This topic is for the purpose of discussing #8191: warmCabin's NES Color a Dinosaur "four color theorem" in 08:11.72

Very creative use of dinosaur coloring, nice work. Although the actual video is too slow to be entertaining.

Singlehandedly brought dinosaurs back from extinction

Great concept.

This was bearable at 2x :)

While browsing the Workbench, I didn't even realize this was about the 4-coloring theorem until I looked very closely at the branch name the 3rd/4th time around, upon which my interest immediately went from 0 to 100. Nice to see there are mathematicians on this site.

MAThVideos: Math demonstrating game movies. When TASing skills are just not enough. Awesome idea. I like how these AFD submissions go into totally different dimensions than you would imagine.

Dang, didn't notice this one during the festivities, thanks for the submission. Unfortunately, I think we have officially gone too far with this gag. Fortunately, I live for going too far with gags.

drawing 14 contains an error. Between the T'rex's legs the background is the same color above and below the horizon.

If the painting speed for big parts wasn't so long, I'd vote "yes". And, on this regard, the entertainment value is bigger at Samsara's submission.

So far it's getting pretty decent feedback! Maybe it doesn't look all that enjoyable directly, but the technical marvel seems to really amuse people.

This movie has been published. The posts before this message apply to the submission, and posts after this message apply to the published movie. ---- [5332] NES Color a Dinosaur "four color theorem" by warmCabin in 08:11.72

drawing 14 contains an error. Between the T'rex's legs the background is the same color above and below the horizon.

This is addressed in the submission notes, by the way:

T-Rex (Drawing 14) Another fake-out horizon line.

See also:

Pterodactyl (Drawing 7) Even though there's a horizon line here, it doesn't go all the way to the right. The background is one region if you fill it.

Oops... 😅

Oops... 😅

It looks like you could swap the yellow belly and the gray crotch's colours and you'd be fixed again?

The eyes being the same color seems also wrong here. Or is it that they can't be independently colored by the game?

Fixed the two drawings. I hope. For the record I did this because there was a lengthy discussion about the mistakes in Discord. It's likely improvable time-wise.

drawing 14 contains an error. Between the T'rex's legs the background is the same color above and below the horizon.

This is addressed in the submission notes, by the way:

T-Rex (Drawing 14) Another fake-out horizon line.

See also:

Pterodactyl (Drawing 7) Even though there's a horizon line here, it doesn't go all the way to the right. The background is one region if you fill it.

It's a bit confusing since Drawing 7 does have a small bit of background above the horizon line (between the tail and the wing) that is gray instead of blue, but in Drawing 14 the bit between the legs gets coloured with the rest of the background. So it's impossible to have the bit between the legs be a different colour.

Had another look through These eyes in dino 2 are definitely touching. This was discussed (for far too long) on Discord whether the pink segments are touching. I think we concluded that they are as they share an adjacent pixel, with adjacent including diagonals (or, there are other ways to come to that conclusion). If anyone has a rigorous method to determine this I'd like to know. I think it's somewhat unanswerable, as the four color theorem applies to abstract shapes, rather than pixels, especially pixel art with an outline.

Maybe: Two separately colorable regions A and B are deemed to be touching if there is a colorable pixel in A and a colorable pixel in B where the distance between their centers is less than 2.5 pixel lengths. The goal is to consider minimum(-distance) offsets (±1,±1), (0,±2), (±2,0), (±1,±2) and (±2,±1) as touching/adjacent, but (±2,±2), (0,±3) or (±3,0) as NOT touching/adjacent (so as to avoid the problem of "corner adjacency" which is not a valid interpretation of adjacency in the four-color theorem). The former offsets all have distance sqrt(5)≈2.236 or less, and the latter ones have distance sqrt(8)≈2.828 or greater. Possible issues: Colorable regions could have minimum offsets of (0,±1) or (±1,0), which may lead to "crowding" (but I'm not aware of any cases so far). Colorable regions could also have minimum offsets of (±2,±2), (0,±3) or (±3,0) and should be considered touching by common sense (but I'm also not aware of any cases so far). Using this metric, the pink segments have a minimum offset of (±2,±1) and so are deemed touching. Around the 2x2 black square in the same image near the top center, the outside and inside blue segments are not touching, and the left yellow and right pink segments are also not touching, because the minimum offsets are (0,±3) and (±3,0), respectively.

Can you explain this like I'm the intended demographic for Color a Dinosaur? Thanks though, seems sufficient for an answer.

like I'm the intended demographic for Color a Dinosaur?

lol Though I realized shortly after my post that I could have just used overlapping rectangles. So like (words in [square brackets] are optional): Two [colorable] regions are deemed to be touching if there is a 2x3 pixel rectangle [in any position and orientation] that overlaps both of them. In case of certain edge (no pun intended) cases, we could include/replace with a slightly larger rectangle with an additional restriction: Two [colorable] regions are deemed to be touching if there is a 2x3[ and/or 3x3 and/or 3x4] pixel rectangle [in any position and orientation] that overlaps both of them, and no other [colorable] regions.

(very large; recommended to view image in new tab) Here is a PNG that I labelled showing what I believe are all the colorable regions (let me know if I missed any!). The only purpose of this PNG is to identify the colorable regions by labels 1..9a..zA..V (excluding l and I). Generally I let 1 be the background, 2 the ground if separately colorable, and 2/3 the largest region of the dino. Beyond that, there is no rule to the labelling. I tried to keep labels relatively close together, and the sclera usually has an earlier label than the eye itself (sorry for the mediocre positioning). #2, #6 and #14 were the only ones I found that have phantom bounded regions that are forced to be the same color as the background "1" (they are indicated in the PNG). Using 2x3 or 3x2 overlapping rectangles as the basis for when regions touch each other, I made a list of adjacencies. Each pair of regions that touches each other by this rule is specified, with the earlier label (that comes earlier in the ordering 1..9a..zA..V) on the left of ":" and the later label on the right. I have also included a number of possibly debatable "touches" that are not covered by the above rule. All of them concern the eyes. The debatable ones are in brackets () and also has a question mark somewhere in there. #3, #9, #10, #11, #12, #13 and #16 are affected.

#1

1:23456789abcdefghijmnopqrstuvwxyzABDEF
2:347cgopqrstuvwyABDEGHJKLMNOPQRSTUV
3:4
4:5678
7:8abc
8:9
9:a
a:b
c:degh
d:e
e:f
g:h
h:ijk
j:km
k:m
m:n
o:p
q:r
w:x
y:z
B:C
D:EF

#2

1:23456789abmnprtuv
2:34567bcjnprstuw
4:5
6:7
7:89aw
8:9
9:a
b:cdefghijkm
e:f
n:ov
p:qv
s:t
t:u

#3

1:23456789abcdefghijkmnopq(s?)
2:37beijkmnopqrs
3:456
4:5
5:6
7:89abc
8:9
9:a
a:c
b:cd
c:d
e:fgh
f:g
g:h
(r:s?)

#4

1:23456789acdefghijkopqs
2:3567bcdefghijkmnopqrs
3:45
6:7
7:89abc
8:9
9:a
c:d
d:e
f:g
g:h
j:k
k:mno

#5

1:234bcdefghijkmr
2:3579a
3:4569abcdefghmnor
5:67
6:789
7:89
8:9
9:a
i:m
j:m
k:m
m:no
n:op
o:q

#6

1:2345679acegijmnopq
2:36
3:45689abcdefghijmopq
4:56
j:km
k:m
m:n

#7

1:23456789cdhiknopq
2:34569abcdfhjkmqr
3:4
4:5
6:7
7:8
d:ef
f:g
h:ijr
j:k
k:mnopq

#8

1:23456789abcdefghijps
2:36
3:456789abcdefkmnopqsu
5:6u
d:e
f:gkm
g:hm
h:imn
i:jno
j:op
k:m
m:n
n:o
o:p
q:rs
s:t

#9

1:2368efghijkmnoptvy
2:3589abcdg
3:456789abfghijkmntxy
4:5
6:78
7:8
8:bcd
9:a
a:b
c:d
e:f
g:hijkmnt
h:i
i:j
j:k
k:m
m:n
o:p(q?)rt
p:q
r:st
t:uvwx
v:wy
w:xy
x:y

#10

1:23eijkmnopqrvx
2:3456789dfghi
3:456adejkmnopqrsuvwx
4:56
5:6
6:789ab
7:8
8:9
9:bd
a:bcd
b:cd
c:d
d:efg
e:ghi
f:g
g:h
h:i
k:m
m:n
o:p
p:q
q:r
s:tu
t:u
(u:v?)

#11

1:234589bcegiknqst
2:59a
3:458cdefghijkmnopqrst
4:5
5:6789
6:79
7:9
8:9
9:a
b:c
o:(p?)q
(q:r?)
s:t

#12

1:23456789abcdefgmnopq
2:789acdefghikmnopqrst
3:6
4:6
5:67
6:7
7:8t
8:t
a:bc
b:c
c:d
f:ghi
(g:h?)
h:jk
i:jk
(j:k?)
m:n
n:o
o:p
p:q

#13

1:2345fghinopqrstu
2:356789bcdeg
3:5abcdefghiknoprst
4:5
5:6a
6:789ab
7:8
8:9
b:c
c:d
f:g
h:i
i:(j?)k
(k:m?)
n:o
p:q
s:t
t:u

#14

1:23456789abcdefghijprtuvwxyz
2:5bghijkmnoprv
3:4
4:5
5:6b
6:9ab
7:8
8:9a
9:a
b:cdefvwxy
c:d
d:e
h:j
i:k
j:k
k:mn
m:n
n:o
p:qrz
r:sz

#15

1:23defghijkmnopsuv
2:3457bde
3:4789abcdghijkmnopqsuv
4:567
5:7
6:7
7:89
8:9
a:b
c:d
d:e
g:h
m:n
n:o
q:rs
s:t

#16

1:2345689cdejmn
2:3abcdefghijkmn
3:45789a
4:57
5:678
6:8
8:9
9:ab
a:b
(f:g?)
(h:i?)
j:k

From discussion so far, it seems that the eyes should be deemed touching whichever regions make up the sclera? Or anything nearby? In that case, the rule would need a bit of updating. (The goal is to have a rule that aligns with common sense.)

Great list

From discussion so far, it seems that the eyes should be deemed touching whichever regions make up the sclera? Or anything nearby? In that case, the rule would need a bit of updating. (The goal is to have a rule that aligns with common sense.)

I'm not sure what discussion you're referring to. I've only been considering the inner eye, but I suppose if you ignore the sclera, you're starting to treat black as a fifth colour.