A shocking realization
- CanSteam
- Evolved Chao
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A shocking realization
With a little math, a shocking revelation comes out about chao.
Using this little equation:
(9 * 15)[(2 * 2 * 14) + 11 + 14]
9 represents the amount of second evolutions, 15 represents the amount of first evolutions, the two 2's represnt shininess and tone, the 14 represents the colors, the 11 represents jewels, and the 2nd 14 represents shiny jewels
and add
[(2*2*14) + 11 + 14] to account for chaos chao,
you can have 11016 chao of different looking appearance, at the maximum second evolution.
Isn't it kind of weird? 11016 totally unique, fully adult chao.
Using this little equation:
(9 * 15)[(2 * 2 * 14) + 11 + 14]
9 represents the amount of second evolutions, 15 represents the amount of first evolutions, the two 2's represnt shininess and tone, the 14 represents the colors, the 11 represents jewels, and the 2nd 14 represents shiny jewels
and add
[(2*2*14) + 11 + 14] to account for chaos chao,
you can have 11016 chao of different looking appearance, at the maximum second evolution.
Isn't it kind of weird? 11016 totally unique, fully adult chao.
is the pikmin in my avatar on something or...
- Jeffery Mewtamer
- Advanced Chaos Chao
- Posts: 3234
- Joined: Thu Jun 16, 2011 3:59 pm
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Re: A shocking realization
That's a combinatoric explosion for you.
To double check the math:
15 first evolutions times 9 second evolutions + 3 Chaos Chao = 138 complete evolutions.
2*2*14 + 11 + 14 = 81 Breeds.
138 * 81 = 11,178
Our two numbers differ by 2*81 = 162. I think this is due to you only counting one type of Chaos Chao.
Oh, and if you add in medals, that adds another factor of 15, and I think someone once calculated that there were more than a million animal part combinations.
And, if evolution into Amy, Tails, and Knuckles Chao could be triggered(or you just hack it), that raises the 138 in the above calculation to 141. And speaking of hacking, there's Moon Chao plus several other unused Jewel coats.
And we haven't even gotten into how Shiny Jewel Chao look vastly different between Gamecube and SADXPC04 or how Shiny Two-Tones look vastly different between Gamecube and SA2BPC, or the vastly different breeding mechanics of SA1DC and SA1PC11 compared to the various versions of SA2 and SADXGC/SADXPC04.
And you could go a step further with second evolution and include every point on the 41*41 grid you get as the Cartesian product of the run<>power and swim<>fly sliders instead of just it's center, corners, and edge midpoints. 41^2 is 1,681 by the way and would still need to be multiplied by 15 to get the larger evolution number.
Point is, there are easily billions, if not trillions of ways a fully mature Chao can look.
And if that boggles the mind:
There are so many atoms in a grain of sand that, if you enlarged each atom to be as big as the hed of a pin, the grain of sand would be enlarged to a cube a mile on an edge.
More than 1,000 Earths could fit inside Jupiter and the Sun comprises 99% of the solar system's mass.
There are an estimate 100 billion stars in the milky way and 2 Trillion Galaxies in the observable universe.
There are roughly 1.8 trillion pixels in 2 hours of 1080p video running at 120 fps.
There is an estimated 10^80 elementary particles in the observable universe. The number of 16-bit 4*4 bitmap images is 2^256, which is greater.
A Googol is 10^100. A Googolplex is 10^Googol. As minddbogglingly big as a Googolplex is, It is nothing compared to Graham's Number, which is itself nothing compared to infinity.
Oops, I think I got a bit carried away there. To bring things back to SOnic, Blue Sphere levels are made by combining four quadrants from a set of 128, giving a total number of levels equal to 128^4, which is roughly a quarter of a billion. Also, there may be more possibilities for the look of a fully mature Chao than there are galaxies in the observable universe and we haven't even gotten into things that don't affect a Chao's appearance. For example, the 81 phenotypical breeds mentioned above compared with the 4*4*14^2*11^2 genotypical breeds we get if we care about order with heterozygous traits or the fact that their are 6^5 = 7776 possible stat grade combinations for visible stats and that's just counting current grades and not anything hidden in the Chao's genetics.
To double check the math:
15 first evolutions times 9 second evolutions + 3 Chaos Chao = 138 complete evolutions.
2*2*14 + 11 + 14 = 81 Breeds.
138 * 81 = 11,178
Our two numbers differ by 2*81 = 162. I think this is due to you only counting one type of Chaos Chao.
Oh, and if you add in medals, that adds another factor of 15, and I think someone once calculated that there were more than a million animal part combinations.
And, if evolution into Amy, Tails, and Knuckles Chao could be triggered(or you just hack it), that raises the 138 in the above calculation to 141. And speaking of hacking, there's Moon Chao plus several other unused Jewel coats.
And we haven't even gotten into how Shiny Jewel Chao look vastly different between Gamecube and SADXPC04 or how Shiny Two-Tones look vastly different between Gamecube and SA2BPC, or the vastly different breeding mechanics of SA1DC and SA1PC11 compared to the various versions of SA2 and SADXGC/SADXPC04.
And you could go a step further with second evolution and include every point on the 41*41 grid you get as the Cartesian product of the run<>power and swim<>fly sliders instead of just it's center, corners, and edge midpoints. 41^2 is 1,681 by the way and would still need to be multiplied by 15 to get the larger evolution number.
Point is, there are easily billions, if not trillions of ways a fully mature Chao can look.
And if that boggles the mind:
There are so many atoms in a grain of sand that, if you enlarged each atom to be as big as the hed of a pin, the grain of sand would be enlarged to a cube a mile on an edge.
More than 1,000 Earths could fit inside Jupiter and the Sun comprises 99% of the solar system's mass.
There are an estimate 100 billion stars in the milky way and 2 Trillion Galaxies in the observable universe.
There are roughly 1.8 trillion pixels in 2 hours of 1080p video running at 120 fps.
There is an estimated 10^80 elementary particles in the observable universe. The number of 16-bit 4*4 bitmap images is 2^256, which is greater.
A Googol is 10^100. A Googolplex is 10^Googol. As minddbogglingly big as a Googolplex is, It is nothing compared to Graham's Number, which is itself nothing compared to infinity.
Oops, I think I got a bit carried away there. To bring things back to SOnic, Blue Sphere levels are made by combining four quadrants from a set of 128, giving a total number of levels equal to 128^4, which is roughly a quarter of a billion. Also, there may be more possibilities for the look of a fully mature Chao than there are galaxies in the observable universe and we haven't even gotten into things that don't affect a Chao's appearance. For example, the 81 phenotypical breeds mentioned above compared with the 4*4*14^2*11^2 genotypical breeds we get if we care about order with heterozygous traits or the fact that their are 6^5 = 7776 possible stat grade combinations for visible stats and that's just counting current grades and not anything hidden in the Chao's genetics.
Re: A shocking realization
Actually you both are a bit mistaken, but that becomes more blatant when looking at finer, version specific details.
While the (14 * 2 * 2) unquestionably covers the majority of breeds, and 11 is correct for obtainable Jewel Chao on Gamecube, 14 only works for Shiny Jewel Chao if we are counting all possible breeds based on genotype, not phenotype.
If we were looking for genotype, we'd have (14*2*2*12), with the 12 standing for both all the Jewel Coats and the state of lacking a Jewel Coat. But this would obviously be filled with plenty of repeats in appearance. If we're focused on phenotype, we need to pay attention to what game we are playing.
On Gamecube, there are 14 possibly Shiny Jewel Chao, but you cannot count 4 of them:
1. Shiny Jewel Normal is identical to a Shiny Normal Chao
2. Shiny Jewel Brown is identical to a Shiny Two-Tone White Chao
3. Three of the Shiny Jewel Chao are all Invisible Chao
But on PC and later versions, we have a different set of Shiny Jewel Chao, where 11 of them are no longer Transparent or Invisible. From the Gamecube version, we still have the Shiny Jewel Normal, Shiny Jewel Brown, and Shiny Jewel Pink or "Mint" Chao. Currently I haven't found anyone arguing whether or not the PC Shiny Jewel Chao are identical to any existing breeds, but if we assume they aren't, do we count these 11 separately from the 11 on Gamecube?
This puts our breed based on appearance count to [(14*2*2)+11+10] = 77 on Gamecube, [(14*2*2)+ 2+ 12] = 70 or so on PC+, and [(14*2*2)+11+22] = 89. This is by not including any Chao that would be obtained by hacking, which is why there are only 2 Jewel Chao on PC.
There also is the highly neglected factoring in of Baby Chao as a Chao appearance. By that remark, we should count the types of Baby, First Evolutions, Second Evolutions, and Chaos, then multiplied by alignments. So [3*(1+5+(5*9)+1)] = 156 obtainable, noticeable stages in a Chao's life. Resulting in 13,884 possible Chao between platforms.
But clearly this is easily disputed.
----
What I myself find more fascinating is the changing number of accessible Chao per game.
SA-DC: 6 Breeds (2 Breeds through breeding) + 47 Types
SA2-DC: 9 Breeds (1 Breed in-game, 3 Breeds offline, 5 Breeds online) + 156 Types
SA2B: 56 Breeds (50 Breeds in-game per save, 6 Breeds from a different save file, 21 with Tiny Chao Garden access) + 156 Types
I'd mark more but you get the picture. Growth was incredible.
While the (14 * 2 * 2) unquestionably covers the majority of breeds, and 11 is correct for obtainable Jewel Chao on Gamecube, 14 only works for Shiny Jewel Chao if we are counting all possible breeds based on genotype, not phenotype.
If we were looking for genotype, we'd have (14*2*2*12), with the 12 standing for both all the Jewel Coats and the state of lacking a Jewel Coat. But this would obviously be filled with plenty of repeats in appearance. If we're focused on phenotype, we need to pay attention to what game we are playing.
On Gamecube, there are 14 possibly Shiny Jewel Chao, but you cannot count 4 of them:
1. Shiny Jewel Normal is identical to a Shiny Normal Chao
2. Shiny Jewel Brown is identical to a Shiny Two-Tone White Chao
3. Three of the Shiny Jewel Chao are all Invisible Chao
But on PC and later versions, we have a different set of Shiny Jewel Chao, where 11 of them are no longer Transparent or Invisible. From the Gamecube version, we still have the Shiny Jewel Normal, Shiny Jewel Brown, and Shiny Jewel Pink or "Mint" Chao. Currently I haven't found anyone arguing whether or not the PC Shiny Jewel Chao are identical to any existing breeds, but if we assume they aren't, do we count these 11 separately from the 11 on Gamecube?
This puts our breed based on appearance count to [(14*2*2)+11+10] = 77 on Gamecube, [(14*2*2)+ 2+ 12] = 70 or so on PC+, and [(14*2*2)+11+22] = 89. This is by not including any Chao that would be obtained by hacking, which is why there are only 2 Jewel Chao on PC.
There also is the highly neglected factoring in of Baby Chao as a Chao appearance. By that remark, we should count the types of Baby, First Evolutions, Second Evolutions, and Chaos, then multiplied by alignments. So [3*(1+5+(5*9)+1)] = 156 obtainable, noticeable stages in a Chao's life. Resulting in 13,884 possible Chao between platforms.
But clearly this is easily disputed.
----
What I myself find more fascinating is the changing number of accessible Chao per game.
SA-DC: 6 Breeds (2 Breeds through breeding) + 47 Types
SA2-DC: 9 Breeds (1 Breed in-game, 3 Breeds offline, 5 Breeds online) + 156 Types
SA2B: 56 Breeds (50 Breeds in-game per save, 6 Breeds from a different save file, 21 with Tiny Chao Garden access) + 156 Types
I'd mark more but you get the picture. Growth was incredible.
Avatar Art by chocohugs; Signature by Tsui
Re: A shocking realization
This has mostly been genetics discussing, but since evolution traits were also talked about, I am just going to go all the way listing other factors in chao appearances! I will not be doing the math, because I am lame.
Hats, adding in a large new factor for Black Market hats and egg hats
Animal Parts, contributing a huge variety more, with some version dependence, given the difference in animals. The added factor being (Foot slots * head slots * *face slots * ball slot * tail slots * wing slots * arm slots)
Medals from races (can look really awesome on certain chao)
Faces
Really, a huge number. And Crazo, makes me wish even more we that we all could see what could have been next in chao games.
Hats, adding in a large new factor for Black Market hats and egg hats
Animal Parts, contributing a huge variety more, with some version dependence, given the difference in animals. The added factor being (Foot slots * head slots * *face slots * ball slot * tail slots * wing slots * arm slots)
Medals from races (can look really awesome on certain chao)
Faces
Really, a huge number. And Crazo, makes me wish even more we that we all could see what could have been next in chao games.
- AngekFire
- Child Chao
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Re: A shocking realization
I've never thought that there could be so many combinations. Knowing all this just blows my mind!
- CanSteam
- Evolved Chao
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Re: A shocking realization
There's also the case of the Bright Chao, which adds more hundreds of appearances.
edit:
So here's a better equation, in terms of unique appearance alone. (All possible chao to be seen on anything that isnt Dreamcast?)
{B * Z * Y * X * 15 * 9([2 * 2 * 14] + 14 + 11 + 10 + A)} + {B * Z * Y * X * 2([2 * 2 * 14] + 14 + 11 + 10 + A)}
Z represents the chao medals in both SA1 and SA2
Y represents the amount of unique, possible faces there are.
X represents the amount of animal part combinations there are, in both SA1 and SA2.
B represents hats
15 is first evolutions
9 is all second evolutions + just newly evolved
2 in the 2nd term represents baby chao and chaos chao, which never second evolve.
PARENTHESES:
The two 2's represent tone and shine
The first 14 represents color
The 2nd 14 represent bright chao
The 11 represents the amount of jewl coats(which override normal coats, which is why it is outside the brackets)
The 10 represents the amount of unique looking shiny jewels in Gamecube
The A represents the unique looking shiny jewels in Sonic Adventure 2004 PC.
I'm not sure whether to multiply the ZYX or not.
edit: Okay yeah multiply
edit:
So here's a better equation, in terms of unique appearance alone. (All possible chao to be seen on anything that isnt Dreamcast?)
{B * Z * Y * X * 15 * 9([2 * 2 * 14] + 14 + 11 + 10 + A)} + {B * Z * Y * X * 2([2 * 2 * 14] + 14 + 11 + 10 + A)}
Z represents the chao medals in both SA1 and SA2
Y represents the amount of unique, possible faces there are.
X represents the amount of animal part combinations there are, in both SA1 and SA2.
B represents hats
15 is first evolutions
9 is all second evolutions + just newly evolved
2 in the 2nd term represents baby chao and chaos chao, which never second evolve.
PARENTHESES:
The two 2's represent tone and shine
The first 14 represents color
The 2nd 14 represent bright chao
The 11 represents the amount of jewl coats(which override normal coats, which is why it is outside the brackets)
The 10 represents the amount of unique looking shiny jewels in Gamecube
The A represents the unique looking shiny jewels in Sonic Adventure 2004 PC.
I'm not sure whether to multiply the ZYX or not.
edit: Okay yeah multiply
Last edited by CanSteam on Sat Dec 17, 2016 10:33 pm, edited 5 times in total.
is the pikmin in my avatar on something or...
Re: A shocking realization
Interesting thoughts and calculations.
Maybe we should do a creativity contest given that premise. Where we post pictures of our craziest Chaos.
Maybe we should do a creativity contest given that premise. Where we post pictures of our craziest Chaos.
- CanSteam
- Evolved Chao
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Re: A shocking realization
Amount of face combinations: 16
Race Medals : 15
SA1 Animal Combinations: 16 * 13 * 15 * 13
Hats : 14 + 13 + 11
I can't do the Shiny Jewel PC unique right now, nor the SA2 animal combos.
Race Medals : 15
SA1 Animal Combinations: 16 * 13 * 15 * 13
Hats : 14 + 13 + 11
I can't do the Shiny Jewel PC unique right now, nor the SA2 animal combos.
Last edited by CanSteam on Sat Dec 17, 2016 10:57 pm, edited 1 time in total.
is the pikmin in my avatar on something or...
- Jeffery Mewtamer
- Advanced Chaos Chao
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Re: A shocking realization
^^Isn't that the whole point of the Chao Recipes sub-forum?
As for hats, shouldn't Jewel Egg shells be included as well?
And yeah, totally forgot that not all shiny jewels are unique even within the same version.
Real shame that you'd need at least 4 memory cards to have all breeds or all second evolutions in the same game at one time.
As for hats, shouldn't Jewel Egg shells be included as well?
And yeah, totally forgot that not all shiny jewels are unique even within the same version.
Real shame that you'd need at least 4 memory cards to have all breeds or all second evolutions in the same game at one time.
- CanSteam
- Evolved Chao
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Re: A shocking realization
Oh yes, a good point that jewels hae different eggs.
is the pikmin in my avatar on something or...
Re: A shocking realization
You're right. But a lot of the time there are no screenshots. We need a creative chao screenshot mega thread.Jeffery Mewtamer wrote:^^Isn't that the whole point of the Chao Recipes sub-forum?
Re: A shocking realization
Thanks.CanSteam wrote:^ viewtopic.php?f=7&t=5809
- Pratzelwurm
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Re: A shocking realization
All this math... I love it
So many unique chao. Simply one of the many reasons I love them so much.
So many unique chao. Simply one of the many reasons I love them so much.