## UR type 4 - Sorry to bother you guys again

Advanced methods and approaches for solving Sudoku puzzles

### UR type 4 - Sorry to bother you guys again

I have been trying to understand the differences between Unique Rectangles...

I have load 2 examples that came with SudokuSusser.

First one is a Type 4

Remove the 5 from the roof

The Second One is a Type 4b

Remove the 7 from the roof

I can't see the differences between type 4 & type 4b

Thanks
jeanboucher

Posts: 24
Joined: 04 August 2006

There is no real difference. In type 4, the cells with extra candidates shared a box and another group. In type 4b, the cells with extra candidates did not share a box, just a row or column.
Myth Jellies

Posts: 593
Joined: 19 September 2005

I think they're completely different!

In the first case, you're eliminating 5 from r56c5, which would have forced r6c4=4 and r6c6=8, which would have formed the UR {59} in r37c46.
Here you're eliminating candidates from cells NOT ON the UR!

In the 2nd case, you're eliminating 7 from r1c19, which would have forced r1c3=1 and r1c7=5, which would have formed the UR {27} in r13c19.
Here you're eliminating candidates from cells ON the UR!

Although the mechanism works the same, the effects of the 2 cases are totally different - in case 1 after the eliminations the UR could still be useful for other eliminations, while in case 2 the eliminations completely destroy the UR... If implemented in a solver, type 4 should always be performed ahead of type 4b, otherwise it could miss some useful moves...

BTW not sure what do you mean by roof...
udosuk

Posts: 2698
Joined: 17 July 2005

udosuk wrote:BTW not sure what do you mean by roof...

Floor (cells): The cells without the extra candidates.

Roof (cells): The cells with the extra candidates.
Ruud

Posts: 664
Joined: 28 October 2005

udosuk wrote:I think they're completely different!

In the first case, you're eliminating 5 from r56c5, which would have forced r6c4=4 and r6c6=8, which would have formed the UR {59} in r37c46.
Here you're eliminating candidates from cells NOT ON the UR!

The type 4 reduction merely removes the 5's from r7c46. You then get to remove the 5's from r56c5 due to locked candidates in box 8 col 5, but that bonus is beside the point--the type 4 reduction is on the UR.
Myth Jellies

Posts: 593
Joined: 19 September 2005

Sorry guys, I didn't know about the floor/roof terminology...

But still I think there's a subtle difference between the 2 cases...

In case 1, if represented in a "forcing chains" way, it would look like this:

Code: Select all
`r7c4=5 => r3c4=9 => r3c6=5 => r6c6=8 => r7c6=9 => r37c46 forms an UR of {59}r7c6=5 => r3c6=9 => r3c4=5 => r6c4=4 => r7c4=9 => r37c46 forms an UR of {59}Therefore r7c46<>5`

For case 2:

Code: Select all
`r1c19=7 => r1c3=1 & r1c7=5 => r1c91=2 => r13c19 forms an UR of {27}Therefore r1c19<>7`

In the 1st case, the "chains" go through the "floor" cells while in the 2nd case they don't!

Other than that, the effects are more or less the same...

Myth Jellies wrote:You then get to remove the 5's from r56c5 due to locked candidates in box 8 col 5, but that bonus is beside the point

Yes, but even without the bonus "locked candidates", we could obviously remove 5s from r56c5 using the UR... Is there a particular type of UR doing this reduction?
udosuk

Posts: 2698
Joined: 17 July 2005

Maybe, what is confusing me is the reductions that SudokuSusser's giving:

Type 4
R7C4 - can remove <5> from <459> leaving <49>.
R7C6 - can remove <5> from <589> leaving <89>.

Type 4b
R1C1 - can remove <7> from <127> leaving <12>.
R1C9 - can remove <7> from <257> leaving <25>.

Both reductions are identical!

Both of them are destructive!!!
jeanboucher

Posts: 24
Joined: 04 August 2006

I also never made a difference (and did not know, what 4b is): When i see a UR pattern with 2 connected (same row/column) bivalue cells, and one of the UR numbers has a strong link in the opposite cells (again in row/column), i can eliminate the other number there.
Sample:
Code: Select all
`12    1212X   12Y no other 1 in this row`

A 2 in 12X forces a 1 in 12Y => deadly pattern
Code: Select all
`1     22     1`
A 2 in 12Y forces a 1 in 12X => deadly pattern
Code: Select all
`2     11     2`
(note that those patterns are always deadly, i.e. dont allow a unique solution, when none of the 4 numbers are givens)
ravel

Posts: 998
Joined: 21 February 2006

Myth Jellies wrote:There is no real difference. In type 4, the cells with extra candidates shared a box and another group. In type 4b, the cells with extra candidates did not share a box, just a row or column.

jeanbouncer: MJ's statement says it all. There is no real difference. In both cases, the elimination logic is the same. If the floor cells are in the same box, then it's a UR Type 4. If the floor cells are not in the same box, then it's a UR Type 4b. Some people/solvers track this distinction, others don't.

udosuk: To my knowledge, a UR Type 4/4b is based on the contents of the four UR cells and the fact that one of the candidates in the floor cells must occur in one of the roof cells. This eliminates the possibility of the other candidate being present in the roof cells. I don't like your forcing chains examples because they rely on cells exterior to the UR. A forcing chains relationship may not exist in some other puzzle where the UR Type 4/4b would still be valid.
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

Thank you all, for your enlightening reflexions.
jeanboucher

Posts: 24
Joined: 04 August 2006

Yes, I fully understand how the eliminations were made from ravel's description. The forcing chains are indeed irrelevant.

Thanks for all the explanation guys!
udosuk

Posts: 2698
Joined: 17 July 2005

### The difference

The difference between the two is this:

In the "regular" UR, the cells with the extra values are in the same box. They have two sets of shared buddies: One is their line (row or column), and a second is their box.

In the "Type B" UR, the cells with the extra values are not in the same box. They have one set of shared buddies in their line.

If you look at the original thread, you will see that Robert Woodhead made this distinction because the programming logic for the two types is slightly different

By the way, I think the concept of "roof" and "floor" cells is a little dangerous. It is not necessary that a UR have no extra candidates on two adjacent vertices.

In a Type 5, the "floor" cells are on a diagonal. Other Types (6 plus) are constrained by strong links and can have extra candidates on three (or possibly all four?) vertices.

To learn more, take a look at the links in Mike Barker's post (the "sticky" at the top of this forum).

http://forum.enjoysudoku.com/viewtopic.php?p=21804#p21804

Keith
keith
2017 Supporter

Posts: 215
Joined: 03 April 2006