Zendo, play by forum

[Proxima]

White:
EAGER
EE
EEL
EVE
EVEN (and all anagrams of EVEN)
NN
NEVER
PETER
SEE
VV

Black:
DDO
EEVEE
EVAN
EVIN
EVON
EVUN
GIVE
NEV
ODD
OOD
UP

[namida]

ART
ANIMA
ISIS
INDICATING
POLONIUM
HOMOPHONE
ENEMY

[Proxima]

White:
ART
EAGER
EE
EEL
EVE
EVEN (and all anagrams of EVEN)
INDICATING
NN
NEVER
PETER
SEE
VV

Black:
ANIMA
DDO
EEVEE
ENEMY
EVAN
EVIN
EVON
EVUN
GIVE
HOMOPHONE
ISIS
NEV
ODD
OOD
POLONIUM
UP

[namida]

Rule guess: Every vowel that is in the koan must occur an equal number of times to its ordinal value (ie. 1x A, 2x E, etc).

[Proxima]

Well, that was a quick round 

[namida]

Wait, ahouldn't that mean ENEMY is white? Unless you also count 6x Y.

[Proxima]

Yes, Y counts as a vowel in this rule. I assumed you'd included the ENEMY guess specifically to check whether this was the case or not.

(Bonus puzzle: what are the first three positive integers whose English names are white with this rule?)

[namida]

Three, seven, twelve.

I'll start a new round soon.

[Proxima]

Oops, I forgot twelve. Then what's the next?

[namida]

No idea. I'm sure I could get it quickly if I wrote a code snippet to work it out, but that defeats the purpose (plus then there's the question of how exactly one words them - "One Hundred And Twelve"? "A Hundred And Twelve"? "Hundred And Twelve"? etc).

Anyway, let's start a new round. And I promise the rule is much simpler this time.

White:
RATE

Black:
RAGE

[Proxima]

TEAR
RARE
ATE
AGE
GREAT

[namida]

White:
ATE
RARE
RATE

Black:
AGE
GREAT
RAGE
TEAR

[Proxima]

ATG
ATT
RAG
RASE
RAT

[namida]

White:
ATE
ATG
ATT
RARE
RASE
RAT
RATE

Black:
AGE
GREAT
RAG
RAGE
TEAR

EDIT: Made a mistake before. ATT is white, not black.

[Proxima]

EAT
RAGG
RAHE
SAGE
SATE