## Need help explaining a chain

Post the puzzle or solving technique that's causing you trouble and someone will help

### Need help explaining a chain

This solver program produces a chain I need help explaining:

The sudoku:

Code: Select all
` . 6 1 | . 8 . | . . 7  . 7 . | 4 . . | . 8 .  . . . | . . 7 | 9 1 . -------+-------+------  . 5 7 | 9 4 2 | 1 . .  1 9 . | 8 5 3 | . . .  . . 4 | 7 . . | 5 9 . -------+-------+------  . 3 9 | 2 . . | . . 1  . 1 . | . . 4 | . . .  7 . . | 1 3 . | 2 5 . `

The number state:

Code: Select all
`  2|3|4|5|9      6        1 |    3|5        8      5|9 |      3|4    2|3|4          7     2|3|5|9      7    2|3|5 |      4  1|2|6|9  1|5|6|9 |      3|6        8    2|3|5|6   2|3|4|5|8  2|4|8  2|3|5|8 |  3|5|6      2|6        7 |        9        1  2|3|4|5|6 ----------------------------+--------------------------+-----------------------------       3|6|8      5        7 |      9        4        2 |        1      3|6      3|6|8           1      9      2|6 |      8        5        3 |    4|6|7  2|4|6|7      2|4|6     2|3|6|8    2|8        4 |      7      1|6      1|6 |        5        9    2|3|6|8 ----------------------------+--------------------------+-----------------------------     4|5|6|8      3        9 |      2      6|7    5|6|8 |  4|6|7|8    4|6|7          1     2|5|6|8      1  2|5|6|8 |    5|6    6|7|9        4 |  3|6|7|8    3|6|7    3|6|8|9           7    4|8      6|8 |      1        3    6|8|9 |        2        5    4|6|8|9 `

The solver outputs:

Consider the chain (9,2)-4-(7,1)-5-(7,6)-8-(9,6).
When the cell (9,2) contains the value 8, so does the cell (9,6) - a contradiction.
Therefore, the cell (9,2) cannot contain the value 8.

Can someone explain this chain to me ? Have a very hard time finding the logic in this.
mpcsmits

Posts: 4
Joined: 16 March 2006

Hi Mpcsmits.

Welcome to this forum. Let me try to explain the chain. First let's note the following: in box 7, the only two possible places for "4" are the cells r7c1 and r9c2; in row 7, "5" can only be in r7c1 and r7c6; and in box 8, the only two possible places for "8" are the cells r7c6 and r9c6. Consider now the following logic: if r9c2 is "8" then in box 7 candidate "4" can only be in r7c1, which implies that in row 7 candidate "5" can only be in r7c6, and this implies that in box 8 candidate "8" can only be in r9c6: but we started with the assumption that r9c2=8, and we have ended up with two "8s" in row 9, which is a contradiction. So, r9c2 cannot be "8" and so r9c2 is "4".

Hope this help.

Regards, Carcul
Carcul

Posts: 724
Joined: 04 November 2005

If (9,2) is 8, then (7,1) is the only place left for a 4 in that box.
That means that (7,1) is the only place left for a 5 in that row.
Which means that (9,6) is the only place left for an 8 in that column.

The net result is two 8s in one row (9), which is a contradiction. Ergo, (9,2) can't be 8, and must therefore be 4.

(Edit: Oops, too late. Oh well. )
vidarino

Posts: 295
Joined: 02 January 2006

### Re: Need help explaining a chain

mpcsmits wrote:The solver outputs:

Consider the chain (9,2)-4-(7,1)-5-(7,6)-8-(9,6).
When the cell (9,2) contains the value 8, so does the cell (9,6) - a contradiction.
Therefore, the cell (9,2) cannot contain the value 8.

Can someone explain this chain to me ? Have a very hard time finding the logic in this.

Yet another point of view. The link "(9,2)-4-(7,1)" may be written any of several ways:
• "if row9 col 2 does not equal 4, then row 7 col 1 equals 4"
• "if r9c2<>4 then r7c1=4"
• "r9c2<>4 implies r7c1=4"
• "r9c2<>4 -> r7c1=4"
Using the latter, we note either r9c2=4 or r9c2<>4. Thus, there are two implication "chains":
• r9c2=4 -> r9c2<>8
• r9c2<>4 -> r7c1=4 -> r7c1<>5 -> r7c6=5 -> r7c6<>8 -> r9c6=8 -> r9c2<>8
Since the outcome for both implication chains is the same, r9c2<>8.

Ron
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

Never said thanx to thank you. So here you go Thanks !
mpcsmits

Posts: 4
Joined: 16 March 2006

perhaps it is easier to see the XY-Wing:
if r9c2 is 8 then r6c2 is 2 and r9c3 is 6.
And r4c3? there would be no digit for r4c3.
So r9c2 <>8.
But of course you wanted to understand the
nonrepetetive path in the bilocation graph.

Ok, but has anybody a suggestion for the next move?
the 4 in r9c2 does not solve the puzzle
absolute beginner

Posts: 22
Joined: 26 February 2006

absolute beginner wrote:Ok, but has anybody a suggestion for the next move?
the 4 in r9c2 does not solve the puzzle

This is the best i found:
Code: Select all
` *--------------------------------------------------------------------* | 2459   6      1      | 35     8      59     | 34     234    7      | | 259    7      235    | 4      1269   1569   | 36     8      2356   | | 2458   28     2358   | 356    26     7      | 9      1      23456  | |----------------------+----------------------+----------------------| | 368    5      7      | 9      4      2      | 1      36     368    | | 1      9      26     | 8      5      3      | 467    2467   246    | | 238    28     4      | 7      16     16     | 5      9      238    | |----------------------+----------------------+----------------------| | 568    3      9      | 2      67     568    | 4678   467    1      | | 2568   1      2568   | 56     679    4      | 3678   367    369    | | 7      4      68     | 1      3      689    | 2      5      69     | *--------------------------------------------------------------------*`

If r3c4=6:
1. r1c4=3,r1c7=4,r1c8=2
following r3c9<>4, r5c7<>4 and r5c8<>2
2. r3c5=2,r3c2=8,r6c2=2,r5c2=6,r5c7=7,r5c8=4
following r5c9<>4
Then there is no 4 left for column 9.
Therefore r3c5 must be 6, which solves the puzzle.
ravel

Posts: 998
Joined: 21 February 2006

Here is another possible solution:

Code: Select all
` *--------------------------------------------------------------------* | 2459   6      1      | 35     8      59     | 34     234    7      | | 259    7      235    | 4      1269   1569   | 36     8      2356   | | 2458   248    2358   | 356    26     7      | 9      1      23456  | |----------------------+----------------------+----------------------| | 368    5      7      | 9      4      2      | 1      36     368    | | 1      9      26     | 8      5      3      | 467    2467   246    | | 238    28     4      | 7      16     16     | 5      9      238    | |----------------------+----------------------+----------------------| | 4568   3      9      | 2      67     568    | 4678   467    1      | | 2568   1      2568   | 56     679    4      | 3678   367    369    | | 7      48     68     | 1      3      689    | 2      5      469    | *--------------------------------------------------------------------*`

[r3c5]{-2-[r3c2]=2=[r6c2](-2-[r6c1|r6c9])-2-[r5c3]-6-[r4c1]=(Unique Rectangle: r4c1/r4c9/r6c1/r6c9)=6=[r4c9](-6-[r8c9|r9c9])-6-[r2c9|r3c9]=6=[r2c7]}-2-[r2c5]-(Unique Rectangle: r6c5/r6c6/r2c5/r2c6/r1c6)(-5-[r1c4]-3-[r1c7|r1c8])-5-[r1c4|r3c4]=5=[r8c4]-5-[r7c6]=5=[r7c1]=4=[r9c2]-4-[r9c9]-9-[r8c9]-3-[r2c9|r3c9]

which means that, if r3c5=2 then would be no place for candidate "3" in box 3. So, r3c5<>2 which solve the puzzle. I have based this deduction in the observation of the two Almost Unique Rectangles present in cells {r6c5/r6c6/r2c5/r2c6/r1c6} and {r4c1/r4c9/r6c1/r6c9}, and in their relation with cell r3c5. For an explanation of the notation used, please refer here.

Carcul
Carcul

Posts: 724
Joined: 04 November 2005

Here is another Error Net for the same placement as ravel's, which solves the puzzle with cascading naked singles:

r3c5(-2-r3c2=2=r6c2-2-r5c2)=6=r3c4=3=r1c4(-3-r1c7-4-r3c9=4=r5c9)-3-(ALS:r1c7|r1c8)-2-r5c8

... which eliminates all the 2s in row 5, implying r3c5<>2 or r3c5=6
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA