922 Million CryptoPunks
In another universe (parallel or not), there are millions of potential CryptoPunks. Imagine if there was an alien with a beanie, or an ape with mohawks.
So, we took that idea and started thinking, what if we had all possible combination of accessories and punks while preserving 1 rule:
A punk can have up to 7 accessories only.
Starting with the base, we have 5 Types:
- Alien
- Ape
- Zombie
- Female
- Male
Going with 4 different skin tones and the fact that non-humans have the same skin color, we end up with total of 11 different types for a punk.
Moving on, we have accessories. Some of them cannot overlap (e.g. you cannot have 2 different eye colors at once). So, we grouped accessories and ended up with the following categories:
- Eyewear (16 entries)
- Facial Hair (12 entries)
- Hair (25 entries)
- Headwear (16 entries)
- Mouth (6 entries)
- Neckwear (3 entries)
- Skin Feature (3 entries)
- Smoking Device (3 entries)
- Other - Clown Nose (1 entry)
- Other - Earring (1 entry)
- Other - Medical Mask (1 entry)
Next, we need to find out all possible combinations of accessories (with up to 7 accessories in each combination). This calls for use of power sets that give you all possible combinations of a specific size in a given set of elements. By getting all the subsets of the aforementioned categories, we just need to replace each category with corresponding number of accessories in that category and find the total combination of features. Afterward, we just multiply the final value by 11 to get the total number of combinations across multiple types. Let's look at a few cases below:
- Size 0 (No Accessory): { } --> 1 Combination
- Size 1 (Exactly 1 Accessory): {Hair}, {Eyewear}, ... --> (25) + (16) + ... = 87
- Size 2 (Exactly 2 Accessories): {Hair, Eyewear}, {Hair, Facial}, etc. --> {25 x 16} + {25 x 12} + ... -> 400 + 300 + ... = 3111
Doing the calculations, we get the following table:
- Size [0] : 1
- Size [1] : 87
- Size [2] : 3,111
- Size [3] : 59,771
- Size [4] : 678,525
- Size [5] : 4,743,645
- Size [6] : 20,819,917
- Size [7] : 57,554,361
To get the final number, we just need to add all the numbers above and multiply them by 11:
11 x (1 + 87 + 3111 + ... + 57554361) = 922,453,598
Now, let's compare existing punks with potential punks from another universe where each combination has a punk:
| Accessory Count | Total Combinations | Existing Combinations (Coverage) |
|---|---|---|
| 0 | 1 x 11 = 11 | 8 (72.72 %) |
| 1 | 87 x 11 = 957 | 333 (34.80 %) |
| 2 | 3,111 x 11 = 34,221 | 3560 (10.40 %) |
| 3 | 59,771 x 11 = 657,481 | 4501 (0.68 %) |
| 4 | 678,525 x 11 = 7,463,775 | 1720 (0.23 %) |
| 5 | 4,743,645 x 11 = 52,180,095 | 166 (0.0003 %) |
| 6 | 20,819,917 x 11 = 229,019,087 | 11 (0.000005 %) |
| 7 | 57,554,361 x 11 = 633,097,971 | 1 (0.00000016 %) |
There could have been a potentially vast universe of CryptoPunks with 922,453,598 punks (one for every 8.5 people on this planet), however we only have 10,000 of them (< 0.0011 % of the total)!