Another use case I saw mentioned several times was "do search > pick random page > go from there". Which feels like it should be (hopefully) straightforward, too - say, get the first matching GID, get the last matching GID, pick a random number between them, and show the listing of results starting at that GID. (Assuming I understand it correctly, and the "start at this GID" URL doesn't actually require that GID to be included in the results set.)
Might not be perfectly random if the distribution of that tag/search isn't uniform or if the counts are too low... but the impression I get is that people using this pick searches with thousands of pages anyway, and whatever results this would give would be significantly more random than whatever they did before (ie "oh, a thousand pages, let's go to lucky page 777").
These - and others - are all individual use cases of a more general pattern, though. People want to be able to not only search for their results, but also jump deep into those results. The former is handled very, very thoroughly via searching, filtering, etc. The latter used to be handled by page numbers, but has no user-friendly replacement now. URL editing with GIDs exists, yeah, but that's not a user-friendly or permanent solution to expect the average user to do.
...Honestly, the page numbers were never that good of a way to do it anyway. As has been noted, they shift around as time goes on, or tags change, or whatever. But they are phenomenally simple and intuitive. Everyone intrinsically understands them, and they work quickly and easily. Thus people don't mind clicking around a bit to (say) locate October 2014 or whatever.
So maybe the solution is to add the ability to "jump to" a region of results, based on some (limited) criteria. Like the jump to page feature of before, but instead of by page, it provides other options.
* Jump forward/backward N minutes/hours/days/etc. Or to a specific date/time, ie YYYY-MM-DD HH:MM:SS or whatever. One's just a derivative of the other after all; only difference is in convenience.
* Jump forward/backward N results (assuming this doesn't end up creating the same server load as before).
* Jump forward/backward to the next favorite (any color, or a specific color) - lets favorites be used as bookmarks of sorts. Like some people mentioned, this is the kind of site people explicitly try to not have browser bookmarks for, so while a browser bookmark would work good, it's not for everyone.
* Jump to specific GID. (Maybe adding a "click to copy this gallery's GID" on gallery pages? Might be easier on mobile users.)
* Random, as mentioned above.
All of those basically boil down to "pick a GID that satisfies some requirements, and then load the search starting at that GID". And while none of them deliver the exact same functionality as page numbers, those kinds of things provide solutions to the same problems that page numbers were being used to solve to begin with.
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