I decided to compile some trivia from the Zendo, play by forum (https://www.lemmingsforums.net/index.php?topic=1849.0) game. Since the game moves quickly at times, I decided to do this in a separate topic so it's easy to find and update my posts as the game continues.
First, here is a list of all starter koans played so far. (Yes, round 18 had two of each colour.) Of course, there's nothing wrong with playing the same starter koans as a previous round if the rule is different! It's interesting that the only exact repeat pair we've had, WHITE/BLACK, was repeated by the same player (ccexplore).
White Black
1 LEMMINGS CHEAPO
2 INSANE STEVE
3 SIMON GEOO
4 WHITE BLACK
5 STOOGE GATES
6 NEWPORT CARDIFF
7 BROWN BLACK
8 BUDDHA NATURE
9 WHITE BLACK
10 CLAM SPAM
11 YELLOW BLUE
12 QUILLSHWAMMY LIGHTYAGAMI
13 LEMMING PROXIMA
14 ELEUSIS ZENDO
15 NO YES
16 EUCLID EULER
17 ALABAMA MISSISSIPPI
18 FIRE, MIGHTY FIRES, MIGHT
19 EVEN ODD
20 RAT RAGE
21 BUBBLE WRAP
22 ENOUGHTHINGS OTHERSTUFF
23 GABRIEL CHARLES
24 EDITOR DOSBOX
25 BANANA APPLE
26 PINEAPPLE LOOPING
27 THOUSAND MILLION
28 MARY POPPINS
29 BANANA APPLEPIE
30 LEMMING LIME
31 POTATO CARROT
32 JOY SWIM
33 END START
34 WRONG INCORRECT
35 NORTH SOUTH
36 BUCKLE UP
37 TAXING MAYHEM
38 NOT KNOT
39 CITADEL CASTLE
40 TELEPHONE CELLPHONE
41 CASHMERE SILK
42 TRIFURCATION ASYMMETRIC
43 LIGHT BRIGHT
44 FERRET WEASEL
And here's the list of all rules so far! All rules take the form "A koan has the Buddha-nature if and only if..."
1 (Simon): its letters are not all different
2 (Insane Steve): at least half its letters are vowels (Y is consonant)
3 (geoo): all letters have either mirror or rotational symmetry
4 (ccexplore): it contains an odd number of consonants (Y not tested)
5 (möbius): exactly half its letters are vowels (Y not tested)
6 (Minim): each consonant occurs at most once (Y not tested)
7 (Simon): its first letter is alphabetically earlier than all its other letters
8 (Proxima): its length is greater than or equal to alphabetic position of its first letter
9 (ccexplore): it contains at least one letter from "AKHTBN" and letters from that string occur only in that order
10 (Clam): it contains two letters that are consecutive in the alphabet
11 (Nepster): exactly half its letters are from the first half of the alphabet
12 (Proxima): its first and last letters are the same or their alphabetic positions differ by a multiple of 4
13 (Ramon): it contains the letter whose alphabetic position equals its length
14 (Nepster): when arranged alphabetically, its last letter is a vowel (Y is consonant)
15 (Minim): the sum of alphabetic positions of its letters is prime
16 (Proxima): its length is square or triangular, but not both
17 (Clam): its third letter is the same as its third letter when arranged alphabetically
18 (namida): it has at least three letters, and at least half its letters have (alphabetic position - 1) a multiple of the length, with the first letter counting twice if it is a multiple
19 (Proxima): for each n, the nth vowel either does not appear or appears n times (Y is vowel)
20 (namida): every pair of adjacent letters are either the same or from opposite halves of the alphabet
21 (Proxima): it contains an odd number of letters with exactly one vertical line segment
22 (Simon): its first letter is the same as or alphabetically earlier than all its other letters
23 (Proxima): the second and second-last letter are both consonants or both vowels (Y is consonant)
24 (Dullstar): it alternates between vowels and consonants (Y not tested)
25 (namida): its first two vowels are at positions n and 2n for some n (Y is consonant)
26 (Dullstar): it has odd length and its middle three letters are two vowels and a consonant (Y is consonant)
27 (Proxima): it contains a Roman numeral and doesn't contain any invalid Roman numerals
28 (WillLem): its letters are all different
29 (Simon): one letter appears exactly (n/2) times where n is the length
30 (Proxima): it would score at least 10 in English-language Scrabble (ignoring legality)
31 (namida): its second letter appears more often than its first letter does
32 (WillLem): it is an English noun
33 (Ramon): every pair of consecutive letters either both touches or neither touch the corners of their "bounding box" along the edge where they meet
34 (Simon): its letters are either in alphabetical order or reverse alphabetical order
35 (Proxima): it contains more \ diagonal line segments than / ones
36 (Ramon): it contains a pair of consecutive letters that border each other on a QWERTY keyboard
37 (Proxima): its length plus the alphabetic position of its first letter is even
38 (WillLem): it is a word contained in the title of a Lemmings level
39 (Proxima): it is the string CITADEL
40 (kaywhyn): its alphabetic positions sum to 100
41 (Ramon): it contains two of the same letter that are not adjacent
42 (Simon): instances of the letter I are symmetrically positioned
43 (kaywhyn): its length is a prime number
Here's a list with each round linked to.
All rules take the form "A koan has the Buddha-nature if and only if..."
Round 1 (https://www.lemmingsforums.net/index.php?topic=1849.msg46010#msg46010) (Simon): its letters are not all different
Round 2 (https://www.lemmingsforums.net/index.php?topic=1849.msg46013#msg46013) (Insane Steve): at least half its letters are vowels (Y is consonant)
Round 3 (https://www.lemmingsforums.net/index.php?topic=1849.msg46023#msg46023) (geoo): all letters have either mirror or rotational symmetry
Round 4 (https://www.lemmingsforums.net/index.php?topic=1849.msg46072#msg46072) (ccexplore): it contains an odd number of consonants (Y not tested)
Round 5 (https://www.lemmingsforums.net/index.php?topic=1849.msg46091#msg46091) (möbius): exactly half its letters are vowels (Y not tested)
Round 6 (https://www.lemmingsforums.net/index.php?topic=1849.msg46107#msg46107) (Minim): each consonant occurs at most once (Y not tested)
Round 7 (https://www.lemmingsforums.net/index.php?topic=1849.msg46128#msg46128) (Simon): its first letter is alphabetically earlier than all its other letters
Round 8 (https://www.lemmingsforums.net/index.php?topic=1849.msg46152#msg46152) (Proxima): its length is greater than or equal to alphabetic position of its first letter
Round 9 (https://www.lemmingsforums.net/index.php?topic=1849.msg46192#msg46192) (ccexplore): it contains at least one letter from "AKHTBN" and letters from that string occur only in that order
Round 10 (https://www.lemmingsforums.net/index.php?topic=1849.msg46360#msg46360) (Clam): it contains two letters that are consecutive in the alphabet
Round 11 (https://www.lemmingsforums.net/index.php?topic=1849.msg46382#msg46382) (Nepster): exactly half its letters are from the first half of the alphabet
Round 12 (https://www.lemmingsforums.net/index.php?topic=1849.msg46434#msg46434) (Proxima): its first and last letters are the same or their alphabetic positions differ by a multiple of 4
Round 13 (https://www.lemmingsforums.net/index.php?topic=1849.msg46494#msg46494) (Ramon): it contains the letter whose alphabetic position equals its length
Round 14 (https://www.lemmingsforums.net/index.php?topic=1849.msg46536#msg46536) (Nepster): when arranged alphabetically, its last letter is a vowel (Y is consonant)
Round 15 (https://www.lemmingsforums.net/index.php?topic=1849.msg46606#msg46606) (Minim): the sum of alphabetic positions of its letters is prime
Round 16 (https://www.lemmingsforums.net/index.php?topic=1849.msg46665#msg46665) (Proxima): its length is square or triangular, but not both
Round 17 (https://www.lemmingsforums.net/index.php?topic=1849.msg46755#msg46755) (Clam): its third letter is the same as its third letter when arranged alphabetically
Round 18 (https://www.lemmingsforums.net/index.php?topic=1849.msg51616#msg51616) (namida): it has at least three letters, and at least half its letters have (alphabetic position - 1) a multiple of the length, with the first letter counting twice if it is a multiple
Round 19 (https://www.lemmingsforums.net/index.php?topic=1849.msg52071#msg52071) (Proxima): for each n, the nth vowel either does not appear or appears n times (Y is vowel)
Round 20 (https://www.lemmingsforums.net/index.php?topic=1849.msg52087#msg52087) (namida): every pair of adjacent letters are either the same or from opposite halves of the alphabet
Round 21 (https://www.lemmingsforums.net/index.php?topic=1849.msg53954#msg53954) (Proxima): it contains an odd number of letters with exactly one vertical line segment
Round 22 (https://www.lemmingsforums.net/index.php?topic=1849.msg54321#msg54321) (Simon): its first letter is the same as or alphabetically earlier than all its other letters
Round 23 (https://www.lemmingsforums.net/index.php?topic=1849.msg54368#msg54368) (Proxima): the second and second-last letter are both consonants or both vowels (Y is consonant)
Round 24 (https://www.lemmingsforums.net/index.php?topic=1849.msg57927#msg57927) (Dullstar): it alternates between vowels and consonants (Y not tested)
Round 25 (https://www.lemmingsforums.net/index.php?topic=1849.msg57952#msg57952) (namida): its first two vowels are at positions n and 2n for some n (Y is consonant)
Round 26 (https://www.lemmingsforums.net/index.php?topic=1849.msg58150#msg58150) (Dullstar): it has odd length and its middle three letters are two vowels and a consonant (Y is consonant)
Round 27 (https://www.lemmingsforums.net/index.php?topic=1849.msg58638#msg58638) (Proxima): it contains a Roman numeral and doesn't contain any invalid Roman numerals
Round 28 (https://www.lemmingsforums.net/index.php?topic=1849.msg103945#msg103945) (WillLem): its letters are all different
Round 29 (https://www.lemmingsforums.net/index.php?topic=1849.msg103989#msg103989) (Simon): one letter appears exactly (n/2) times where n is the length
Round 30 (https://www.lemmingsforums.net/index.php?topic=1849.msg104019#msg104019) (Proxima): it would score at least 10 in English-language Scrabble (ignoring legality)
Round 31 (https://www.lemmingsforums.net/index.php?topic=1849.msg104185#msg104185) (namida): its second letter appears more often than its first letter does
I'll come back and edit this post periodically to add new rounds, and might also add who made the correct guess (in most cases, it's the person who starts the next round, but not always).