## Help with a proof?

Advanced methods and approaches for solving Sudoku puzzles

### Re: Help with a proof?

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

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`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:
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`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: 602
Joined: 15 April 2013

### Re: Help with a proof?

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: 719
Joined: 22 May 2010

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