What if...?

[helen2010]

Taka

Inspirational and very funny.  Will shamelessly appropriate this and another of your insights.   My gender is now 42 and my identity is m2me.  I am unique, I am non binary and I am proud to be me.  Now that I have the answer I need to go find more questions.

Safe travels

Aisla

[Kaelin]

I'm going to lose my mind unless I know someone "got" my last post, so I'm going to spoil it here.  I also want to improve upon the original idea slightly.

The numbers I listed (0ll0, 0lll00, 0lllll0000, 0lllllll000000, and 0lllllllllllll000000000000) become 6, 28, 496, 8128, and 33550336 in base 10.  These are the first five so-called "perfect" numbers, which are integers equal to the sum of all factors besides itself.  For example, 6 = 1 + 2 + 3, and 28 = 1 + 2 + 4 + 7 + 14.  As best we can tell, all perfect numbers are of the form (2^p - 1)*2^(p - 1), where p is a prime number.  This is does not work for *all* prime numbers (it works for a lot of small primes, including 2, 3, 5, 7, 13, 17, 19, but the success rate drops off a cliff after that).

A thing about (2^p - 1)*2^(p - 1) is that when it is written in binary, it becomes (10^p - 1)*10^(p - 1).  Interestingly:

p = 2: (10^2 - 1)*10^(2 - 1) = (100 - 1)*10^1 = 11*10 = 110
p = 3: (10^3 - 1)*10^(3 - 1) = (1000 - 1)*10^2 = 111*100 = 11100
p = 5: (10^5 - 1)*10^(5 - 1) = (100000 - 1)*10^4 = 11111*10000 = 111110000

For any perfect number written in binary, it will lead off with p 1s and finish with (p - 1) 0s.  Naturally, this makes binary perfect numbers a provocative gender non-binary example: people would at someone and tally many "male" and "female" factors in a row and try to read someone as having that gender, but it'll throw them off when they see a ton of factors in a row the other way.

A quibble is that these numbers have one more 1 than 0 in them, which biases the male side, so I padded the numbers with a 0 in front to balance things out.  It is sort of cheating, but binary integers as a whole have one more 1 than 0 digit on average.  This is due to numbers not being written with "leading zeros."  This helps make our numbers look more compact, but it is akin to writing "1" for male and "0" for female, but skipping all female factors until the first male factor is recorded.  This would lead approximately to the same "one extra 1" bias.  To undo this effect, adding an extra leading 0 would basically work, but following the nature of the bias, the more proper thing would be to say our binary perfect numbers should be written with n leading zeros 2^(-n-1) of the time.  In other words, for the perfect number 11100, we should write:

11100 for 1/2 of people with 11100 at the end
011100 for 1/4 of people with 11100 at the end
0011100 for 1/8 of people with 11100 at the end
00011100 for 1/16 of people with 11100 at the end, and so on.

Considering this modified form is less restrictive, I advocate for it in retrospect.

[Taka]

thank you for the math lesson.
i'm sorry i didn't figure it out before you presented the answer, i forgot about the whole challenge because of a report that was due on friday...

i'm slightly worried about the ones who'd prefer many more 1s than 0s. what kind of numbers would that make?

[Kaelin]

For convenience, I'm going to write 1 instead of l.

There isn't really a common thread or pattern between binary numbers with a lot of 1s instead of 0s unless each identity has a fixed number of digits and we group together the 1s and 0s.  For the most part, we can really only say that identities with more 1s will tend to be larger than identities with more 0s.  The fact that perfect numbers worked out so... perfectly... is due to the structure of such numbers: they not only happen to fit a very specific form, but that form translates very well into binary.

If we are saying there are 31 factors that are either 0 or 1, each identity can be a 31-bit number made of 0s and 1s.  If we feel we need to reserve 2 digits for some factor (for varying degrees of male or female), then we'll just have it gobble up two of the total 31 positions.  There are basically two ways of communicating who people are from an information standpoint.  One is to just treat people as people and just send a 31-bit identity for everyone.  The other approach is to assume that people are fundamentally one gender or another and then to list the "exceptions."  With the latter approach, we will need 1 bit to designate gender, 4 to describe how many exceptions there are, and there will need to be 5 additional bits for each exception.

31-bit approach: each person needs 31 bits, full stop

Exceptions approach, with 0: 5 bits
Exceptions approach, with 5: 30 bits
Exceptions approach, with 10: 55 bits
Exceptions approach, with 15: 80 bits

There are efficiency savings if someone has 0 - 5 male bits or 26 - 31 male bits.  On the other hand, there are significant efficiency losses if someone has between 6 - 25 male bits.  It also places more of a communication burden on those who are mixed, although many will simply say "I'm a blend" and leave it at that.  Of all 31 bit combinations, about 99.98% fall in the 6-25 male range, so it is very little potential for savings here.  The only way the "exceptions" approach could potentially provide benefits is if people tend to hang around the extremes (the 5 or less exception area).  When stereotypes are invoked for this purpose, while there are relatively few male-primaries who wear dresses or work stay-at-home parents and relatively few female-primaries who hunt or work as CEOs, there are tons of female-primaries who don't stay at home or don't wear dresses and male-primaries who don't hunt or don't work as CEOs.  There are so many dimensions with mixed outcomes (at least for one gender) that people will tend to not hug the extremes often enough.  In practice, the "exceptions" system does not pay attention to every exception (it is so unwieldy) but rather specific exceptions that may only be recorded for one gender, like clothes for men and administration aspirations for women -- the focus isn't really on someone's gender or even their gender role alignment but rather their defiance of gender mores and taboos.  It only ever looks at a sliver of someone's gender expression, and yet it is the only way that gender binary can continue to survive.