Odd Even puzzle

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Odd Even puzzle

Postby CathyW » Thu Mar 09, 2006 11:41 am

No takers so far in the Sudoku Variants section.

Help please! http://forum.enjoysudoku.com/viewtopic.php?t=3430
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Postby Carcul » Thu Mar 09, 2006 1:19 pm

Hi CathyW.

First, let's notice that we have a possible Almost Unique Rectangle in cells r4c6/r4c9/r6c6/r6c9, and this is the first solution that I have spoted:

[r5c7]-2-[r4c9|r6c9]-8-[r2c9]=8=[r2c8]-8-[r7c8]=8=[r7c7]=6=[r7c2](-6-[r5c2])-6-[r8c1|r9c1]-2-[r6c1]-{Nice Loop: [r5c4]-6-[r6c6]=(AUR:r4c6|r4c9|r6c6|r6c9)=6|2=[r4c6|r6c6]-2-[r4c5]-4-[r8c5]=4=[r8c4]-4-[r1c4|r3c4]-6-[r5c4]}-6-[r5c4]=6=[r6c6]-6-[r6c1]-4-[r5c2]-2-[r5c7],

which implies r5c7<>2 and that solve the puzzle. I have based this deduction in the recognition that a weak link with label "2" from r5c7 to r46c9 originates an AUR that allow us to write the nice loop that eliminates "6" from r5c4.

Here is an alternative solution:

1. [r1c2]-6-[r7c2]=6=[r7c7]=8=[r7c8]-8-[r2c8]-2-[r2c3]-6-[r1c2], => r1c2<>6.

2. [r6c1]-6-[r6c6]=6=[r5c4]-6-[r1c4]=6=[r1c1]-6-[r6c1], => r6c1<>6.

3. [r7c2]=6=[r7c7]=8=[r7c8]-8-[r2c8]-2-[r2c3]-6-[r5c3|r6c3]=6=[r5c2]-6-[r7c2], => r7c7<>2, r2c9<>2.

4. [r1c2]-2-[r7c2]=2=[r7c8]-2-[r2c8]=2=[r2c3]-2-[r1c2], => r1c2<>2.

5. [r5c2]-2-[r5c7]=2=[r8c7]-2-[r8c1]=2=[r7c2]-2-[r5c2], => r5c2<>2 which solve the puzzle.

Regards, Carcul
Last edited by Carcul on Thu Mar 09, 2006 9:49 am, edited 1 time in total.
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Postby ravel » Thu Mar 09, 2006 1:27 pm

Code: Select all
 *-----------------------------------------------------------*
 | 246   246   1     | 26    8     3     | 5     7     9     |
 | 3     5     26    | 1     7     9     | 4     28    268   |
 | 7     8     9     | 246   5     246   | 1     3     26    |
 |-------------------+-------------------+-------------------|
 | 1     9     3     | 5     24    248   | 7     6     248   |
 | 5     246   268   | 2468  9     7     | 28    1     3     |
 | 246   7     268   | 3     1     2468  | 9     5     248   |
 |-------------------+-------------------+-------------------|
 | 9     26    4     | 7     3     5     | 268   28    1     |
 | 268   3     5     | 248   24    1     | 26    9     7     |
 | 28    1     7     | 9     6     28    | 3     4     5     |
 *-----------------------------------------------------------*
A solution with coloring:
The 2 in r2c3 can be eliminated, because it would kill all 2's in box 4:
r2c3=2 => r56c3<>2
(r2c3=3) => r2c89<>2 => r3c9=2 => r46c9<>2 => r5c7=2 => r5c2<>2
(r5c7=2) => r78c7<>2 => r7c8=2 => r7c2<>2 => r89c1=2 => r6c1<>2

BTW: What is an Odd/Even puzzle?
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Postby CathyW » Thu Mar 09, 2006 5:55 pm

Thank you Carcul and Ravel - complicated solutions indeed, especially for a puzzle intended to be solved on paper. I still haven't got my head round "almost unique" (a tautology if ever there was one!). I managed to follow Ravel's reasoning though wasn't sure how colouring helped in this situation.

The puzzle was one of the variation Sudokus published for the World Championships that are taking place this weekend. Certain cells are shaded grey and must contain the even numbers, and the non-shaded cells must contain the odd numbers. The original puzzle can be viewed in pdf format here http://wpc.puzzles.com/sudoku/WSC_sample.pdf but, as I said in the Sudoku Variants section, don't look too closely at the first page!
Last edited by CathyW on Thu Mar 09, 2006 5:21 pm, edited 1 time in total.
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Postby ravel » Thu Mar 09, 2006 8:18 pm

Sorry for the word "colouring", it is just my internal name for all that can be done with chains in one digit. "Grouped colouring" would have been more precise.
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