Monolithic Center Nakade Eyes of Size Greater than 7?

[Robert Jasiek]

It is generally assumed to be a truth that monolithic center nakade
eyes of size greater than 7 do not exist. Has this been proven?!

[Erik van der Werf]

On 09/23/2011 07:49 PM, Robert Jasiek wrote:
> It is generally assumed to be a truth that monolithic center nakade
> eyes of size greater than 7 do not exist. Has this been proven?!

I'm not sure what you mean by 'monolithic'. For the argument below I'll
just assume it to mean that the eye-surrounding group doesn't have any
defect or liberty shortage (which perhaps could be exploited to
consecutively play two or more attacker stones inside the eye or cause
some part of the eye space to become false). Also, all play should be
local alternating in the eye.

We can then define a nakade eye shape of size n (n>1) as a shape where
the attacker can force a capture that reduces to a nakade eye shape of
size n-1 and the defender has no play inside that would create a
non-nakade eye shape of size n-1.

Consequently, if no nakade eye shape exists of size n, then no nakade
eye shape can exist of size greater than n.

Directly after a nakade eye shape size is reduced by one, it contains no
stones (because the eye space is empty directly after capture). We
therefore just have to proof that, in the center, a completely empty
7-point eye shape cannot be nakade (because it needs more than one
attacker stone inside).

In the center, empty eye shapes with more than one split point give miai
for two eyes, so those can be discarded directly.

These are the 4 candidate shapes of size 7 with a single split point:

##
###
##

#
###
###

#
###
###

####
###

In the center only the top one can become seki, the rest trivially gets
2 eyes. So we found none at size=7 and therefore none will exist at size>7.


Erik

[Robert Jasiek]

Erik van der Werf wrote:
>I'm not sure what you mean by 'monolithic'.

Thomas Wolf has introduced this term for "adjacent to only one
[surrounding] string".

[...]

>Consequently, if no nakade eye shape exists of size n, then no nakade
>eye shape can exist of size greater than n.
>
>Directly after a nakade eye shape size is reduced by one, it contains no
>stones (because the eye space is empty directly after capture). We
>therefore just have to proof that, in the center, a completely empty
>7-point eye shape cannot be nakade (because it needs more than one
>attacker stone inside).

[...]

I think your sketch of a proof is correct, thank you!

> #
>###
>###

Except that this you might mean to be

##
###
##

[Erik van der Werf]

On 09/24/2011 07:36 AM, Robert Jasiek wrote:
> Erik van der Werf wrote:
>> I'm not sure what you mean by 'monolithic'.
>
> Thomas Wolf has introduced this term for "adjacent to only one
> [surrounding] string".

Ok, makes sense, requiring only one surrounding string is a good way to
rule out defects. For some purposes you could even make it a bit more
strict by using the definition of 'plain eye': one surrounding string
and all empty intersections must be adjacent to that string.

>
> [...]
>> Consequently, if no nakade eye shape exists of size n, then no nakade
>> eye shape can exist of size greater than n.
>>
>> Directly after a nakade eye shape size is reduced by one, it contains no
>> stones (because the eye space is empty directly after capture). We
>> therefore just have to proof that, in the center, a completely empty
>> 7-point eye shape cannot be nakade (because it needs more than one
>> attacker stone inside).
> [...]
>
> I think your sketch of a proof is correct, thank you!

:-)

>
>> #
>> ###
>> ###
>
> Except that this you might mean to be
>
> ##
> ###
> ##

Something seems to have gone wrong with the formatting. On my screen it
looked fine, but when I open the source of the message, or reply to
yours, then somehow extra spaces appear. Just tried groups.google.com
and there it looks fine again...

I'll try again, but now with underscores instead of spaces:

_##
###
##_ (seki)

_#_
###
### (2 eyes)

__#
###
### (2 eyes)

###_
#### (2 eyes)


Erik