So +1 get 10 points for answers and questions, and -1 give -2 points. It is easy to see that users only have even reputation numbers now. Or did I miss a reputation-changing mechanism? (If not, why not +5 for good answers / questions, and -1 for bad ones?)

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    $\begingroup$ if you downvote an answer 1 rep.point is subtracted from your own score; and if your answer is accepted you gain 15 points, so there is no parity conservation. $\endgroup$ – Carlo Beenakker Nov 21 '19 at 9:24
  • $\begingroup$ @CarloBeenakker I guess that if you expand your comment a bit (and perhaps add a link to some FAQ posts listing all possible ways to gain/lose reputation such as "How does “Reputation” work?") this would be perfectly acceptable answer. (So maybe you could post it as such.) $\endgroup$ – Martin Sleziak Nov 21 '19 at 11:12
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    $\begingroup$ Since every user starts with +1 reputation, if only possible reputation changes were by an even number, you would expect reputation to be always odd. Of course, users with association bonus start at 101, but that's the same parity. (Maybe it's odd to start at 1 reputation point.) $\endgroup$ – Martin Sleziak Nov 21 '19 at 11:14
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    $\begingroup$ @CarloBeenakker Thanks - that's the part I forgot about! $\endgroup$ – Dominic van der Zypen Nov 22 '19 at 1:16

This was basically answered in the comments, let me summarize the main points here.

Various ways of gaining/losing reputation are listed both in the help center and in this FAQ post: How does “Reputation” work? There are a few where the reputation change is odd:

  • If your answer is accepted, it gives you +15 reputation points.
  • If you downvote some answer, it means -1 reputation for you.
  • The only possible bounty amounts are multiples of 50. So if a bounty is auto-awarded, the reputation earned by the answerer is therefore a multiple of 25 which can be odd - depending on the size of the original bounty.. (More details on bounties and auto-awarding can be found in the help center and in this FAQ post: How does the bounty system work?)

Some additional points:

  • If the only possible reputation changes were by an even number, you would expect the total reputation of a user to be odd, since a new user starts with 1 reputation point. (Or with 101, if the user got association bonus.
  • A user cannot have total reputation below 1 reputation point. So even if somebody gets enough downvotes that would get them to negative numbers, only part of the reputation is deduced. (It is cut off after the user gets to 1 reputation point.)
  • There is also the daily reputation cap on reputation from upvotes/downvotes and suggested edits at 200 reputation points. That means that sometimes part of the reputation increase is not counted - if the reputation from votes and suggested edits would exceed 200 reputation points. (For example, from an upvote you get only 6 reputation points if you were already at 194 on that day.) However, this cannot influence parity, since the total is 200 reputation points.

I will also add that you can audit your reputation on this link: https://mathoverflow.net/reputation (It is similar to your reputation tab, but this link is probably better if you want to check many various days.) There you can see your reputation after every day, when it was odd/even and how it changed. (The reputation breakdown is shown using the current rules - both the upvotes on answer and upvotes on questions add +10.)

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    $\begingroup$ If someone puts up a 50-point bounty, and fails to award it to anyone, the software awards half of it to someone, so that's another way to get an odd increase, right? $\endgroup$ – Gerry Myerson Nov 22 '19 at 9:05
  • $\begingroup$ @GerryMyerson Yes, that's entirely correct - and this is mentioned in the third bullet point of the current revision (and also of the original revision). (And thanks for correcting the typo in my post.) $\endgroup$ – Martin Sleziak Nov 22 '19 at 9:09
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    $\begingroup$ Sorry, don't know how I missed that. But shouldn't it say, "the reputation change may be odd"? If the bounty offered is 100, the auto-award is 50. $\endgroup$ – Gerry Myerson Nov 22 '19 at 9:10
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    $\begingroup$ @GerryMyerson Again, what you're saying is perfectly correct. (And it's a bit embarrassing that I have not noticed that when writing the answer.) I have rephrased the bullet point concerning auto-awarded bounties. $\endgroup$ – Martin Sleziak Nov 22 '19 at 9:14

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