## Logical puzzle "Catch unavoidable!"

Notes on possible new logic puzzles

### Logical puzzle "Catch unavoidable!"

Hi, people!
I've found that manual searching for another unavoidable (permutation) of the given (strongly minimal) UA set is rather amusing logical game. You need not to have exact understanding - what is UA set to solve this puzzle, but you should follow simple logical rules.

For example, let's consider strongly minimal UA set U14, having 2 permutations (2-valent strongly minimal UA set) having such first permutation:
Code: Select all
`4 5 7|. . 9. 1 9|. . 36 3 .|. . 4-----+-----5 6 .|. . .. . .|. . .. 7 1|. . .`

We must find the second permutation of this UA set, i.e. we must rearrange given digits in such way that following conditions are met:
1. New configuration must contain digits in the same cells comparing with given configuration.
2. Sets of digits used for each row/column/box must be the same.
3. New configuration must not contain the same digits in the same cells comparing with given configuration.

Certainly everyone can simply find another permutation by means of computer, but finding it manually, in "pencilmark style" is rather interesting.
Here is one possible solution way.

1. We can see, that bottom row (r6) contain 2 digits only - "7" and "1". New permutation must contain these digits in the same cells of the same row, but in reversing order. The same consideration can be used for digits "5" and "6" in the r4 row. So, we know 4 digits of the target permutation:
Code: Select all
`? ? ?|. . ?. ? ?|. . ?? ? .|. . ?-----+-----6 5 .|. . .. . .|. . .. 1 7|. . .`

2. Let's consider column c3. It must contain digits "1", "7", "9". We already have digit "7" in the r6c3 cell. So, digit "7" must go away from the cell r1c3 (from the column c3) to another cell of the row r1. Cell r1c2 of the row r1 is the only possible new place for the digit "7", because columns c1 and c6 don't contain this digit. So, we came to this configuration:
Code: Select all
`? 7 ?|. . ?. ? ?|. . ?? ? .|. . ?-----+-----6 5 .|. . .. . .|. . .. 1 7|. . .`

3. When we placed digit "7" in the cell r6c3, we pushed out "old" digit "1" from this cell. This digit must go in another cell of the column c3, because this column must contain digit "1" somewhere. Cell r2c3 of the column c3 is the only possible new place for the digit "1", because row r1 don't contain this digit. So, we came to this configuration:
Code: Select all
`? 7 ?|. . ?. ? 1|. . ?? ? .|. . ?-----+-----6 5 .|. . .. . .|. . .. 1 7|. . .`

4. Column c3 can contain digit "9" only in the cell r1c3, because positions of the rest digits ("1" and "7") are already known. We have
Code: Select all
`? 7 9|. . ?. ? 1|. . ?? ? .|. . ?-----+-----6 5 .|. . .. . .|. . .. 1 7|. . .`

5. When we placed digit "1" in the cell r2c3, we pushed out "old" digit "9" from this cell. This digit must go in another cell of the row r2, because this row must contain digit "9" somewhere. Cell r2c6 of the row r2 is the only possible new place for the digit "9", because column c2 cannot contain this digit. So, we came to this configuration:
Code: Select all
`? 7 9|. . ?. ? 1|. . 9? ? .|. . ?-----+-----6 5 .|. . .. . .|. . .. 1 7|. . .`

6. Row r2 can contain digit "3" only in the cell r2c2, because positions of the rest digits ("1" and "9") are already known. We have
Code: Select all
`? 7 9|. . ?. 3 1|. . 9? ? .|. . ?-----+-----6 5 .|. . .. . .|. . .. 1 7|. . .`

7. Column c2 can contain digit "6" in the cell r3c2 only, because positions of the rest digits ("1", "3", "5", "7") are already known. We have
Code: Select all
`? 7 9|. . ?. 3 1|. . 9? 6 .|. . ?-----+-----6 5 .|. . .. . .|. . .. 1 7|. . .`

7. If we check out row r3, we can see that cell r3c1 can contain digit "3" or digit "4", but digit "3" is absent in the column c1. So, cell r3c1 can contain digit "4" only. We have
Code: Select all
`? 7 9|. . ?. 3 1|. . 94 6 .|. . 3-----+-----6 5 .|. . .. . .|. . .. 1 7|. . .`

8. It is easy to find 2 missing digits in the cells r1c1 and r1c6 (we should check out columns c1 and c6). So, we have found puzzle solution:
Code: Select all
`5 7 9|. . 4. 3 1|. . 94 6 .|. . 3-----+-----6 5 .|. . .. . .|. . .. 1 7|. . .`

If you found this example being too simple, you can solve another puzzle (strongly minimal UA set U27):
Code: Select all
`. . .|6 . .|4 . .. . 6|. 2 1|. 3 97 8 9|3 4 .|6 1 2-----+-----+-----. 3 7|4 . 8|. . .. 6 1|. . 3|. . .8 . 4|1 6 7|. . .`

Serg
Serg
2018 Supporter

Posts: 727
Joined: 01 June 2010
Location: Russia

### Re: Logical puzzle "Catch unavoidable!"

I am too lazy to solve your second puzzle manually. Seems most of the people are lazy too. One weak point is that the verification of the solution is compatible in difficulty to the solution process.

Starting from a predefined "source" grid, reach a predefined "target" grid which is a simple morph of the source, say with bands 1 and 2 exchanged, moving trough unlimited number of valid intermediate grids, where each "move" is a permutation of values within a minimal unavoidable set.
Level2: Find the optimal algorithm(s) (sequence of minimal number of moves). It is "source" and "target" dependent.
Level3: Classify the grids by these optimal algorithms
dobrichev
2016 Supporter

Posts: 1794
Joined: 24 May 2010

### Re: Logical puzzle "Catch unavoidable!"

Hi, dobrichev!
dobrichev wrote:I am too lazy to solve your second puzzle manually. Seems most of the people are lazy too. One weak point is that the verification of the solution is compatible in difficulty to the solution process.

Are you too lazy to solve sudoku puzzles manually as well? This game ("Catch unavoidable") resemble me sudoku early days, when you can solve sudoku by simple logic rules only, without modern sophisticated (advanced) solving methods. Do you really treat verification as very hard job? Maybe you are really too lazy .
Starting from a predefined "source" grid, reach a predefined "target" grid which is a simple morph of the source, say with bands 1 and 2 exchanged, moving trough unlimited number of valid intermediate grids, where each "move" is a permutation of values within a minimal unavoidable set.
Level2: Find the optimal algorithm(s) (sequence of minimal number of moves). It is "source" and "target" dependent.
Level3: Classify the grids by these optimal algorithms

This theme was discussed a year ago at setbb.com/sudoku forum (thread "Solution grids and UA connections" initiated by blue - he found very interesting results).

Serg
Serg
2018 Supporter

Posts: 727
Joined: 01 June 2010
Location: Russia

### Re: Logical puzzle "Catch unavoidable!"

Hi, Serg.
dobrichev wrote:the verification of the solution is compatible in difficulty to the solution process.

Serg wrote:Do you really treat verification as very hard job?

No. There are simple UA which are easy to permute and easy to verify. There are more complex UA which are difficult to permute and difficult to verify.

The proposed theme actually isn't discussed. Blue did great job in coding the connections and in almost proving the connectivity of the entire space.
What is the minimal number of UA moves to exchange 2 bands? Which grids can be transformed so?
What is the minimal number, guarantying transformation of any grid (absolute max of grid-dependent mins)?
Which sequence better describes the grid - this with minimal number of moves, this with shortest UA used, or this with maximum number of unaffected cells/regions?
Is it possible to switch 2 bands w/o changing anything in the third band during the sequence? Or at least in one of its boxes?

No matter. I submitted this as an example of a complex problem with easily verifiable result.

Yes, I am lazy, maybe too lazy. I still have no code for path tracing and still talk "in principle".

I am poor solver. Solving moderate sudokus manually is problem to me. Recently I am enjoying solving the "hardest" from the papers (not from this forum) sudokus w/o pencilmarks.

I like your idea to play with something simple w/o computer assistance.

Cheers,
MD
dobrichev
2016 Supporter

Posts: 1794
Joined: 24 May 2010

### Re: Logical puzzle "Catch unavoidable!"

Hi, dobrichev!
The idea, that any solution grid can be transformed to arbitrary another grid by a sequence of UA sets permutation plus traditional VPTs, is very impressive. I think, blue could investigate this theme in more detail. I am too deeply involved into 16-clue puzzle search to study other serious themes.

Again unavoidables! Why strongly minimal UA set are bivalent? I feel it can be proved, but I don't know in what way yet.

BTW, weakly minimal UA sets can be used for this game too. One should find all possible permutations of UA set in this case. It resembles solving sudoku having multiple solutions. Such game variant is more complex. I had to use shameful T&E method to solve such multisolution puzzle.

I am poor sudoku solver too. The most complicated method I can use is search of hidden trios/quads. Some time ago I solved sudoku regularly (I prefer to solve my own sudoku puzzles because they have predictable for me complexity level), but during last months I have no time to do it. (Maybe sudoku properties investigation is more interesting for me.)