A UR Conjecture

Advanced methods and approaches for solving Sudoku puzzles

A UR Conjecture

How different are Type 3 and Type 4 UR reductions?

Keith's conjecture: For every Type 3 reduction there is a Type 4 reduction that achieves the same result.

Consider the following:
Code: Select all
`+--------------+ |  -   -    -  | | ab   -   ab  | |  -   -    -  | +--------------+ | abc  -   abd | |  -   -    -  | |  -   -    -  | +--------------+ `

Others have named <abc> and <abd> the "roof cells".
Code: Select all
`+--------------+ |  -   -    -  | | ab   -   ab  | |  -   -    -  | +--------------+ | ab   -   ab  | |  -   -    -  | |  -   -    -  | +--------------+ `

and <a> and <b> are the "deadly values".
One unique solution fragment would be:
Code: Select all
`+--------------+ |  -   -    -  | | ab   -   ab  | |  -   -    -  | +--------------+ | c    -   d   | |  -   -    -  | |  -   -    -  | +--------------+ `

which is fine. Except, if this is indeed the case, neither a Type 3 nor a Type 4 reduction would be possible.

Both the Type 3 and the Type 4 reductions are possible only if, in the solution, one of the roof cells is a deadly value.

The Type 3 reduction is this:
Code: Select all
`+--------------+ |  -   -    -  | | ab   -   ab  | |  -   -    -  | +--------------+ | abc  -   abd | |  -   -    -  | |  -   cd   -  | +--------------+ `

One of the roof cells must be <c> or <d>. Together with <cd> the roof cells form a pair, so <cd> can be eliminated from all other cells in the house. But then, one of the roof cells must be <a> or <b>.

The Type 4 elimination says that the roof cells are a strong link on one of the deadly values, so the other can be eliminated from the roof cells.

As a practical matter, it seems that if a Type 3 reduction is possible, an easier to recognize Type 4 is also there.

I am not proposing the reverse: That every Type 4 has a companion Type 3..

Keith
keith
2017 Supporter

Posts: 215
Joined: 03 April 2006

Re: A UR Conjecture

keith wrote:As a practical matter, it seems that if a Type 3 reduction is possible, an easier to recognize Type 4 is also there.

An example of a UR Type 3 without a concurrent Type 4:
Code: Select all
`4.8.....2.....3.....19..8......3.......1...4...5..2....3.546..91....9..67.....4..After SSTS: 4      679    8      | 67     1567   17     | 135679 135679 2 5     -2679   69     |-24678 -12678  3      |*1679  *1679   147 3      267    1      | 9      2567   47     | 8      567    457----------------------+----------------------+--------------------- 269    14     7      | 468    3      458    | 12569  125689 158 269    8      3      | 1      69     57     | 269    4      57 69     14     5      | 4678   6789   2      | 13679  136789 1378----------------------+----------------------+--------------------- 8      3      2      | 5      4      6      |*17    *17     9 1      5      4      | 2378   278    9      | 23     238    6 7      69     69     | 238    128    18     | 4      2358   358 UR(17) Type 3 in r27c78, quantum cell r2c78 with r2c3 forms a naked pair <69>, implies r2c245<>69`
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

Keith's folly!

ronk,

Thank you! I expected to be shot down in flames, but not so soon!

Keith
keith
2017 Supporter

Posts: 215
Joined: 03 April 2006