denis_berthier wrote:What's the U4?

- Code: Select all
`. . . | .`

1 . . | 2

2 . . | 1

---------+---

. . . | .

denis_berthier wrote:What's the U4?

- Code: Select all
`. . . | .`

1 . . | 2

2 . . | 1

---------+---

. . . | .

- coloin
**Posts:**1702**Joined:**05 May 2005

denis_berthier wrote:It is now clear that UR1.1:

- depends on the assumption of uniqueness,

- depends on an assumption on clues (which amounts to fixing the names of some digits).

When you know that all UR cells are "unclued" and one or more UR digits are missing, you may "freely invent" the missing UR digits and apply uniqueness techniques to that UR ... without making an assumption as to the uniqueness of the solution grid.

- ronk
- 2012 Supporter
**Posts:**4764**Joined:**02 November 2005**Location:**Southeastern USA

Let's get this straight - if we can show there is a unique way to fill a qualifying rectangle with two digits how does this ASSUME uniqueness?

The subject of uniqueness is restricted to just the four cells in question, not the entire puzzle.

If it's possible to eliminate one of a pair of digits in the rectangle from deductions arising from external givens, AFAIAC we have demonstrated the composer has provided sufficient clues to distinguish between the two digits in that rectangle and that it's not isolated from the rest of the puzzle - no more.

Having established that fact, we can use the knowledge that this would be impossible if the rectangle was confined to a two-digit solution unless one of the cells was a given.

Whether this is a theorem or a hypothesis I leave to the mathematicians, but in the history of Sudoku, no-one has ever found a counter example.

The subject of uniqueness is restricted to just the four cells in question, not the entire puzzle.

If it's possible to eliminate one of a pair of digits in the rectangle from deductions arising from external givens, AFAIAC we have demonstrated the composer has provided sufficient clues to distinguish between the two digits in that rectangle and that it's not isolated from the rest of the puzzle - no more.

Having established that fact, we can use the knowledge that this would be impossible if the rectangle was confined to a two-digit solution unless one of the cells was a given.

Whether this is a theorem or a hypothesis I leave to the mathematicians, but in the history of Sudoku, no-one has ever found a counter example.

Last edited by David P Bird on Sat Nov 08, 2008 12:31 am, edited 1 time in total.

- David P Bird
- 2010 Supporter
**Posts:**1008**Joined:**16 September 2008**Location:**Middle England

ronk wrote:When you know that all UR cells are "unclued" and one or more UR digits are missing, you may "freely invent" the missing UR digits and apply uniqueness techniques to that UR ... without making an assumption as to the uniqueness of the solution grid.

You mean making an assumption of "local uniqueness"?

- denis_berthier
- 2010 Supporter
**Posts:**1258**Joined:**19 June 2007**Location:**Paris

denis_berthier wrote:You mean making an assumption of "local uniqueness"?

No, when a UR digit is missing, although "global uniqueness" may be true or false, "local uniqueness" is known to be true.

David P Bird wrote:Let's get this straight - if we can show there is a unique way to fill a qualifying rectangle with two digits how does this ASSUME uniqueness?

Are you responding to anyone or any statement in particular?

Last edited by ronk on Sat Nov 08, 2008 12:27 am, edited 1 time in total.

- ronk
- 2012 Supporter
**Posts:**4764**Joined:**02 November 2005**Location:**Southeastern USA

denis_berthier wrote:ronk wrote:When you know that all UR cells are "unclued" and one or more UR digits are missing, you may "freely invent" the missing UR digits and apply uniqueness techniques to that UR ... without making an assumption as to the uniqueness of the solution grid.

You mean making an assumption of "local uniqueness"?

... assumption to which, refering to my last post in the previous page, I could then give the more precise name of no-isomorphism-uniqueness.

- denis_berthier
- 2010 Supporter
**Posts:**1258**Joined:**19 June 2007**Location:**Paris

ronk wrote:denis_berthier wrote:You mean making an assumption of "local uniqueness"?

No, when a UR digit is missing, although "global uniqueness" may be true or false, "local uniqueness" is known to be true.

We've been cross-posting.

I don't know what you mean. Like global uniqueness, "local uniqueness" is known to be true only if you somehow make an assumption about it.

UR1 or UR1.1 (I don't know which you're referring to) has been proven only under some assumption of uniqueness.

- denis_berthier
- 2010 Supporter
**Posts:**1258**Joined:**19 June 2007**Location:**Paris

denis_berthier wrote:I don't know what you mean. Like global uniqueness, "local uniqueness" is known to be true only if you somehow make an assumption about it.ronk wrote:No, when a UR digit is missing, although "global uniqueness" may be true or false, "local uniqueness" is known to be true.

No again, local uniqueness is known because at least one of the UR digits is missing.

denis_berthier wrote:UR1 or UR1.1 (I don't know which you're referring to) has been proven only under some assumption of uniqueness.

Since I'm speaking of UR digits being missing, isn't it obvious I'm referring to UR1.1

- ronk
- 2012 Supporter
**Posts:**4764**Joined:**02 November 2005**Location:**Southeastern USA

I wasn't going to bother chipping in on this debate but here's my pennyworth.

In the following M and N are integers (including zero).

Unique Rectangle Case

In the case of the pattern

Eliminating 3 leads to 2M solutions through interchangeability (2M cannot equal 1 so no unique solution can be found here)

Placing 3 leads to N solutions ( A unique solution might be found here if N=1)

Deadly Pattern (no digits givens)

Eliminating 3 leads to a deadly pattern (No valid solutions may be found here)

Placing 3 leads to N solutions (A unique solution might be found here if N=1)

In the following M and N are integers (including zero).

Unique Rectangle Case

In the case of the pattern

- Code: Select all
`12 | 12`

12 | 123

Eliminating 3 leads to 2M solutions through interchangeability (2M cannot equal 1 so no unique solution can be found here)

Placing 3 leads to N solutions ( A unique solution might be found here if N=1)

Deadly Pattern (no digits givens)

- Code: Select all
`1 | 2`

2 | 13

Eliminating 3 leads to a deadly pattern (No valid solutions may be found here)

Placing 3 leads to N solutions (A unique solution might be found here if N=1)

- Glyn
**Posts:**357**Joined:**26 April 2007

What ronk is saying is, that e.g. in my sample above (after the elimination of 8 in r7c3) you can place 2 in r9c8 without assuming uniqueness.denis_berthier wrote:UR1 or UR1.1 (I don't know which you're referring to) has been proven only under some assumption of uniqueness.

- eleven
**Posts:**1783**Joined:**10 February 2008

In spite of continued claims to the contrary, all that has been given here as a valid proof is RedEd's. It assumes uniqueness.

For the "believers" (David's word) of UR1 or UR1.1 independent of some uniqueness assumption, as the arguments remain a matter of faith and not logic, I suggest you discuss this in another thread.

For the "believers" (David's word) of UR1 or UR1.1 independent of some uniqueness assumption, as the arguments remain a matter of faith and not logic, I suggest you discuss this in another thread.

- denis_berthier
- 2010 Supporter
**Posts:**1258**Joined:**19 June 2007**Location:**Paris

But this is just a simple corollary to RedEd's proof of a trivial fact (which btw also is needed for UR type 1, so i never will understand, how you could accept this and UR 1.1 not without it).denis_berthier wrote:In spite of continued claims to the contrary, all that has been given here as a valid proof is RedEd's. It assumes uniqueness.

RedEd, please can you formulate another trivial proof to make Denis a believer this time also ?

- eleven
**Posts:**1783**Joined:**10 February 2008

I thought it was clear: no proof has been provided, neither for UR1 nor for UR1.1, without some uniqueness assumption.

- denis_berthier
- 2010 Supporter
**Posts:**1258**Joined:**19 June 2007**Location:**Paris

denis_berthier wrote:I thought it was clear: no proof has been provided, neither for UR1 nor for UR1.1, without some uniqueness assumption.

Until you make a distinction between an observable property of uniqueness and the assumption of uniqueness, you're never going to get to the correct answer.

Allan Barker wrote:Maybe tomorrow I or someone could start a new thread, then I would move this there.denis berthier wrote:For the "believers" (David's word) of UR1 or UR1.1 independent of some uniqueness assumption, as the arguments remain a matter of faith and not logic, I suggest you discuss this in another thread.

Considering all the discussion here already, it seems much too late for that.

- ronk
- 2012 Supporter
**Posts:**4764**Joined:**02 November 2005**Location:**Southeastern USA

denis_berthier wrote:I thought it was clear: no proof has been provided, neither for UR1 nor for UR1.1, without some uniqueness assumption.

You just dont see it. This time i will not write it for you, because you dont accept my proofs without being able to say, what would be wrong with it. So think !

- eleven
**Posts:**1783**Joined:**10 February 2008