Excellent explanation of a fun problem!<p>Amazing that the optimal path to land creates a perfect "J". It's solutions like this that leave me in awe-- another felt connection to mathematical truth through an emotional reaction to simplicity. It's empowering to understand a piece of it, but humbling to know it's only part of a larger system that I can't fathom. I think that's the loop that beckons mathematicians.
continuous time games are very messy. Defining the correct notion of Nash equilibrium can be very hard since a player might "respond" to another player's strategy epsilon time after that player "moves". Of course when there is a Nash equilibrium that makes sense, it's all the more interesting.
I first saw this puzzle when it was the monthly "Ponder This" challenge from IBM Research. If you like it, you might enjoy looking through their archives: <a href="http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/pages/index.html" rel="nofollow">http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/pages/in...</a>
Nice! I worked out something much more basic for how to outrun a crashing spaceship: <a href="http://www.win-vector.com/blog/2012/06/how-to-outrun-a-crashing-alien-spaceship/" rel="nofollow">http://www.win-vector.com/blog/2012/06/how-to-outrun-a-crash...</a>
I feel like a zigzag pattern would work here as well. Keep the monster switching directions. Essentially, the monster is balanced perfectly on the opposite side of the lake from the one you are headed towards.
This problem is similar to the one OP had to solve in his interview:
<a href="https://news.ycombinator.com/item?id=6583580" rel="nofollow">https://news.ycombinator.com/item?id=6583580</a><p>DAMN..
umm another naive solution for this would be to go from the center to any arbitrary end, do not cross on to the lake just wait around 2-3 meters from the bank and wait for the monster.
Once the monster reaches the bank just in a straight line to the other side.<p>This way you will have to cover 2<i>r distance where as the monster will cover 3.14</i>r (pi*r ) distance .<p>You will be able to out run a monster around 1.57 times your speed.
Sure it is not that good enough but if your calculus isn't that good (To be honest calculus is the last thing on your mind when a monster is chasing you) and the monster is slow then you will be able to out run him.