This post by @harphony reminded me of this one game that was played at my school. It was probably only played at my school, being an invention by somebody in my cohort - heck, it might have been me - so in this telling I may have slightly adjusted some details such as to not accidentally doxx myself. If you think you recognise it and know where it's from, no you don't. So with that said, it may or may not have been called:

Not Octopus

Not Octopus is a variant of Octopus Tag, which is itself a variant of Red Rover (or British Bulldog, or whatever your school called it). To recap the rules of Octopus Tag:

1. The active play area is a marked rectangle, longer than it is wide.
2. Some small number of players (~1) are octopi, who start in the middle of this area. Everybody else are fish and start at one of the ends.
3. The octopi periodically call the fish across, at which point they must attempt to cross the play area without being tagged by an octopi.
4. While doing this, somebody will inevitably fall over and be injured.
5. If tagged, a fish becomes... seaweed? A crab? An octopus (and the octopi described earlier are called "sharks"? In any case they must now stand in one place for the remainder of the game and gain the ability to help the octopi tag players.
6. Tagged players may be permitted to pivot on one leg.
7. It is usually not specified how often they are allowed to change which leg that is.
8. I mean, what is walking if not repeatedly pivoting on one leg, alternating once per step?
9. The last remaining fish after many rounds traditionally become the new octop(i/us) for the next game.

Octopus Tag is terrible. It's very easy to run into people at great speed and get injured, it's very easy to get tagged early and have basically nothing to do for rest of the game, and it's also very easy to not get tagged for an incredibly long time if the octopi have no sense and only chase after the fast kids.

And so Not Octopus is invented. Not Octopus is not itself a general solution to the problems of Octopus Tag. It has two caveats that it only really works under. The first is that while Octopus Tag can take just about any rectangular play area, Not Octopus is only ever played on this exact space:

It's a synthetic playing field. The play area is the lined box, about 55 meters by 30 meters. This synthetic field has two important properties:

1. There are various markings on the field (not pictured), which include lines at regular intervals running parallel to the short ends.
2. 55 by 30 meters is kind of a large space to play Octopus Tag on!

These points combined lead to the first major innovation of Not Octopus. To give the not octopi a chance of catching anybody in such a large area, the tagged players ("not crabs") get an upgrade: they have full freedom of movement, so long as they remain on the (usually unmarked) line parallel to the field's ends that runs through where they were tagged. Basically they can run side to side, guided by the marked lines. Kinda like how crabs move (see, the theming makes sense). Tagged players can now defend an entire line across the playing area, not just a single point.

This makes the not crabs much harder to avoid as a not fish, creates a constantly shifting playing space that's somewhat interesting to navigate, and gives not crabs something meaningful to do. It naturally leads to some strategy for the not fish - if a fast runner has been tagged (which the not octopi now have an actual incentive to do!), their line becomes a major obstacle on the field, which you might try to safely cross by coordinating with the other not fish to split up or cause a distraction. On such a large field, there ends up being many of these points, and though the state of play builds up faster with upgraded tagged players, it doesn't become entirely overwhelming too quickly.

But it is pretty quick, and now that tagged players are harder to avoid it's much easier to get tagged early. So arises the second innovation of Not Octopus: one must be tagged on three separate runs in order to become a not crab. The first two times a not fish is tagged, they move to the side of the playing area and proceed to the end that way, safe from being tagged a second time in the same run. The third time, they become a not crab. This ensures that everybody gets at least a few runs before being out of the competition, and provides a sense of rising tension as the state of the game progresses invisibly. This also leads to the second caveat: it's pretty hard to independently keep track of everybody's tag count, so the people playing need to be the sort who won't try to fudge the numbers.

The desire to make the number of tags more meaningful, and easier to keep track of, leads to the third and final innovation of Not Octopus: the not octopi may elect to call across only those on a specific number (for convenience, named not bass, not trout, and not cod for 0, 1, and 2 respectively), or potentially some combination of numbers (not necessarily simultaneously!). This is borrowed from rules typical to some implementations of Red Rover, where the tagging team may call across a specific player, or everybody with some trait, or such. In those implementations, doing so is almost entirely pointless because it removes opportunities to tag players without actually giving anything in return. In Not Octopus, it gives information (who has x number), which allows the not octopi to meaningfully focus their efforts on players with specific game states; and the ability to call one group across and then another while the first is still running allows for some weird staggered-spawning strategies. Despite this generally it's still better to call everyone ("entire ocean" - look, the naming scheme doesn't work here), so there's an extra rule to force it: you can't do the same set twice in a row.

As a bonus side-effect, not having to run on every round provides a welcome break from running the kind of long distance used for the game. It also splits up which side people are running from after enough rounds, which makes navigating the space even more interesting.

Not Octopus is, yes, slightly complicated for a kids' game. Then again, all kids seem to somehow have complete, if not completely correct, knowledge of how to play Monopoly. Turns out they can get the hang of it. I was never a very sporty kid in school, but I didn't mind Not Octopus, especially compared to everything else with running in it - being appealing to kids who weren't fast runners while still being obviously physical exercise surely contributed to it being a popular choice with teachers in outdoor exercise lines.

I guess the point here, if anything, is that you can get to some pretty interesting places by adjusting game rules rather than just taking them as written (or, as unwritten, as they so often are). And if you've got, say, a 30 by 50 metre field and a large group of patient kids, maybe try Not Octopus... and let them decide which of the rules I've described here should be changed.

You know now that I think about it, it really wasn't any safer than regular Octopus Tag.