[Noire ♜]

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Well, the thing about the switching of game modes idea you would have to switch the section it is under in the "Garry's Mod find server tab" each time you swap the gamemode. It would probably require more effort than anyone that has access to the backend of the servers wants to exert. 

As far as the deathrun and murder servers we currently have, and I'll keep this tangent short considering it's getting a bit off-topic from the original idea, they can be active but only when they are loved and cared for. We saw that when Titan was admin over DR or when, and god forgive me for saying this, [redacted] was the figurehead of murder. However, in their current state, they are DOA beside the few nights where a group of friends wants to hop on and kick-start the server's activity again. 

In all honesty, a minigames/changing game mode server would be totally kick-ass, in practice, this would not work due to the lack of interest in a rotating game modes server from people outside of our current community, because, as you stated, communities don't change game modes like that. It would be a very neat idea to put into practice though because it would give nigh constant diversity when it comes to what games you play. For the record, I am not saying that the rotations would be restricted to jailbreak and bhop. I feel as though it should include various game modes, including old ones that have died out a long time ago, i.e. Extreme Football Throwdown and fan favourites like Melonbomber and Suicide Barrels.

It could be done under a completely separate game tag, like 'Dinkleberg's Rotating Gamemodes' or something but you run into the issue of no one finding the server and it dying at the bottom of the server list. Due to this, I doubt Dink will put in the effort of creating an entirely new server for gamemodes like the ones I listed above because it would be a net negative income for him. So, I suggested the consolidation of the dying servers and this proposed rotating game mode server into one conglomerate to help facilitate the idea of a rotating gamemode server into a more grounded reality. 

If we did do something like that, it would even free up another server slot for something like DarkRP to come back into the Dinkleberg's Gmod community.

TL;DR
Jack of all trades, master of none. Oftentimes better than a master of one.

All of that was just a bunch of brainstorming though, so, take it with a grain of salt.