I have started a bounty on this question, it is about ten minutes that my bounty has ended; and I have received a message: $$\text{Your bounty on question} \\ \text{"Counting the Number of Integral Solutions to} \ x^2+dy^2=n" \\ \text{is completed.} \\ \text{You must award it to an answer within 24 hours}.$$

Note that each of the answer to that question exists before I start a bounty on it; and none of them satisfies me. Does the term must means that there is a force to award this bounty on one of the existing answers?

Note that long time ago, I have shown my interest on this special problem; as you could see here, and after ending this bounty I will increase the amount of bounty on the same question.

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    $\begingroup$ You don't have to award the bounty to anyone. The phrasing means that IF you want to award the bounty THEN you must do so within 24 hours. Basically it is a reminder about the rules of the bounty system. If you don't act within this time window, you forfeit the right to award it to anyone. And to reiterate, the points were irretrievably gone the moment you started the bounty. $\endgroup$ – Jyrki Lahtonen Sep 23 '17 at 16:48
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    $\begingroup$ On a side note, this is an occasion for the typo is this automatic message "is colmpleted" to be corrected by whomever is able to. $\endgroup$ – Evargalo Sep 27 '17 at 15:58
  • $\begingroup$ On another side note, the open quotation marks could do with being edited in that message as well (at the moment they appear as close quotation marks). $\endgroup$ – Dan Rust Sep 28 '17 at 15:42

Yes, there is an automatic process that will take care of the bounty if you do not award it yourself. From the help-center:

If you do not award your bounty within 7 days (plus the grace period), the highest voted answer created after the bounty started with a minimum score of 2 will be awarded half the bounty amount (or the full amount, if the answer is also accepted). If two or more eligible answers have the same score (their scores are tied), the oldest answer is chosen. If there's no answer meeting those criteria, no bounty is awarded to anyone.

If the bounty was started by the question owner, and the question owner accepts an answer posted during the bounty period, and the bounty expires without an explicit award then we assume the bounty owner liked the answer they accepted and award it the full bounty amount at the time of bounty expiration.

It seems in your case there is no answer matching the criteria and thus the bounty will not be awarded to any answer. It will just disappear; you will not get back the points.

At some point in the past there was a discrepancy between the actual behavior and the documented behavior regarding not considering answers given before the bounty was active, yet I believe this was fixed by now.

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    $\begingroup$ It makes no sense why tie-breaking favours the older answer. The newer answer had less time to garner the same score, so it should be favoured. Was there any good reason given for the tie-breaker? $\endgroup$ – user21820 Sep 25 '17 at 5:32
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    $\begingroup$ I don't know if there was ever any reason given. I can see your point. A goal might have been to curb "bounty-grabbing." (If so, I am not sure it'd be effective.) If you care about this you likely should post to Meta Stack Exchange about it. $\endgroup$ – quid Sep 25 '17 at 17:12
  • $\begingroup$ Well it's one of those things that are odd but too minor to have significant effect and hence difficult to justify making changes just to fix it. And I don't want to go to Meta SE for that. Anyway bounty-grabbing can't be the reason, because it would make people post earlier sacrificing quality, rather than later. $\endgroup$ – user21820 Sep 25 '17 at 17:16

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