The New Sudoku Players' Forum

[ravivprasad]

Need help on two 23-clue puzzles

I would be grateful for help on the following two 23-clue puzzles.

The first one I am trying to solve is:


a b c d e f g h i
r 8 6 3, - - -, - - -
s - - -, - 4 2, - - 5
t - - -, - - -, 9 - -
u - 4 8, - - -, - - 7
v - - -, 5 1 8, - - -
w 5 - -, - - -, 3 2 -
x - - 7, - - -, - - -
y 1 - -, 9 8 -, - - -
z - - -, - - -, 1 6 4


I have reached:

a b c d e f g h I
r 8 6 3, [17] [59] [59], [27] 4 [12]
s 7 [19] [19], [38] 4 2, 6 [38] 5
t [24] [25] [45], [13678] [367] [1367], 9 [378] [13]
u [69] 4 8, [236] [2369] [369], 5 1 7
v 3 7 2, 5 1 8, 4 9 6
w 5 [19] [169], [467] [679] [4679], 3 2 8
x [246] [235] 7, [1246] [256] [1456], 8 [35] 9
y 1 [35] [456], 9 8 [456], [27] [357] [23]
z [29] 8 [59], [237] [2357] [357], 1 6 4



Is it possible to proceed without using Nishio, Trial-and-Error, etc.? I would be grateful for any hints as to how to proceed.

According to the puzzle setter, the solution, which can be obtained by pure logic alone and without using T&E, is


a b c d e f g h I
r 8 6 3, 1 5 9, 7 4 2
s 7 9 1, 8 4 2, 6 3 5
t 4 2 5, 6 3 7, 9 8 1
u 9 4 8, 2 6 3, 5 1 7
v 3 7 2, 5 1 8, 4 9 6
w 5 1 6, 7 9 4, 3 2 8
x 6 3 7, 4 2 1, 8 5 9
y 1 5 4, 9 8 6, 2 7 3
z 2 8 9, 3 7 5, 1 6 4



The second 23-clue puzzle I am trying to solve is:


a b c d e f g h I
r - - -, - - 2, - - -
s - - -, - 6 -, 3 5 -
t 1 6 8, - - -, - - -
u 4 - 6, - 1 -, - - -
v 5 - -, - 7 -, - - 8
w - - -, - 3 -, 6 - 9
x - - -, - - -, 4 2 1
y - 7 9, - 2 -, - - -
z - - -, 6 - -, - - -


I have reached:


a b c d e f g h I
r [37] [345] [3457], [79] [49] 2, [1789] [16789] [467]
s 9 [24] [247], [18] 6 [18], 3 5 [47]
t 1 6 8, [3579] [459] [3579], [279] [79] [247]
u 4 [2389] 6, [289] 1 [89], [257] [37] [2357]
v 5 [1239] [123], [249] 7 6, [12] [134] 8
w [78] [128] [127], [2458] 3 [458], 6 [14] 9
x 6 [358] [35], [3579] [589] [3579], 4 2 1
y [38] 7 9, [1345] 2 [1345], [58] [368] [356]
z 2 [14] [14], 6 [58] [35], [5789] [3789] [357]


I would be grateful for any hints as to how to proceed. Do I have to resort to Trial-and-Error at this stage?

According to the puzzle setter, the solution, which can be obtained by pure logic alone and without using T&E, is

a b c d e f g h I
r 3 5 4, 7 9 2, 1 8 6
s 9 2 7, 8 6 1, 3 5 4
t 1 6 8, 3 4 5, 9 7 2
u 4 8 6, 2 1 9, 7 3 5
v 5 9 3, 4 7 6, 2 1 8
w 7 1 2, 5 3 8, 6 4 9
x 6 3 5, 9 8 7, 4 2 1
y 8 7 9, 1 2 4, 5 6 3
z 2 4 1, 6 5 3, 8 9 7


Ravi Visvesvaraya Prasad
New Delhi, India

r@50g.com p@r67.net

[emm]

the solution, which can be obtained by pure logic alone and without using T&E

Logic and T&E is a much debated matter and there's confusion in the meaning of the terms. T & E techniques eg forcing chains are actually logical although some people consider them to be guessing.

There are some simple moves you could do to get the puzzles a bit further ahead.

Puzzle 1 - there’s a hidden pair 4,6 in row 8 and then another in box 6.
Puzzle 2 – there’s a hidden pair 1,8 in row 1 and 1,4 in row 8. You can remove all other candidates from those cells. Then there are locked candidates - the 9s in box 2, the 3s & 5s in box 2, the 2s in box 3, and the 5s in box 7 ,8, 9.

I put the puzzles into Sudoku Susser which says you need forcing chains to solve Puzzle 1 eg
If r4c6 = 9 => r9c1 = 2 => r3c1 = 4 => r7c1 = 6 => r8c3 = 4 => r8c6 = 6 => r4c6 = 39
Since this is a contradiction r4c6 cannot = 9 so must be 6

and in the case of Puzzle 2 Trebors Tables with over 20,000 implications! This is obviously way beyond us but if you’re interested to see the moves you can download the solver from the Sudoku Susser site http://www.madoverlord.com/projects/sudoku.t

[Hanyou Hottie]

Hi there. I was playing around with the first puzzle you provided, and I noticed that you can solve it without needing to use forcing chains. I don't particularly like the idea of using trial and error, so I always try to find ways around using things like forcing chains.

The technique I used can be found inthis thread. I'm just learning about the uniqueness techniques and find them very interesting.

Here's what I found (cells with candidate 1s are in green, the blue cells are the uniqueness square I found):
img444.imageshack.us/img444/1289/uniqueness2rx.png { broken link }

Based on the rules described in the thread I linked to, you can remove candidate 9 from both of the lower highlighted cells. (The thread describes the process much better than I could, so I decided to link to it rather than try to describe it myself.) After that you get a few singles, then need to use locked candidates once, and then it unravels into singles.

I also have a question for you. How did you manage to figure out that cells r5c7, r5c8, and r5c9 were 4, 9, and 6, respectively? The farthest I could get on the puzzle (before using uniquness) is like this:

Code: Select all

 *--------------------------------------------------------------------*
 | 8      6      3      | 17     59     59     | 247    47     12     |
 | 7      19     19     | 368    4      2      | 68     38     5      |
 | 24     25     45     | 13678  367    1367   | 9      378    136    |
 |----------------------+----------------------+----------------------|
 | 69     4      8      | 236    2369   39     | 5      1      7      |
 | 3      7      2      | 5      1      8      | 46     49     69     |
 | 5      19     169    | 467    679    4679   | 3      2      8      |
 |----------------------+----------------------+----------------------|
 | 46     235    7      | 1246   256    1456   | 28     3589   239    |
 | 1      235    46     | 9      8      46     | 27     357    23     |
 | 29     8      59     | 237    2357   357    | 1      6      4      |
 *--------------------------------------------------------------------*

The 4, 6, and 9 are a naked triple, but I couldn't figure out which cell is which number.

[PaulIQ164]

I fact, once yo've used the uniqueness test on the 1/9s, I think you can make some simple eliminations in box 7, then use it again on the 3/5s in rows 7&8, columns 2&8.

[emm]

Thanks for this example HH and Paul – it seems like such a simple key to an otherwise impossible puzzle.

Is this the rule?

2 candidates a and b form a pair in box X.
They are also candidates in the 2 corresponding cells in box Y.
If a is also locked in box Y then you can eliminate b from the two cells in that box.

Does it crop up very often?

[PaulIQ164]

That's pretty much it. It's very hard to explain in full, but the essence is that you can never have a situation like this:

Code: Select all

.1.¦..2
...¦...
.2.¦..1


Because then you could interchange those 1s and 2s and have another valid puzzle. The two boxes in question don't have to be next to each other, but they have to be in the same 'band' of 3 boxes. There are a number of ways that you can deduce things from this rule, but they all have the same essence.

[Hanyou Hottie]

One of the other variations of the uniqueness test is as follows:

Code: Select all

+---------+---------+
|.  .  .  |.  .  .  |
|.  23 .  |.  23 .  |
|.  23 .  |.  236.  |
+---------+---------+

I think this is the original uniqueness variation actually. With this one you don't need to look at any of the other candidates in either box. It simply states that you can set the one cell with a 6 candidate to 6, because doing otherwise would create two different solutions to the puzzle. Like so:

Code: Select all

+---------+---------+
|.  .  .  |.  .  .  |
|.  2  .  |.  3  .  |
|.  3  .  |.  2  .  |
+---------+---------+
and
+---------+---------+
|.  .  .  |.  .  .  |
|.  3  .  |.  2  .  |
|.  2  .  |.  3  .  |
+---------+---------+

Back to the variant we used in this thread; you pretty much have the definition right, except "a" can either be locked in box "Y", or the row/column it's in. Either case satisfies the rule. I'm not really sure how often it's usefull though, as I've just discovered the technique myself.

[PaulIQ164]

Exactly. All the variations on this test stem from avoiding those situations shown above at any cost! Remember though, that if one of the numbers in the forbidden square (as I have decided to call it) is a clue number, then it's no longer forbidden, as the clue fixes you to one of the two possibilities.

[ravivprasad]

Many thanks to everyone for all their help on this.

Is there any way of solving these two puzzles without invoking uniqueness?

Ravi Visvesvaraya Prasad

[Sue De Coq]

Here are the simplest solutions I could find. I think the first puzzles highlights the potential virtues of the Uniqueness strategy (if you are able to ignore the ethical concerns).


Puzzle 1 (First 16 moves):

Code: Select all

1. The cell r4c8 is the only candidate for the value 1 in Row 4.
2. The cell r4c7 is the only candidate for the value 5 in Row 4.
3. The cell r6c9 is the only candidate for the value 8 in Row 6.
4. The cell r9c2 is the only candidate for the value 8 in Row 9.
5. The value 2 in Box 3 must lie in Row 1.
- The move r3c9:=2 has been eliminated.
The value 2 in Box 4 must lie in Row 5.
- The move r4c1:=2 has been eliminated.
The value 3 in Box 4 must lie in Row 5.
- The move r4c1:=3 has been eliminated.
The value 4 in Box 3 must lie in Row 1.
- The move r3c8:=4 has been eliminated.
The value 5 in Box 2 must lie in Row 1.
- The moves r3c5:=5 and r3c6:=5 have been eliminated.
The value 6 in Row 5 must lie in Box 6.
- The moves r5c1:=6 and r5c3:=6 have been eliminated.
The value 7 in Box 4 must lie in Row 5.
- The move r6c2:=7 has been eliminated.
The value 7 in Box 8 must lie in Row 9.
- The move r8c6:=7 has been eliminated.
The value 9 in Box 7 must lie in Row 9.
- The moves r7c1:=9 and r7c2:=9 have been eliminated.
The value 9 in Row 5 must lie in Box 6.
- The moves r5c1:=9, r5c2:=9 and r5c3:=9 have been eliminated.
The value 2 is the only candidate for the cell r5c3.
6. The values 5 and 9 occupy the cells r1c5 and r1c6 in some order.
- The moves r1c5:=7, r1c6:=1 and r1c6:=7 have been eliminated.
The values 3, 6 and 8 occupy the cells r2c4, r2c7 and r2c8 in some order.
- The moves r2c4:=1, r2c4:=7, r2c7:=7 and r2c8:=7 have been eliminated.
The value 1 in Box 1 must lie in Row 2.
- The moves r3c2:=1 and r3c3:=1 have been eliminated.
The value 7 in Box 1 must lie in Row 2.
- The moves r3c1:=7 and r3c2:=7 have been eliminated.
The values 4 and 6 occupy the cells r8c3 and r8c6 in some order.
- The moves r8c3:=5, r8c6:=3 and r8c6:=5 have been eliminated.
The values 1, 7 and 9 occupy the cells r2c2, r5c2 and r6c2 in some order.
- The move r5c2:=3 has been eliminated.
The value 7 is the only candidate for the cell r5c2.
7. The value 3 is the only candidate for the cell r5c1.
8. The cell r2c1 is the only candidate for the value 7 in Row 2.
9. The value 3 in Box 8 must lie in Row 9.
- The moves r7c4:=3, r7c5:=3 and r7c6:=3 have been eliminated.
The values 4 and 6 occupy the cells r7c1 and r8c3 in some order.
- The move r7c1:=2 has been eliminated.
Consider the chain r4c6~6~r4c1-6-r7c1-6-r8c3-6-r8c6.
When the cell r4c6 contains the value 6, so does the cell r8c6 - a contradiction.
Therefore, the cell r4c6 cannot contain the value 6.
- The move r4c6:=6 has been eliminated.
Consider the chain r3c9-1-r1c9-2-r1c7~2~r7c7-8-r2c7-6-r3c9.
When the cell r3c9 contains the value 6, it likewise contains the value 1 - a contradiction.
Therefore, the cell r3c9 cannot contain the value 6.
- The move r3c9:=6 has been eliminated.
The cell r2c7 is the only candidate for the value 6 in Box 3.
10. The value 4 is the only candidate for the cell r5c7.
11. The value 9 is the only candidate for the cell r5c8.
12. The value 6 is the only candidate for the cell r5c9.
13. The cell r7c7 is the only candidate for the value 8 in Column 7.
14. The cell r1c8 is the only candidate for the value 4 in Row 1.
15. The cell r7c9 is the only candidate for the value 9 in Row 7.
16. The value 2 in Row 8 must lie in Box 9.
- The move r8c2:=2 has been eliminated.
Consider the chain r3c4~1~r3c9~3~r2c8-3-r2c4-8-r3c4.
When the cell r3c4 contains the value 1, it likewise contains the value 8 - a contradiction.
Therefore, the cell r3c4 cannot contain the value 1.
- The move r3c4:=1 has been eliminated.
Consider the chain r6c4-4-r6c6~4~r8c6-4-r8c3-6-r6c3~6~r6c4.
When the cell r6c4 contains the value 6, it likewise contains the value 4 - a contradiction.
Therefore, the cell r6c4 cannot contain the value 6.
- The move r6c4:=6 has been eliminated.
Consider the chain r2c4-3-r2c8~3~r7c8-3-r7c2-2-r9c1-9-r4c1~9~r4c6~3~r4c4.
When the cell r4c4 contains the value 3, so does the cell r2c4 - a contradiction.
Therefore, the cell r4c4 cannot contain the value 3.
- The move r4c4:=3 has been eliminated.
Consider the chain r9c4~2~r9c1-9-r4c1~6~r4c4~2~r9c4.
When the cell r9c4 contains the value 2, the chain is self-contradicting.
 Therefore, the cell r9c4 cannot contain the value 2.
- The move r9c4:=2 has been eliminated.
Consider the chain r1c9-1-r1c4~7~r9c4~3~r2c4-3-r2c8~3~r3c9-1-r1c9.
When the cell r1c9 contains the value 1, the chain is self-contradicting.
 Therefore, the cell r1c9 cannot contain the value 1.
- The move r1c9:=1 has been eliminated.
The value 2 is the only candidate for the cell r1c9.


Puzzle 2 (First 7 moves):

Code: Select all

1. The cell r7c1 is the only candidate for the value 6 in Row 7.
2. The cell r5c6 is the only candidate for the value 6 in Row 5.
3. The value 1 in Box 2 must lie in Row 2.
- The move r1c4:=1 has been eliminated.
The value 1 in Box 8 must lie in Row 8.
- The move r9c6:=1 has been eliminated.
The value 2 in Box 3 must lie in Row 3.
- The move r2c9:=2 has been eliminated.
The value 3 in Box 2 must lie in Row 3.
- The move r1c4:=3 has been eliminated.
The value 4 in Box 8 must lie in Row 8.
- The moves r9c5:=4 and r9c6:=4 have been eliminated.
The value 4 in Box 2 must lie in Column 5.
- The moves r1c4:=4, r2c4:=4, r2c6:=4, r3c4:=4 and r3c6:=4 have been eliminated.
The value 4 in Box 3 must lie in Column 9.
- The moves r1c8:=4 and r3c8:=4 have been eliminated.
The value 5 in Box 2 must lie in Row 3.
- The moves r1c4:=5 and r1c5:=5 have been eliminated.
The value 5 in Box 5 must lie in Row 6.
- The moves r4c4:=5 and r4c6:=5 have been eliminated.
The value 7 in Box 6 must lie in Row 4.
- The move r6c8:=7 has been eliminated.
The value 7 in Box 8 must lie in Row 7.
- The move r9c6:=7 has been eliminated.
The value 8 in Box 2 must lie in Row 2.
- The moves r1c4:=8 and r1c5:=8 have been eliminated.
The value 8 in Box 8 must lie in Column 5.
- The moves r7c4:=8, r7c6:=8, r8c4:=8, r8c6:=8 and r9c6:=8 have been eliminated.
The value 9 in Box 8 must lie in Row 7.
- The moves r9c5:=9 and r9c6:=9 have been eliminated.
The value 9 in Box 1 must lie in Column 1.
- The moves r1c2:=9 and r2c2:=9 have been eliminated.
The values 1, 6 and 8 occupy the cells r1c7, r1c8 and r1c9 in some order.
- The moves r1c7:=7, r1c7:=9, r1c8:=7, r1c8:=9, r1c9:=4 and r1c9:=7 have been eliminated.
The value 6 is the only candidate for the cell r1c9.
4. The cell r8c8 is the only candidate for the value 6 in Row 8.
5. The value 9 in Row 3 must lie in Box 3.
- The moves r3c4:=9, r3c5:=9 and r3c6:=9 have been eliminated.
The values 1 and 8 occupy the cells r2c4 and r2c6 in some order.
- The moves r2c4:=7, r2c4:=9, r2c6:=7 and r2c6:=9 have been eliminated.
The cell r2c1 is the only candidate for the value 9 in Row 2.
6. The values 1 and 4 occupy the cells r8c4 and r8c6 in some order.
- The moves r8c4:=3, r8c4:=5, r8c6:=3 and r8c6:=5 have been eliminated.
The value 5 in Box 9 must lie in Row 8.
- The moves r9c7:=5 and r9c9:=5 have been eliminated.
The values 1, 2 and 4 occupy the cells r9c1, r9c2 and r9c3 in some order.
- The moves r9c1:=3, r9c1:=8, r9c2:=3, r9c2:=5, r9c2:=8, r9c3:=3 and r9c3:=5 have been eliminated.
The value 2 is the only candidate for the cell r9c1.
7. The value 5 in Box 8 must lie in Row 9.
- The moves r7c4:=5, r7c5:=5 and r7c6:=5 have been eliminated.
Consider the chain r5c4~2~r5c7~1~r6c8-4-r5c8-4-r5c4.
When the cell r5c4 contains the value 2, it likewise contains the value 4 - a contradiction.
Therefore, the cell r5c4 cannot contain the value 2.
- The move r5c4:=2 has been eliminated.
Consider the chain r4c2~2~r2c2~4~r2c9-4-r3c9-2-r4c9.
When the cell r4c2 contains the value 2, so does the cell r4c9 - a contradiction.
Therefore, the cell r4c2 cannot contain the value 2.
- The move r4c2:=2 has been eliminated.
Consider the chain r1c2~3~r1c1-3-r8c1-8-r7c2-5-r1c2.
When the cell r1c2 contains the value 3, it likewise contains the value 5 - a contradiction.
Therefore, the cell r1c2 cannot contain the value 3.
- The move r1c2:=3 has been eliminated.
Consider the chain r4c4~9~r1c4~7~r7c4-7-r7c6-9-r4c6.
When the cell r4c4 contains the value 9, so does the cell r4c6 - a contradiction.
Therefore, the cell r4c4 cannot contain the value 9.
- The move r4c4:=9 has been eliminated.
Consider the chain r2c4-1-r2c6~8~r4c6-9-r5c4~4~r8c4-1-r2c4.
When the cell r2c4 contains the value 1, the chain is self-contradicting.
 Therefore, the cell r2c4 cannot contain the value 1.
- The move r2c4:=1 has been eliminated.
The value 8 is the only candidate for the cell r2c4.