All right, I’m going to post a solution to the puzzle now. It’s not the shortest path solution, it’s just the easiest solution in the sense that pretty much anyone can understand why it works. I’m not a fan of overcomplicating things. Anyway, spoilers.

This is just how I envisioned things, in terms of orbits. Say the radius of the lake is R. Then say you travel in an orbit around the center of the lake at a distance of R/4 from the center, going as fast as you can. The monster can travel 4 times as fast as you, and it’s traveling in an orbit 4 times as long as you, so its orbit will be perfectly synchronized with yours. Does that make sense? You’ll both be at the same angle relative to the center, just on different circles.

Note that if you follow an orbit with a radius greater than R/4, the monster doesn’t have to travel as fast as it can to match your orbit (which minimizes the distance between it and you). But, if you follow an orbit with a radius less than R/4, the monster can’t keep up. Makes sense, right? If at R/4, you’re both synchronized with you both moving at max speed, if you take a smaller orbit, you must be “faster” along your circle than the monster is.

So what you do is, swim out towards the monster to a point just *barely* less than R/4 from the center. Then start swimming in a circle around the center. The monster will try to follow you along the edge of the lake. But as explained above, you’ll be “faster” than the monster, so the angle between you and the monster relative to the center of the lake will gradually increase.

Keep doing this until the angle between you and the monster is 180 degrees. When you get to this point, you make a dash for the point on the shore directly opposite the monster. The distance you need to travel to reach that point is just a tad over 3/4 R (The radius of the lake minus the radius of the orbit you’re on). The distance the monster needs to travel is π R (a semicircle around the shore from where it is now). Assuming you chose a good “tad” (less than 0.14 R / 4), you can then beat the monster to the spot you choose, since the monster is only 4 times as fast as you, and the distance you need to go is less than 4 times the distance the monster needs to go.

The key to this puzzle I think is realizing that you’re “faster” than the monster rotationally within a circle of radius R/4 so you can get a head start of that distance on the monster. Conversely, you’re “slower” rotationally outside radius R/4, so moving in anything but a straight line towards the shore outside this point is useless – the monster will gain on you.

I dunno, I think this is the simplest solution, only takes basic math, doesn’t make the problem more complex than it needs to be. It’s the one Joe arrived at, pretty much, when he told me. Dave says the explanation is bad because it’s unique to the parameters of the problem, but if you understand the key to the puzzle, you see that there can be no other parameters. If the monster is only 2 times as fast as you, it’s trivial – just make a dash for the shore opposite the monster. If it’s 5 times as fast as you, the problem is impossible. But I guess you can think about that.