678    4578    3       | | 4       2       3679    | 135789  1357    35789   | 1678    1578    15689   | | 789     389     5       | 134789  6       3789    | 12478   12478   1489    | +-------------------------+-------------------------+-------------------------+| 1269    3469    123469  | 1236789 1237    36789   | 5       13478   1468    | | 1569    7       1369    | 135689  135     4       | 1368    138     2       | | 1256    3456    8       | 123567  12357   3567    | 13467   9       146     | +-------------------------+-------------------------+-------------------------+| 12567   456     12467   | 234567  8       3567    | 9       12345   145     | | 12578   458     1247    | 23457   9       357     | 12348   6       1458    | | 3       45689   2469    | 2456    245     1       | 248     2458    7       | +-------------------------+-------------------------+-------------------------+`

there are ? 5 bilocals at 29/39, 23/32, 37/38, 78/87, 92/93,

Now the question I have .....

is it possible to do the "FORCING-T&E applied to bivalue candidates" technique here ?
coloin

Posts: 1946
Joined: 05 May 2005

### Re: Is there a puzzle with no (3D-)trivalue cell?

coloin wrote:I am not sure if there are puzzles without a bivalue/bilocal set of 2 similar pm values in a box,row or column ... Edit.. is this what you mean ?

Problem of language, here. I don't understand either what you mean: "two similar pm values.."
So let me say what I meant:
rc-bivalue: a (rc-)cell that contains only two candidates
rn-bivalue: a rn-cell that contains only two candidates; i.e. two candidates conjugated in a row
cn-bivalue: a cn-cell that contains only two candidates; i.e. two candidates conjugated in a column
bn-bivalue: a bn-cell that contains only two candidates; i.e. two candidates conjugated in a block
3D-bivalue = rc-bivalue or rn-bivalue or bn-bivalue or cn-bivalue
Replace bivalue everywhere by trivalue and two by three and you get the corresponding definitions.

coloin wrote:Golden Nugget SE 11.9 has ? 6 [ 24/25, 41/63, 42/53, 51/62, 54/66, 81/92] !!! [wow]
Code: Select all
`+---+---+---+|...|...|.39||...|..1|..5||..3|.5.|8..|+---+---+---+|..8|.9.|..6||.7.|..2|...||1..|4..|...|+---+---+---+|..9|.8.|.5.||.2.|...|6..||4..|7..|...|+---+---+---+  Golden Nugget+----------------------+----------------------+----------------------+| 25678  14568  124567 | 268    2467   4678   | 1247   3      9      | | 26789  4689   2467   | 23689  23467  1      | 247    2467   5      | | 2679   1469   3      | 269    5      4679   | 8      12467  1247   | +----------------------+----------------------+----------------------+| 235    345    8      | 135    9      357    | 123457 1247   6      | | 3569   7      456    | 13568  136    2      | 13459  1489   1348   | | 1      3569   256    | 4      367    35678  | 23579  2789   2378   | +----------------------+----------------------+----------------------+| 367    136    9      | 1236   8      346    | 12347  5      12347  | | 3578   2      157    | 1359   134    3459   | 6      14789  13478  | | 4      13568  156    | 7      1236   3569   | 1239   1289   1238   | +----------------------+----------------------+----------------------+`

So, Golden Nugget has
- no rc-bivalue cell
- rn bivalue cells: r2n3,

but a lot of:
rc-trivalue cells: r1c4, r2c7, r3c4...
rn-trivalue: r1n1, r1n5, r2n8,..
cn-trivalue: c1n8, c2n8, c2n9,...
bn-trivalue: b1n1, b1n5...

So, Golden Nugget has both 3D-bivalue and 3D-trivalue cells.
What I'm looking for is a puzzle that has none of them. There are puzzles that have no 3D-bivalue cell (I think I've seen some already) but my (weak) conjecture is: there is no puzzle that has no 3D-bivalue cell AND no 3D-trivalue cell.

coloin wrote:Here is a 20 clue puzzle I found with ED=10.4/10.4/9.7...
Code: Select all
`+---+---+---+|.1.|..2|..3||42.|...|...||..5|.6.|...|+---+---+---+|...|...|5..||.7.|..4|..2||..8|...|.9.|+---+---+---+|...|.8.|9..||...|.9.|.6.||3..|..1|..7|+---+---+---+  20C  +-------------------------+-------------------------+-------------------------+| 6789    1       679     | 45789   457     2       | 4678    4578    3       | | 4       2       3679    | 135789  1357    35789   | 1678    1578    15689   | | 789     389     5       | 134789  6       3789    | 12478   12478   1489    | +-------------------------+-------------------------+-------------------------+| 1269    3469    123469  | 1236789 1237    36789   | 5       13478   1468    | | 1569    7       1369    | 135689  135     4       | 1368    138     2       | | 1256    3456    8       | 123567  12357   3567    | 13467   9       146     | +-------------------------+-------------------------+-------------------------+| 12567   456     12467   | 234567  8       3567    | 9       12345   145     | | 12578   458     1247    | 23457   9       357     | 12348   6       1458    | | 3       45689   2469    | 2456    245     1       | 248     2458    7       | +-------------------------+-------------------------+-------------------------+`

there are ? 5 bilocals at 29/39, 23/32, 37/38, 78/87, 92/93,

ok, I see.
What you call bilocal at 29/39 is what I call a cn-bivalue cell: c9n9 (for values r2, r3)
What you call bilocal at 23/32 is what I call a bn-bivalue cell: b1n3 (for values r2c3, r3c2 - or s6, s8)
...
SudoRules finds more, because some pairs can be bivalue in two different ways:
Hidden Text: Show
f-10764 (bivalue 0 237 238 232)
f-10766 (bivalue 0 992 993 299)
f-10768 (bivalue 0 415 495 354)
f-10770 (bivalue 0 929 939 399)
f-10772 (bivalue 0 323 332 413)
f-10774 (bivalue 0 929 939 439)
f-10776 (bivalue 0 237 238 432)
f-10778 (bivalue 0 748 767 467)
f-10780 (bivalue 0 378 387 493)
f-10782 (bivalue 0 992 993 479)

Here, candidates n1r3c7 and n1r3c8 are both rn-bivalue in r3n1 and bn-bivalue in b3n1. Note that, which way they are bivalue is irrelevant to Forcing-T&E (but may be relevant in other patterns).

Finally, this puzzle has both 3D-bivalue and 3D-trivalue cells (r1c3, r1c5...). I'm looking for the opposite.

coloin wrote:Now the question I have .....
is it possible to do the "FORCING-T&E applied to bivalue candidates" technique here ?

Yes, it is possible (and SudoRules will try only 7 pairs, irrespective of the ways in which they are bivalue).
But FORCING-T&E doesn't give anything in the present case. This should not be a surprise: no technique is guaranteed to solve any puzzle. And as I wrote previously FORCING-T&E is equivalent (elimination-wise) to forcing-braids - which, in turn, don't seem to be more powerful than braids.

However, you are perfectly right to allude to FORCING-T&E in this thread, because it was the motivation for my original question.
It is easy for me to extend FORCING-T&E to both 3D-bivalue and 3D-trivalue cells. But I wondered if I would also need to extend it to 3D-quadrivalue cells.
denis_berthier
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Posts: 1983
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Location: Paris