Making a puzzle harder by adding a number

Everything about Sudoku that doesn't fit in one of the other sections

Postby ronk » Fri Sep 05, 2008 4:09 pm

coloin wrote:I think the question is whether a distant UR can be found after removing an [essential] clue.

If by "essential" you mean a clue required to yield a single solution, I'm confused.

I thought the (perhaps) unstated premise of this thread title is that the puzzle (with the removed clue) still has a unique solution.
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Postby Pat » Sun Sep 07, 2008 3:53 am

udosuk wrote:if you remove a clue, you lose the knowledge if your current puzzle state still has a unique solution.

depending on your luck, you might reach a solution in which the cell where you removed a clue attains a value identical to the clue you removed. In that case you can be sure the solution you got is correct (because the original clueset, before you removed a clue, should guarantee you a unique solution).



in the new puzzle (one less clue),
i may indeed reach an answer which matches the (unique) answer of the original puzzle
    this does not guarantee that the new puzzle has only one answer
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Postby eleven » Sun Sep 07, 2008 5:09 am

Pat wrote:
udosuk wrote:depending on your luck, you might reach a solution in which the cell where you removed a clue attains a value identical to the clue you removed. In that case you can be sure the solution you got is correct
in the new puzzle (one less clue),
i may indeed reach an answer which matches the (unique) answer of the original puzzle
    this does not guarantee that the new puzzle has only one answer
A problem is, that after applying a uniqueness technique, which is not allowed because of a given, you might get a puzzle, where not only this given is reinserted (guaranteeing at most one solution), but also other candidates are eliminated/inserted, so that it has no solution at all.
But if you arrive at a solution you can be sure. that all was correct (as udosuk claimed).
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Postby Pat » Sun Sep 07, 2008 5:41 am

that's the part i don't see

say i got one answer

and yes it does match the answer of the original puzzle

how does this prove that the new puzzle doesn't have more answers??
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Postby coloin » Sun Sep 07, 2008 5:46 am

Yes, if you get the removed clue back - the puzzle can have 0 or 1 valid solutions.

The removed clue - reinserted defines a valid solution.

But it might be 0 [less than 1][it cant be more than 1]

StrmCkrs question perhaps should be "Can we take a minimal puzzle, remove a clue and then find a UR. [A UR which wasnt there before and doesnt envolve the removed clue]"

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Postby Pat » Sun Sep 07, 2008 5:58 am

coloin wrote:Yes, if you get the removed clue back - the puzzle can have 0 or 1 valid solutions.

yes, that part i saw

my question is, how did you "get the removed clue back" -- did you assume a unique answer?
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Postby coloin » Sun Sep 07, 2008 6:12 am

Well you are assuming that the original puzzle has one solution only.

I dont know how you get the removed clue back - it has to be a guess because the puzzle with a clue removed has multiple solutions.

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Postby udosuk » Sun Sep 07, 2008 9:50 am

Pat wrote:in the new puzzle (one less clue),
i may indeed reach an answer which matches the (unique) answer of the original puzzle
    this does not guarantee that the new puzzle has only one answer

From what I see there is no new puzzle. It doesn't matter the puzzle state with one less clue has a unique solution or not. People only care about the old puzzle.

It is always just a mean of working out the solution of the old puzzle. Say you're stuck hopelessly in a diabolical puzzle. You see that removing one given clue will create a Unique Rectangle. Even though you know the puzzle state without that clue would probably have more than one solutions, you blindly remove that clue and apply the uniqueness techniques. There will be 3 outcomes:

1. You reach a contradiction. Tough luck, go back to the state before you removed that clue.

2. You reach a state in which the cell where you removed a clue attains a value different to the original given clue. You know even if you reach a solution from that state, it won't be the correct one for the original puzzle. Tough luck, go back to the state before you removed that clue.

3. You somehow reach a state in which the cell where you removed a clue attains a value identical to the original given clue. BINGO! You are on the right track to the correct solution (of the original puzzle). At the very least you're in a better position than before you apply this maneuver.

Like I said this approach is not logically better than any guessing/trial-and-error approach. But say in a time-based competition where finding solution is the goal, when things are getting desperate it's definitely a viable option to try, albeit the same can be said for any other guessing/trial-and-error moves.:idea:
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Postby Pat » Sun Sep 07, 2008 10:23 am

udosuk wrote:
Pat wrote:in the new puzzle (one less clue),
i may indeed reach an answer which matches the (unique) answer of the original puzzle
    this does not guarantee that the new puzzle has only one answer

From what I see there is no new puzzle. It doesn't matter the puzzle state with one less clue has a unique solution or not. People only care about the old puzzle.


thanks, udosuk, i now see what you mean, you're solving the original puzzle, no new puzzle, sorry i had missed the point earlier---
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Postby eleven » Mon Sep 08, 2008 3:35 am

udosuk wrote:Like I said this approach is not logically better than any guessing/trial-and-error approach.
Its worse in my eyes.
It should be said explicitely, that this approach is logically not correct, when the removed clue is not redundant.

If you arrive at the solution nevertheless, you made a lucky mistake or - iow - let a technique guess for you, which you must not apply in a puzzle, which at this time definitely has multiple solutions.
And - different to other possible guesses - this one is not educated at all.
If you just guess, that one of 2 possible numbers is right, your chances are about 50% and - if you are lucky - you at least have another solution number (and, if the number was selected properly, will bring much more progress).
The approach we discussed here is useless, when it does not lead to a solution. Also if you only try it, if it inserts both the removed and another number with an - very probably invalid - uniqueness technique, in the most cases it will turn out to lead nowhere.
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Postby udosuk » Mon Sep 08, 2008 9:15 am

I wrote:3. You somehow reach a state in which the cell where you removed a clue attains a value identical to the original given clue. BINGO! You are on the right track to the correct solution (of the original puzzle). At the very least you're in a better position than before you apply this maneuver.

I must say I've overlooked another possibility - that when you applied those uniqueness techniques (wrongfully) you could have done some irrepairable damages (i.e. incorrect eliminations) to other cells. So even in this case the removed clue is reinserted, you could still be leading to a contradiction on this track. So you're not necessary in a better position than before.

I agree to eleven that this is not a very recommendable maneuver. I don't think I'll ever suggest it to any player.

The only scenario I think this will work is, if you have a software in hand where you can always plug in a puzzle state and check for uniqueness. But then it wouldn't be what considered by most players as "sudoku solving". (Personally I don't have any moral issue about using help from machines provided most of the logical moves are figured out by the human brain.)
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Postby StrmCkr » Mon Sep 08, 2008 5:36 pm

i guess i should have been a little more consciece about what i was implying on the add a clue and remove.


if a postion of cells is alinged to form a pattern known as mutiple solutios is lacking a cells to complete the pattern can a cell be "temperorarly" removed.

the pattern utilized. to make an exclusion.

then return the displaced clue back to its orgininal location.

the second thing about the rating system is where it rates the "unique" moves lower then a harder move.

aslo a note/
removing the move with a lower rating by placing an extra candiate implys a technique is no longer applicable at a specific stage in solving. thus a clue must be removed by a technqiue with a second option.
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Postby Pat » Tue Sep 09, 2008 6:42 am

StrmCkr wrote:
if a postion of cells is alinged to form a pattern known as mutiple solutios is lacking a cells to complete the pattern can a cell be "temperorarly" removed.

the pattern utilized. to make an exclusion.

then return the displaced clue back to its orgininal location.


no,
the "temporary" removal of a clue might create a puzzle with more than one answer,
you cannot rely on uniqueness-of-answer in the new puzzle.
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Postby StrmCkr » Tue Sep 09, 2008 9:45 pm

its not a new puzzle it is the same puzzle.

im citing from the given puzzles i listed where they both are identical.

1 contains a clue that blocks the ur. the other doesnt.

regardless if it has mutiple solutions or not a
muti sub grid possition cannt exsit in any valid puzzle.

meaning that the propostion of clues blocking the completion can be validated in mutiple directions.

we just know 1 of them.

where are the others?
by opening up that position can show where they are.
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Postby eleven » Wed Sep 10, 2008 5:35 am

Maybe an example helps. This is a puzzle from the Help forum:
Code: Select all
 +-------+-------+-------+
 | . . 3 | . . . | 9 . 1 |
 | . 5 . | . . 9 | . 8 . |
 | . . 9 | . 1 . | 2 . 6 |
 +-------+-------+-------+
 | . . . | 5 . 8 | . . 2 |
 | . . 5 | . . . | 8 . . |
 | 9 . . | 7 . 4 | . . . |
 +-------+-------+-------+
 | 3 . 1 | . 7 . | 6 . . |
 | . 7 . | 9 . . | . 2 . |
 | 8 . 2 | . . . | 5 . . |
 +-------+-------+-------+
 *--------------------------------------------------*
 | 2    468  3    | 468  5    67   | 9    47    1   |
 | 1    5    467  | 46  @2   @9    | 347  8     34  |
 | 47   48   9    | 348  1    37   | 2   #5    #6   |
 |----------------+----------------+----------------|
 | 46   1    47   | 5    369  8    | 347  3469  2   |
 | 467  3    5    | 1   @69  @2    | 8    4679  79  |
 | 9    2    8    | 7    36   4    | 13  #136  #5   |
 |----------------+----------------+----------------|
 | 3    49   1    | 2    7    5    | 6    49    8   |
 | 5    7    46   | 9    8    136  | 134  2     34  |
 | 8    69   2    | 36   4    136  | 5    1379  79  |
 *--------------------------------------------------*

If 9 in r2c6 would not be a given, you could eliminate 9 from r5c5 (to avoid the deadly 29 pattern). In this case you would be lucky, if you do it. In fact the 9 is redundant, the puzzle is unique without it.

But same for the 6 in r3c9. If it would not be given, you could eliminate 6 from r6c8. But this is part of the solution.

That shows, that you cannot temporarily remove a given, make a uniqueness based elimination and reinsert the number then.
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