The idea is, each number is expressed as a sum of n factorials, with n being the number of digits in the number post-conversion. You start with the highest factorial that you can subtract out of the original number and work your way down.
1 becomes 1, because 1 = 1!, so the new number says “1x(1)”.
2 becomes 10, because 2 = 2!. The new number says “1x(2x1) + 0x(1)”.
3 becomes 11, because it’s 2 + 1. The new number says “1x(2x1) + 1x(1)”.
21 becomes 311: 4! is 24, so that’s too big, so we use 3!, which is 6. 3x6 = 18, so our number begins as 3XX.
That leaves 3 left over, which we know is 11. The new number says “3x(3x2x1) + 1x(2x1) + 1x(1)”.
Step 1: Find the area of each chunk. The biggest chunk is your main chunk.
Step 2: Find the distance between the closest edge of main chunk and the center of each other chunk individually.
Step 3: Discontinuity of each chunk is area of chunk * distance from main chunk / total area.
Step 4: Total discontinuity is sum of each chunk’s discontinuity.
Bolded parts are important. If you use the center of the main chunk, larger main chunk radii make other chunks seem more discontinuous than they should be. If you use the closest edge of other chunk’s, you don’t account for the entire area of the other chunk.
This will give you a number that is bigger when there are more and/or bigger pieces that are further away, and smaller when the opposite is true, normalized for the total area of the country so bigger countries aren’t penalized just because they’re bigger.
While I agree with this mostly (permanent is probably too long, maybe X months after you pay for the worker’s medical bills), that wouldn’t have helped in this instance, since it was a stray dog.