1343
Selective rage
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You can get some odd fractions by two parents having similar lineages. Like, if your mother is Irish, and your great-grandmother on your father's side is Irish, you would be five-eighths Irish. I'm having trouble finding a combination that gives you thirds, though.
Doesn’t exist, 3 is prime. No combination of 2^-n will get you a 3 in the denominator.
…unless somewhere along the tree there’s a person who shows up twice.
Gotcha. Three-eighths is roughly one-third, so I guess that? One-quarter German on one side, one-eighth on the other?
That rounding error would be small enough that most people would consider it less bad than incest, maybe.