Type 3 Discontinuous Nice Loop

Advanced methods and approaches for solving Sudoku puzzles

re: terminology

Postby Pat » Wed Mar 12, 2008 12:22 pm

ronk wrote:We already have the fundamental terms forcing chains, nice loops and AICs. We certainly don't need another.



hey ronk,

as i mentioned above, i've just been looking at some definitions and i see that a "forcing chain" must have 2 (or more) implication-streams -- whereas my chain ( either one of them ) is a single implication-stream and thus not a "forcing chain" -- so, what would you call it ?

thanks in advance

Pat
User avatar
Pat
 
Posts: 3438
Joined: 18 July 2005

Re: re: terminology

Postby ronk » Wed Mar 12, 2008 1:32 pm

Pat wrote:my chain ( either one of them ) is a single implication-stream and thus not a "forcing chain" -- so, what would you call it ?

There are a minimum of two implication streams associated with every valid elimination. A chain, not to be confused with nets such as Sudoku Explainer's region forcing chains and cell forcing chains, has exactly two implication streams.

Consider this r1c3<>1 elimination example.
Code: Select all
 13   8    34-1 | 5    6    2    | 9    7    34
 2    359 *3459 | 37   79   1    |*348  6    348
 36   369  7    | 4    8    39   | 2    5    1
----------------+----------------+---------------
 4    36   368  | 2    1    5    | 7    9    68
 9    2   *18   | 6    3    7    |*148  48   5
 156  7    156  | 9    4    8    | 16   3    2
----------------+----------------+---------------
 8    1    39   | 37   279  6    | 5    24   349
 356  4    3569 | 1    259  39   | 368  28   7
 7    3569 2    | 8    59   4    | 36   1    369

The nice loop (NL) expression is:
Code: Select all
 r1c3 -1- r5c3 =1= r5c7 =4= r2c7 -4- r2c3 =4= r1c3, implies r1c3<>1

A nice loop is meant to be read both left-to-right (L->R) and right-to-left (R->L); read L->R, you get one implication stream; read R->L, you get the second.

One can start anywhere in the loop, so let's start with r2c7. Ultimately either r2c7=4 or r2c7<>4:
Code: Select all
L->R: r2c7=4 -> r2c3<>4 -> r1c3=4 -> r1c3<>1
R->L: r2c7<>4 -> r5c7=4 -> r5c7<>1 -> r5c3=1 -> r1c3<>1

In both cases, r1c3<>1. Therefore, r1c3<>1.

Now start with the left end of the NL instead. Ultimately either r1c3=1 or r1c3<>1.
Code: Select all
L->R: r1c3=1 -> r5c3<>1 -> r5c7=1 -> r5c7<>4 -> r2c7=4 -> r2c3<>4 -> r1c3=4 -> r1c3<>1
R->L: r1c3<>1

In both cases, as before, r1c3<>1. Therefore, r1c3<>1.

When the implication streams start at the elimination cell, i.e., the discontinuity of the nice loop, we simply don't bother to write the second implication stream ... but that doesn't mean the second one doesn't exist. From this POV, therefore, your "single implication chains" are forcing chains.
ronk
2012 Supporter
 
Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

re: terminology

Postby Pat » Thu Mar 13, 2008 1:37 pm

      hey ronk, thanks so much for your detailed response -- i've been solving SuDoku for 3 years now, and following the Forum almost as long, yet i've probably never read Jeff's definitions until now
    and gave up on the "nice loop" definition,
    thus i was unaware that the loop i described qualifies as a "nice loop"
i now see that my implication-stream
does also qualify as a "single implication network"---



ronk wrote:There are a minimum of two implication streams associated with every valid elimination


i may seem to be quibbling here,
but the fact is that i spelled out a single implication-stream
which loops back to bite its head with a contradiction,
thus proving the exclusion.
this single implication-stream is sufficient to prove the exclusion.
( assuming Proof-By-Contradiction is acceptable to you. )



ronk wrote:A forcing chain has exactly two implication-streams


at least 2 implication streams; sometimes 3, rarely more.
( Jeff's definitions from 2006 )



thanks again !

Pat
User avatar
Pat
 
Posts: 3438
Joined: 18 July 2005

Postby ronk » Thu Mar 13, 2008 2:28 pm

Pat, I guess I wasted my time, but I'm not really surprised. After all, you're the only person I know to use the terms duo and trio.:)
ronk
2012 Supporter
 
Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

Previous

Return to Advanced solving techniques