## "Lucky-cell" sudokus

Everything about Sudoku that doesn't fit in one of the other sections

### "Lucky-cell" sudokus

Let a "lucky-cell sudoku" be, roughly, one such that a single lucky guess at the start leads directly to the solution using only the simplest possible moves. More precisely ....

Define a simple move to be the deduction of one correct candidate by a naked- or hidden-single.

Define a "lucky cell" to be a cell with two candidates, such that
(1) guessing the correct candidate leads to a solution by simple moves only, and
(2) guessing the incorrect candidate leads to a contradiction by simple moves only.

Define a "lucky-cell sudoku" to be a unique-solution sudoku which, at the start, has a lucky cell.

(The sudoku discussed at http://forum.enjoysudoku.com/viewtopic.php?t=1458 is a lucky-cell sudoku. It has 30 starting hints, and the given contradiction-path has 18 simple moves. Generally, the sum of these numbers, #hints + contradiction-path-length, might be an interesting total to try to minimize.)

Anyone know of other lucky-cell sudokus, especially with fewer hints and notably short contradiction-paths?
r.e.s.

Posts: 337
Joined: 31 August 2005

### Re: "Lucky-cell" sudokus

r.e.s. wrote:Let a "lucky-cell sudoku" be, roughly, one such that a single lucky guess at the start leads directly to the solution using only the simplest possible moves. More precisely ....

Define a simple move to be the deduction of one correct candidate by a naked- or hidden-single.

Define a "lucky cell" to be a cell with two candidates, such that
(1) guessing the correct candidate leads to a solution by simple moves only, and
(2) guessing the incorrect candidate leads to a contradiction by simple moves only.

Define a "lucky-cell sudoku" to be a unique-solution sudoku which, at the start, has a lucky cell.

(The sudoku discussed at http://forum.enjoysudoku.com/viewtopic.php?t=1458 is a lucky-cell sudoku. It has 30 starting hints, and the given contradiction-path has 18 simple moves. Generally, the sum of these numbers, #hints + contradiction-path-length, might be an interesting total to try to minimize.)

Anyone know of other lucky-cell sudokus, especially with fewer hints and notably short contradiction-paths?

are you sure, you don't want unlucky sudokus instead ?
By your definition I assume that any random sudoku
should have a >99% chance of being "lucky-cell"
dukuso

Posts: 479
Joined: 25 June 2005

### Re: "Lucky-cell" sudokus

dukuso wrote:By your definition I assume that any random sudoku
should have a >99% chance of being "lucky-cell"

Care to explain why you assume that?
r.e.s.

Posts: 337
Joined: 31 August 2005

### Re: "Lucky-cell" sudokus

r.e.s. wrote:
dukuso wrote:By your definition I assume that any random sudoku
should have a >99% chance of being "lucky-cell"

Care to explain why you assume that?

just from my experience with those little beasts.
Most have a "forced" placements by basic rules.
Most can even be solved by a chain of these.

Now, just find a placement with 2 candidates (much more likely
than a forced placement) in a forced puzzle and you're
probably done.
And many problems can be solved by just one bifurcation,
I had a program to count the bifurcations, might check this.

Ahh, I'm not happy I wrote 99%, though.
When you ask me right now, I'd say 90%
dukuso

Posts: 479
Joined: 25 June 2005

### Re: "Lucky-cell" sudokus

dukuso wrote:
r.e.s. wrote:Care to explain why you assume that?

just from my experience with those little beasts.
Most have a "forced" placements by basic rules.
Most can even be solved by a chain of these.

Something tells me you didn't read the part that requires all exclusions to be by naked- or hidden-singles only.

dukuso wrote:Now, just find a placement with 2 candidates (much more likely than a forced placement) in a forced puzzle and you're
probably done.

And many problems can be solved by just one bifurcation,
I had a program to count the bifurcations, might check this.

Again, using naked- or hidden-singles only?

dukuso wrote:Ahh, I'm not happy I wrote 99%, though.
When you ask me right now, I'd say 90%

By the time you read the italicised part of the definition, I think you'll want to revise your claims rather drastically.
r.e.s.

Posts: 337
Joined: 31 August 2005

no, I wasn't happy that I had changed my 99% to a smaller value,
but didn't want to change it again

here is a statistics about sudokus with 17 clues
posted by Nick70 some time ago:

9276 diff 00 [trivial]
5994 diff 01 [single unit candidate]
778 diff 02 [naked pair]
511 diff 03 [hidden pair]
4 diff 04 [x-wing]
1037 diff 05 [turbot fish]
17 diff 06 [naked triplet]
2 diff 07 [swordfish]
8 diff 08 [hidden triplet]
499 diff 09 [xy-wing]
61 diff 10 [skewed swordfish]
2 diff 14 [jellyfish]
931 diff 15 [forcing chain]
72 diff 16 [multi forcing chain]
5 diff 20 [guessing]

as you see, about 50% sudokus were solved by hidden single,
naked single.

And this is for locally minimal sudokus. But a random sudoku
has about 40 clues and is _very_ likely to be solvable with
trivial methods. A quick examination gives about 0.997
as probability.

Such a sudoku is also likely to have a cell with 2 candidates.
Then one must lead to the solution and the other to a dead end
with trivial methods.

-Guenter
dukuso

Posts: 479
Joined: 25 June 2005

dukuso wrote:no, I wasn't happy that I had changed my 99% to a smaller value, but didn't want to change it again

If you think there are so many lucky-cell sudokus, why not just post one so we can see where the misunderstanding lies?
r.e.s.

Posts: 337
Joined: 31 August 2005

just the first random sudoku I found:

Code: Select all
`8.7...594.4527.361.36.5.2.741269.8.35.9.4..72.7851.9..25..8..3....725..87.1934.25`

cell (1,4) should be lucky. There could be others, I didn't check.
dukuso

Posts: 479
Joined: 25 June 2005

dukuso wrote:just the first random sudoku I found:

Code: Select all
`8.7...594.4527.361.36.5.2.741269.8.35.9.4..72.7851.9..25..8..3....725..87.1934.25`

cell (1,4) should be lucky. There could be others, I didn't check.

All thirteen two-candidate cells appear to be "lucky"! I am now going for lunch, which is to consist of my hat. Thanks for your patience.
r.e.s.

Posts: 337
Joined: 31 August 2005