[RuggeroAltair]

The determinant factor is not the direction, but the current angle of both entities.

The monster will always have to run towards the direction where the angle between itself and the boat is smaller, otherwise it'd be choosing a longer path.

In facts, if at some point the monster decides to go clockwise, the boat could simply choose the symmetrical direction compared to the relative position of the monster.

This being sad, I'm not sure I am convinced that no other trajectory can be faster, since many of the arguments seem to be a little too wavy.

[squeakynick]

How can I help convince you?

- I hope you agree that, until you breach the circle of safety, it does not matter what the strategy, as you have control to adjust relative positions to anywhere? So, it's all about how to get from the edge of the circle of safety to the shore.

- I hope you can agree that the path from the circle of safety to the shore is a straight one? After all, if you elected for a curved path you would be wasting time. (Any curved path you took between point A and B is inefficient as it would be longer than the straight line connecting the two points. If it is longer, it would take more time. This is bad!)

- So now, we know we need to connect a line from the circle to the shore. We are starting with the monster as far away from us as possible, and we then select where we draw the line.

- In the article I show that the line can be anywhere in a 90 degree sector (it's pretty obvious that outside of these 90 degrees is not too).

- With these constraints, I use trigonometry and calculus to find the maximum.

I'll be happy to re-clarify things in more detail if you let me know which of these arguments is "a little too wavy".

[RuggeroAltair]

I liked the website and the idea. I didn't mean 'wavy' as offensive, but just that this above may be correct assumptions, but I fail to see whether or not are mathematically rigorous.

For example. I agree that within the circle of safety you can readjust your position however you want, given enough time for you to readjust.

What I don't necessarily see, is that one straight line is the best way to maximize the escape. For example, I could consider going at pi/2 for a bit, having the monster trying to catch up counter clock wise, and then slightly bend toward pi/2 + alpha.

This may be wrong, and maybe it's easy to see, but I failed at understanding whether or not that is possible from the explanation. After all, it's a similar strategy to the pi/2 one. I.e., the monster will have to get there covering more space. This would be very different from aiming at that point when you are at the edge of the safety circle, since now the monster is at a different relative position. So it depends on whether or not there is no solution that gives you some extra time.

For example, why is a spiral (that keeps being a spiral after leaving the safety circle) worse than a J shape? (same point as above but with a curve instead of two straight lines at an angle).

Another point is the optimal strategy for the monster. If the monster waits for a second waiting for you to move, would pi/2 still be the best strategy?

To be honest I think that the solution sounds perfectly reasonable. I just thought I wasn't mathematically 100% convinced.

And I liked the website and the other puzzles (I'm still thinking about the Ace of Spade, on which I have a question).

[squeakynick]

Any path that is not straight could have been reached sooner by going on the straight line connecting: The shortest distance between two opposite corners on a rectangle is the diagonal line between them. Any curve you draw connecting them is longer. Any curved path you run gets you to the same point later than you could have got to it at if you went directly there. Outside of the circle of safety, it has to be a straight path.

A spiral is also not going to work outside the circle of safety, as the monster has a higher angular velocity; he moves around the shore faster than your projection to the shore.