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Today, a new user John created an account on PSE, and he got an association bonus. He answered a question. Few days later, after logging in, his reputation went up! (He did not do any other things other than answering, and he did not receive spam or offensive flags.)

How many actions at least have to occur to result the following reputation total, or it is impossible, with the following conditions?

Define an action be a upvote, a downvote, an accept, or a bounty rewarded on John's answer.

Easy 1:

126 rep, non-bounty question, answered 1 day ago.

Easy 2:

202 rep, non-bounty question, answered 1 day ago.

Medium 1:

315 rep, non-bounty question, answered 1 day ago.

Medium 2:

226 rep, bounty question, not accepted, answered 2 days ago.

Hard 1:

512 rep, non-bounty question, answered 1 day ago.

Hard 2:

501 rep, bounty question, accepted, answered 1 day ago.

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    $\begingroup$ Voting to migrate to meta :-P $\endgroup$ – Rand al'Thor May 15 at 10:54
  • $\begingroup$ @Randal'Thor This is a puzzle, not a meta question. $\endgroup$ – Culver Kwan May 15 at 10:54
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    $\begingroup$ I know. I was joking, hence the emoticon. $\endgroup$ – Rand al'Thor May 15 at 10:55
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I'm fairly sure this is it:

126 rep: 1 upvote, 1 accept
202 rep: 1 accept, 2 downvotes, 9 upvotes
315 rep: 22 upvotes, 3 downvotes
226 rep: +75 bounty (1/2 * 150), 5 upvotes
512 rep: 40 upvotes, 1 accept, 2 downvotes
501 rep: +225 (1/2 * 450) bounty, 1 accept, 16 upvotes

Explanation:

There are only two odd reputation changes, accept and half-bounty, and three even reputation changes, upvote, received downvote and bounty.
226: This could have been (1/2*250), but half-bounties are only rewarded for answers with a score of at least +2.
501: The bounty must have been placed by someone other than OP. Otherwise, the bounty would be 100%, since it's accepted.

The answer assumes that:

"posted one day ago" means there are two UTC-days of possible voting, allowing for a maximum reputation increase of +415, disregaring bounties.


Footnote: If you want to make a similar, but more difficult, challenge, I suggest you include the more obscure behaviours of the reputation system. Did you know that undownvote can cost reputation?

I got downvoted on Oct 6 and Oct 27. I hit the rep cap later those days, so I didn't lose any rep from the downvotes. Both downvotes got reversed on Oct 28, but I hit the rep cap that day as well, so I didn't gain any rep from the undownvotes. Today, my rep got recalculated. The system decided none of this was fair and made the original downvotes count as -2 each.

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  • $\begingroup$ 512 would hit the rep cap. 315 also. $\endgroup$ – msh210 May 15 at 9:23
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    $\begingroup$ Are you sure about that? Note that 512 means the increase is 411. And posting the answer one day ago allows for two days of voting. This means the reputation from voting alone can be 400. $\endgroup$ – CG. May 15 at 9:47
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    $\begingroup$ Oh, you're right. $\endgroup$ – msh210 May 15 at 10:07
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    $\begingroup$ Correct! You get an upvote and a checkmark. So you got a total of 65 rep. for answer this question! $\endgroup$ – Culver Kwan May 15 at 10:49
  • $\begingroup$ I added a footnote. Something to keep in mind if you want to make a harder version of this challenge :P $\endgroup$ – CG. May 15 at 11:07

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