CaleBob CSP parity detection

Here's my take on a recent idea from Cale for checking parity for CSP. Mostly taken from the description of the video.

YouTube version: https://youtu.be/Iw79RKQwMbY


It relies on being flexible about how you order pieces, but in a way that's pretty easy to use and remember. You divide each set of 8 pieces into sets of 4 (quads), and then splitting those into pairs. The way you split it into pairs depends on the full quad, and isn't just taking the first 2 pieces and the last 2 pieces, but I think that actually makes it easier. There's very little to remember to use this system, and you break everything down into small easy steps and track parity as you go through them all.

I discuss how this works, then I give some examples, and finish up with some tips that I think will help people. Feedback is welcome.


Here's a summary of what to do:
0) Label each position for edges 1-8 for every cubeshape. I suggest starting in an obvious place on each shape and going clockwise from there.

1) The pieces in positions 1-4 form the first quad, the rest form the second quad.

2) For each quad, split it into two pairs. If there are two pairs which have the same or opposite side colours ("nice" pairs, e.g. YR + (WR, YO, WO), use those (4/0, 2/2 or 0/4 split of the pieces from the M/S rings). Otherwise, one (nice) pair has the same side colour, the other pair is the remaining two pieces ("bad" pairs, e.g. YR + (YG, YB, WG, WB)).

3) Parity of the quad is the sum of the following
 -For each pair in the wrong order, flip parity. The order I use is Y-W, R-O, G-B for nice pairs and (RO)-(GB) for bad pairs.
 -If the pairs are alternating flip parity. That is: label the pairs A, B so the first piece is in A: ABAB flips parity, AABB and ABBA don't flip parity.

4) Parity of all 8 edges is sum of parities of each quad plus the parity of the number of Y (or W) pieces in either quad, that is if it's an odd number of Y pieces in each quad you flip parity. Basically, you need to include a term that relates to which pieces ended up in each quad, the partition term I talked about, and it just happened to work out really easily here.

5) Repeat 0-4 for corners, taking the U/D sticker and one of the side stickers: pick between the clockwise or anticlockwise one (but be consistent), then you have two stickers like you do for edges and it works the same way.

6) Total parity is the sum of parities for edges and corners


Examples:
Order: Y>W R>O G>B (RO)>(GB)
NN=NICE/NICE, NB=NICE/BAD
q=quad, p=pair

layout of examples:
scramble
nice q1p1-q1p2 q2p1-q2p2
q1p1
q1p2
q1 alternating (layout of pairs)
q2p1
q2p2
q2 alternating (layout of pairs)
q1 yellows (partition)
total (mod 2)

WO YB YR WG  YG WR YO WB
NN OR-BG GB-RO
OR 1
BG 1
ABAB 1
GB 0
RO 0
ABBA 0
WYYW 0
TOTAL 1

YB WR WB YR  YO WG YG WO
NN BB-RR OO-GG
YW 0
WY 1
ABBA 0
YW 0
WY 1
ABAB 1
YWWY 0
TOTAL 1

YB YG WR YO  WG WB YR WO
NN BG-RO GB-RO
BG 1
RO 0
AABB 0
GB 0
RO 0
AABB 0
YYWY 1
TOTAL 0

WR WG YO WO  WB YG YR YB
NB RG-OO BB-GR
RG 0
YW 0
AABB 0
WY 1
GR 1
ABBA 0
WWYW 1
TOTAL 1

YR WG YB WB  WR WO YO YG
NB RG-BB RG-OO
RG 0
YW 0
AABB 0
RG 0
WY 1
ABBA 0
YWYW 0
TOTAL 1

WG YB YG YR  WO WB WR YO
NB GG-BR OO-BR
WY 1
BR 1
ABAB 1
WY 1
BR 1
ABBA 0
WYYY 1
TOTAL 0



Tips:
-Check what pieces are in the first quad by quickly looking through it backwards, then go forwards and check the parity once you know the pairs.

-Do the second quad for that piece type immediately afterwards since you should then know what pieces are in it.

-For 4/4 cases, you can count the Y pieces parity for the edges and corners at the same time since they form one layer

-Each of the first 3 pieces of the second quad tells you one of the three things you need to check, the 4th can be ignored.
Axxx: you know the order of one pair
AAxx: order of one pair, not alternating
ABxx: order of both pairs