## Type 3 Discontinuous Nice Loop

Advanced methods and approaches for solving Sudoku puzzles

### re: terminology

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

Pat

Posts: 3879
Joined: 18 July 2005

### Re: re: terminology

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:
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`L->R: r2c7=4 -> r2c3<>4 -> r1c3=4 -> r1c3<>1R->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<>1R->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
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Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

### re: terminology

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

Pat

Posts: 3879
Joined: 18 July 2005

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

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