## Nice loop deductions

Post the puzzle or solving technique that's causing you trouble and someone will help

### Nice loop deductions

After reading some posts in this forum, I decided to try out Nice Loops for some of the more difficult puzzles. Given the following candidates grid:
Code: Select all
`+----------------------+----------------------+----------------------+| 2      568    19     | 7      568    68     | 13     4      39     | | 17     56     3      | 245    9      246    | 1267   8      67     | | 78     468    49     | 3      268    1      | 5      26     679    | +----------------------+----------------------+----------------------+| 6      38     157    | 1489   178    79     | 348    359    2      | | 18     9      125    | 1248   3      2468   | 468    7      456    | | 4      238    27     | 289    2678   5      | 368    369    1      | +----------------------+----------------------+----------------------+| 359    24     8      | 6      1257   79     | 2347   1235   3457   | | 35     7      26     | 1258   4      28     | 9      12356  356    | | 59     1      246    | 259    257    3      | 2467   256    8      | +----------------------+----------------------+----------------------+`

are these valid nice loop deductions?

[r2c7]=1=[r1c7]=3=[r1c9]=9=[r1c3](-9-[r3c1]=9=[r3c9])=1=[r2c1]=7=[r3c1]-7-[(r3c9)]-6-[r2c7] => r2c7<>6
[r2c7]=1=[r1c7]=3=[r1c9]=9=[r1c3](-9-[r3c1]=9=[r3c9])=1=[r2c1]=7=[r3c1]-7-[(r3c9)]-6-[r3c8]-2-[r2c7] => r2c7<>2
[r2c7]=1=[r1c7]=3=[r1c9]=9=[r1c3](-9-[r3c1]=9=[r3c9])=1=[r2c1]=7=[r3c1]-7-[(r3c9)]-6-[r2c9]-7-[r2c7] => r2c7<>7

GreenLantern

Posts: 26
Joined: 19 August 2005

### Re: Nice loop deductions

GreenLantern wrote:[r2c7]=1=[r1c7]=3=[r1c9]=9=[r1c3](-9-[r3c1]=9=[r3c9])=1=[r2c1]=7=[r3c1]-7-[(r3c9)]-6-[r2c7] => r2c7<>6
[r2c7]=1=[r1c7]=3=[r1c9]=9=[r1c3](-9-[r3c1]=9=[r3c9])=1=[r2c1]=7=[r3c1]-7-[(r3c9)]-6-[r3c8]-2-[r2c7] => r2c7<>2
[r2c7]=1=[r1c7]=3=[r1c9]=9=[r1c3](-9-[r3c1]=9=[r3c9])=1=[r2c1]=7=[r3c1]-7-[(r3c9)]-6-[r2c9]-7-[r2c7] => r2c7<>7

They all appear to follow Theorem 5 for discontinuous nice loops in a post I read by Jeff. I don't understand why u included the part (-9-[r3c1]=9=[r3c9]). For example in the first loop

[r2c7]=1=[r1c7]=3=[r1c9]=9=[r1c3]=1=[r2c1]=7=[r3c1]-7-[r3c9]-6-[r2c7] => r2c7<>6
seems just fine.
lb2064

Posts: 19
Joined: 29 March 2006

### Re: Nice loop deductions

The purpose of (-9-[r3c1]=9=[r3c9]) is to first "reduce" the trivalue cell at r3c9 to bivalue so that it can be used as a node between two weak links in accordance to nice loop propagation rules as I understand them. The part of my nice loop that I'm not that certain about is the strong link to reduce r3c9. In all the examples I've read, I've only seen weak links used to reduce polyvalue cells.

GreenLantern

Posts: 26
Joined: 19 August 2005

### Re: Nice loop deductions

GreenLantern wrote:.....The part of my nice loop that I'm not that certain about is the strong link to reduce r3c9. In all the examples I've read, I've only seen weak links used to reduce polyvalue cells.

You're right about r3c9. The node has to be bivalue for two weak links to pass thru it. I missed that . So the chains don't work then. Maybe one of the experts will weigh in soon.
lb2064

Posts: 19
Joined: 29 March 2006

I think this one works.

[r4c7]-3-[r1c7]-1-[r1c3]=1=[r2c1]-1-[r5c1]-8-[r4c2]-3-[r4c7] => [r4c7] <> 3
lb2064

Posts: 19
Joined: 29 March 2006

### Re: Nice loop deductions

I was actually thinking about using that type of loop except that I wasn't sure if it was allowed since the starting/ending node is a polyvalue cell between two weak links.

GreenLantern

Posts: 26
Joined: 19 August 2005

### Re: Nice loop deductions

GreenLantern wrote:I was actually thinking about using that type of loop except that I wasn't sure if it was allowed since the starting/ending node is a polyvalue cell between two weak links.

If u look at Jeff's document this seems to be fine since [r4c7] is the target and the weak links meet at this cell making it a discontinous nice loop. He classifies this under Theorem 4.

I've also seen people write this discontinous nice loop as:

3-[r1c7]-1-[r1c3]=1=[r2c1]-1-[r5c1]-8-[r4c2]-3

which then means any cell that the dangling 3's see cannot exist; which would be cell [r4c7] in this case.
lb2064

Posts: 19
Joined: 29 March 2006

### Re: Nice loop deductions

Looking back at Jeff's post, I see that you are correct. The example he gives shows a candidate being removed from a trivalue cell between two weak links with the same label.

GreenLantern

Posts: 26
Joined: 19 August 2005