Help with a proof?

Advanced methods and approaches for solving Sudoku puzzles

Re: Help with a proof?

Postby Sudtyro2 » Sat Dec 07, 2013 12:10 pm

Hi blue,
A little different on this one...still learning as we go!

Your AIC first suggests using:
Code: Select all
r2c1-r2c46=(cp)r13c5
.
(cp)r13c5 still obeys the conjugate-pair hypothesis, although it has both candidates as peers to the EE cell, instead of just one, which the hypothesis (in saying “at least”) does not disallow. I seem to recall occasionally seeing this possibility in other grids, but could always find an alternative chain or network.

Similarly slippery for r8c1, one could maybe go for a [c5] SIS:
Code: Select all
r8c1-r8c789=r7c789-r7c5
                    ||
               (cp)r13c5
                    ||                   
               (cp)r49c5

Again, no conflict with the hypothesis, but it might be interesting if someone could find other network diagrams having only one peer candidate per conjugate pair. Edit to fix typo.

Code: Select all
| .  5  5 | .  5  . | 5  5  5 |
| 5  .  5 | 5  .  5 | 5  5  . |
| .  5  5 | .  5  . | 5  5  . |
+---------+---------+---------+
| .  5  . | 5  5  5 | 5  5  5 |
| .  5  . | 5  .  5 | .  .  5 |
| .  5  5 | 5  .  . | 5  5  . |
+---------+---------+---------+
| .  .  5 | 5 -5  5 | 5  5  5 |
| 5  .  5 | 5  .  . | 5  5  5 |
| .  5  5 | 5  5  5 | .  .  . |
Last edited by Sudtyro2 on Mon Dec 30, 2013 1:06 am, edited 1 time in total.
Sudtyro2
 
Posts: 446
Joined: 15 April 2013

Re: Help with a proof?

Postby JC Van Hay » Sat Dec 07, 2013 2:05 pm

Hi Sudtyro2,

While you were examining Blue's example, I reformulated to myself your conjecture. But in view of your last post, I added what is underligned.
This would limit the quest for a counter-example to a case where the "Fishes" B and C (see below) are at least 333 Swordfishes which is quite improbable.
But who knows!
  1. A digit z is excluded from the set D of candidates for a single digit d if it is not included in at least one solution of D.
  2. If the set D contains a conjugate pair {a,A}, then the exclusion of z can be justified by the following "network" :

      a-...-b=B-z
      ||
      A-z

      or

      a-...-b=B-z
      ||
      A-...-c=C-z

      where B and C are subsets of D.
  3. Sudtyro2's Conjecture : In the second case, it is possible to find a "network" such that at least one of the subsets B and C either belongs to or contains a (grouped) conjugate pair.
  4. More generally, the same kind of conjecture could be formulated if the exclusion is justified from a multilocal whether the set D contains or not a conjugate pair.
JC Van Hay
 
Posts: 692
Joined: 22 May 2010

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