The New Sudoku Players' Forum

[Yogi]

....3..4...2....7..1.5.....7......8....6..4.....1......6...21..4...7.3...........

Code: Select all

+---+---+---+
|...|.3.|.4.|
|..2|...|.7.|
|.1.|5..|...|
+---+---+---+
|7..|...|.8.|
|...|6..|4..|
|...|1..|...|
+---+---+---+
|.6.|..2|1..|
|4..|.7.|3..|
|...|...|...|
+---+---+---+


[pjb]

Good one for replacement method:
Replace all 589, 58, 59, 89, 5, 8, or 9 with xyz
assign x, y, z to r1c1, r1c2, r1c6

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 x        y      6      | 7      3      z      | 2      4      1     
 3xyz     4      2      | xyz    1      6      | xyz    7      3xyz   
 3xyz     1      7      | xyz    2      4      | 6xyz   36xyz  36xyz   
------------------------+----------------------+---------------------
 7       3xyz    1      | 2      4      3xyz   | 6xyz   8      36xyz   
 2       3xyz    xyz    | 6      xyz    7      | 4      1      3xyz   
 6       3xyz    4      | 1      xyz    3xyz   | 7      3xyz   2     
------------------------+----------------------+---------------------
 xyz     6       3      | 4      xyz    2      | 1      xyz    7     
 4       2       xyz    | xyz    7      1      | 3      6xyz   6xyz   
 1       7       xyz    | 3      6      xyz    | xyz    2      4     

then apply basic methods to give

Code: Select all

 x      y       6      | 7      3      z      | 2      4      1     
 3z     4       2      | xy     1      6      | yz     7      6xz   
 3z     1       7      | xy     2      4      | 6yz    3x     36xz   
------------------------+----------------------+---------------------
 7      3x      1      | 2      4      3x     | 5z     y      36z   
 2      3x      y      | 6      z      7      | 4      1      3x   
 6      z       4      | 1      y      3x     | 7      3x     2     
------------------------+----------------------+---------------------
 y      6       3      | 4      x      2      | 1      z      7     
 4      2       x      | z      7      1      | 3      6      y   
 1      7       z      | 3      6      y      | x      2      4     


Since the y at r4c8 corresponds to the given 8, replacing the y's by 8 leads to x = 5 and z = 9, solving puzzle

Phil

[P.O.]

basics:

Hidden Text: Show

Code: Select all

( n1r5c8   n7r1c4   n1r4c3   n1r9c1   n2r4c4   n2r3c5   n7r9c2
  n7r3c3   n1r8c6   n1r2c5   n2r8c2   n4r6c3   n4r3c6   n4r2c2
  n6r9c5   n6r6c1   n6r1c3   n7r6c7   n7r7c9   n1r1c9   n2r5c1
  n2r1c7   n4r9c9   n4r4c5   n6r2c6   n7r5c6   n2r6c9   n2r9c8
  n4r7c4   n3r9c4 )

intersections:
((((5 0) (2 7 3) (5 8 9)) ((5 0) (2 9 3) (3 5 8 9)))
 (((3 0) (4 2 4) (3 5 9)) ((3 0) (5 2 4) (3 5 8 9)) ((3 0) (6 2 4) (3 5 8 9)))
 (((3 0) (2 1 1) (3 8 9)) ((3 0) (3 1 1) (3 8 9))) ( n3r7c3 ))


Code: Select all

589   589   6     7     3     89    2     4     1             
389   4     2     89    1     6     589   7     3589           
389   1     7     5     2     4     689   369   3689           
7     359   1     2     4     359   569   8     3569           
2     3589  589   6     589   7     4     1     359           
6     3589  4     1     589   3589  7     359   2             
589   6     3     4     589   2     1     59    7             
4     2     589   89    7     1     3     569   5689           
1     7     589   3     6     589   589   2     4             

8r7c5 => r6c8 <> 3,5,9
let A be r7c5=8 - r8c4{n8 n9} - b2n9{r2c4 r1c6} - b1n9{r1c12 r23c1} - r7c1{n89 n5}
let B be c3n5{r89 r5} - r5c5{n58 n9}
 A - B - r5c9{n59 n3}
 A - B - r6c5{n89 n5}
 A - r7c8{n5 n9}

=> r7c5 <> 8
ste.


[Yogi]

I have to say WOW to Phil for this replacement idea, which I’ve never heard of before. Lack of a program which will substitute letters for candidates is inhibiting, but the printer makes it workable. I initially wondered why the cells in row1 were given single letters while all others were given multiples, but eventually clicked that being the last three unsolved cells in the house, r1c1, r1c2 & r1c6 must be finally solved for different values which we are here assigning as the fixed unknowns x, y & z. We then set out to solve these using normal Basics. Was it a typo that you converted the 5 at r3c4 to xyz but not the 8 at r4c8, or am I missing something there?

[pjb]

Yes, apologies, the 8 at r4c8 should be xyz. A transcription error as my solver (philsfolly.net.au) can't export the grid in letter form and I had to painstakingly replace the numbers with the xyz in the grid.
Phil

[Yogi]

Brilliant! I got the same eliminations leading to r4c8 = y = 8. Struggled with the further steps until I realised that r3c4, now down to 8X, was a given as 5, so x = 5 and z = 9, solving the puzzle. Now I need to find another puzzle which I can practice this technique on.

[pjb]

Hi Yogi
Heaps of examples on my website
Phil

[Yogi]

That was great. Had a go at the first one there. I wound up creating a template to print off to help as a sort of worksheet. I succeeded in solving it with my worksheet on the second try. This led me to wonder if, although the method is quite simple and straightforward (if a little long-winded) it was possible that it would fail or not proceed to completion if you started in the wrong place. After all, all you are looking for is a house which has only 3 remaining unsolved cells in it, and quite possibly there will be more than one of these available in the puzzle you are working on. Anyway, thanx!

[eleven]

just for fun

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+--------------------+-------------------+-------------------+
|  589   589    6    | 7     3     89    | 2     4     1     |
|  389   4      2    | 89    1     6     | 589   7     3589  |
|  389   1      7    | 5     2     4     | 689   369   3689  |
+--------------------+-------------------+-------------------+
|  7     359    1    | 2     4     359   | 569   8     3569  |
|  2    @3589 #*59+8 | 6   #*59+8  7     | 4     1    a359   |
|  6     3589   4    | 1    *59+8  3589  | 7    *59+3  2     |
+--------------------+-------------------+-------------------+
|#*59+8  6      3    | 4    #59-8  2     | 1    *59    7     |
|  4     2     *589  | 89    7     1     | 3     569   5689  |
|  1     7     *589  | 3     6     589   | 589   2     4     |
+--------------------+-------------------+-------------------+

7-link oddagon 59 in *-ed cells, extra candidates 8r5c3,r56c5,r7c1, 3r6c8
skyscraper 8 r5c35,r7c15, external 8r5c2

8r56c5 = 8r7c5,r89c3 - r5c3,r56c5,r7c1 = oddagon 59 (*) = 3r6c8 - r5c9 = (3-8)r5c2 = skyscraper 8r57 (#) - 8r89c3 = 8r7c1 => -8r7c5, stte

[denis_berthier]

I have to say WOW to Phil for this replacement idea, which I’ve never heard of before. Lack of a program which will substitute letters for candidates is inhibiting, but the printer makes it workable.

The method is due to eleven and has been known for quite some time.
There're no real need to use letters. As I've shown, you can keep the original 3 digits instead; at the end you'll have to apply some permutation of these digits, based on the givens.
SudoRules has all the functions necessary to help make the changes.

[eleven]

Yes, i had the idea of replacement, but i never thought of replacing givens. marek stefanik was the first one, whom i saw applying that.