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While space was being cleared for a new building, a bunch of scitales came out of the water to munch on the worker's lunches, before falling asleep in the way of everything. You, having a knack for puzzling scenarios, have been sent out to drive away the serpents with a basilisk-whistle. However, you realize that the scitales will only be driven deeper into the clearing site. Unless you make some paths

You could borrow a tractor from the worker in order to wear paths through the shrubbery. If one of the scitales awoke with a path by its head, it's certain to take the path instead. Then it should follow the path along, following any bends like in snake games and picking randomly at intersections. You aren't sure what'd happen at a dead end, so best avoid those. The workers have also mapped out where the snakes are sleeping (see below)

Scitales can generally move at around 1/4 of their length every second. They're also heavy sleepers. The only way to wake them up is all at once with your basilisk whistle

Because you've only got a rather mundane tractor to make the paths, it'll need to be in a continuous path from begining to end. But in the meantime it's rather versatile, being able to turn 90 degrees or reverse easily

There are already some preexisting paths. One leads into the water, and the other leads off into the worksite from the tractor's path. Luckily, the latter is already blocked off, so you don't need to worry about it

However, with all the panicked motion collisions could happen, either between snakes or with a snake hitting its own body. Either way, this is certain to cripple the snake, which would make it far more costly to move

Is there any way to make paths to ensure that the scitales never collide? If not, what is the lowest collision chance that can be managed?

The Map The Map

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    $\begingroup$ Please tell me this is a game somewhere, because if it’s not, it’s about to be. 🙃 $\endgroup$ Sep 22 '21 at 22:15
  • $\begingroup$ It's unclear how this works - both of the answers have interpreted the question differently, and I have a third interpretation altogether. (Can you not just drive by each one, wait until it's entered the water, and then continue on?) $\endgroup$
    – Deusovi
    Sep 23 '21 at 4:18
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    $\begingroup$ @Desuovi Another concern I have is whether the scitales all start moving at the same time, whether their speeds are different and (with the edit - should never edit a question after it has been answered!) whether the path cleared can have branches. $\endgroup$ Sep 23 '21 at 8:44
  • $\begingroup$ If you mow out a strip of, say, 3 x 5, do the snakes just wiggle around on it making a decision at each lattice point? $\endgroup$
    – Dr Xorile
    Sep 23 '21 at 19:42
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    $\begingroup$ Also, does the area underneath the snakes at the beginning count as mowed or unmowed? $\endgroup$
    – Dr Xorile
    Sep 23 '21 at 23:55
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I don't have a rigorous justification for this, but it seems likely that any extra paths I make to let snakes avoid each other can also be used by scitales to seek each other out,

unless they lead to the water.

With that in mind, I

link all the scitale heads together in a giant loop, adding only branches that lead to the water to increase the chance of survival. We'll number them 1 through 5, starting from the bottom scitale and working clockwise around our loop.
enter image description here

There are several cases to consider:

Consider scitales 2, 3, and 4. If they're to have any hope of survival, none can start heading towards any other. Four of the eight possibilities achieve this: we'll call them AAA, AAC, ACC, and CCC according to the anti/clockwise direction of each scitale.
Case AAA: 1 is guaranteed to escape if its first turn is a right, and can actually slip between 2 and 3 if it takes two lefts and 2 doesn't turn towards it, but if it takes a left and then a right it's guaranteed to hit 3. Since 3 shepherds it leftwards either way, 5 has nothing to worry about no matter where it goes.
Chance of success: 1/2 + 3/16 = 11/16
Case AAC: Same as before, except now 5's choices are constrained. It can slip out ahead of 4 as long as it doesn't take two rights.
Chance of success: (1/2 + 3/16) * 3/4 = 33/64
Case ACC: Same as before, except now 1 can't possibly hit 3 - it only needs to dodge 2.
Chance of success: (1/2 + 3/8) * 3/4 = 21/32
Case CCC: Same as before, but 1 can't hit 2 either!
Chance of success: 3/4
Total chance of success: 1/8 * (11/16 + 33/64 + 21/32 + 3/4) = 167/512, or slightly less than a third.

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It's possible to drive the scitales into the water

with probability 1: The middle scitale enters the cleared cycle of length 8; since all scitales have length 4 this scitale will not collide with itself. As time goes on, since a scitale chooses randomly at intersections, the probability of the middle scitale choosing the path into the water approaches 1 geometrically.

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  • $\begingroup$ I edited the question, and this is no longer a valid solution $\endgroup$ Sep 23 '21 at 8:11
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I had to make several assumptions:

  • The snakes are smart enough to not try to move into another snake. So if its only path is blocked, then it is like a deadend, and the run fails.
  • The snakes might try to move into a tail, since they are clever enough to know that a tail is about to be an empty space.
  • But not so smart: two snakes could both decide to move into the same space vacated by a tail. Stupid snakes.
  • The snakes can't plan ahead. So two snakes could just decide to move into the same space with the catastrophic results the OP outlined: drama, snake removal, stench, rotting flesh, etc. In short, a fail.
  • The area that the snakes are lying on cannot be mowed (I could go either way on this: maybe flattened is just as good?).
  • If a snake gets it's head into the water, the rest of the body will follow and that snake will be gone (collisions with the body can still occur while it's heading in).

Since we need a single path and we're not allowed dead ends, we need a plan. Collisions will always be possible, so we need to give the snakes every opportunity to choose life.

My first idea was to just:

Mow everywhere. This turns out not to be that great. The snakes just aren't that smart. My simulation gives about 25% probability of success.

My next idea was:

Mow an edge along the shoreline and give all the snakes a path to get there. Also, since we're not allowed dead ends, put a little 2x2 square at the end by its nose. That will give it enough space to turn around if it needs to. My view was that the details probably wouldn't matter much, so here's an example:
Mowing path
Which has a probability of success of around 85%. We'll call that a new lower-bound...

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