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MFT - Lake Escape - April 10, 2012
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Math Fact Tuesday! More of a puzzle, but if I give you the answer it'll become a fact, right??

This puzzle (credited to Martin Gardner) can be stated as follows (text source):

We are in a rowboat in the center of a circular lake. On the edge of the lake is a mean goblin who wants to eat us; and if he catches us, he will do so!! The goblin can't swim and won't go into the lake (and doesn't have a boat), but he can run K times as fast as we can row.

We, however, can run significantly faster than the goblin can, so if we are able to reach a point at the edge of the lake without the goblin being there, then we will be able to escape.

Question 1a: For K = 1, is there a strategy we can use to escape?

Question 1b: What about K = 3? or K = pi? or K = 4? or K = 5?

Question 2: What is the best possible value of T so that:
  • if K>T then the goblin has a strategy to catch you (and eat you),
  • if K<=T then we have a strategy that allows us to escape from the lake without being eaten.

A very nice demonstration (game) due to Alexey Izvalov is below. The levels are as follows:
  • Level 1 - Goblin's speed is 3 times faster than your rowing speed (K = 3).
  • Level 2 - Goblin's speed is 3.25 times faster than your rowing speed (K = 3.25).
  • Level 3 - Goblin's speed is 3.5 times faster than your rowing speed (K = 3.5).
  • Level 4 - Goblin's speed is 3.75 times faster than your rowing speed (K = 3.75).
  • Level 5 - Goblin's speed is 4 times faster than your rowing speed (K = 4).
If you think the answer in Question 2 satisfies T>=4, then you should be able to beat level 5! Otherwise, good luck trying!

Now for a solution.

One (optimal) strategy that works for the rower is to trace out a letter J (under the assumption that the goblin does not change direction). If the goblin changes direction at some point, you can modify your strategy to do even better. Rather than rewriting out the entire solution for this strategy I'll just refer you to this link: http://domino.watson.ibm.com/Comm/wwwr_ponder.nsf/solutions/May2001.html

Then the answer to Question 2 (after a bit of work) is the solution T to the pair of equations:
      cos(B) = 1/T,
      sin(B) = (1/T)*(pi + B).
Solving this gives: T=4.6033388...

What this means is that you can win the game above on level 5! (and if there were a level 6 with K=4.6, you can likely win with some careful mouse movements).

If you would like to see exactly how an optimal solution works for K=4.6033388... then visit the link (http://openmap.bbn.com/~tomlinso/chaser/chaser.html) and view the simulation applet on the right.

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this is how we can survive from zombie apocalypse using our math... screw you math-hater...

round round baby round round
spinning out on me
I dont need no goblin
eat my @ss for free...

spinning around so the goblin would be a little slower than the boat, until you are nearly half a spin away from him, and then make a run for it... completed the first four like that.

I can't even beat level 2?!

I have a variation of the pigeonhole problem (in more ways than 1)
suppose I have n zones, each with a value x_n attached to them and suppose I want to put these regions into m regions, 1<m<n such that if more than one zone is in a given region the value of the region is equal to the value of the sum of the values of the zones making up that region. How do I ensure that I group the zones together into the m regions such that I get the smallest possible varience in values across the regions?

PS I struggled with lvl 1 of that game (despite trying to use the 'J' tactic)

I am not sure it is possible to beat level 5, I reached level 5 but couldn't win. It would be possilbe if me nd the goblin were points, but in the game we are bigger than points.

It is possible, I just did it. Took me 22 tries though..

29 seconds for the whole game. Boom.

I beat level 5.

153 points

Ironically I got level 5 in one try, the others took me more effort.

K = 2π?

Hi! It's very pleasant for me to see that my very first flash game is liked in the math community :)

Later I included this puzzle together with some other interesting ones into my another game, "5 Brain Teasers". Have fun!

The one complaint I have is that if the cursor is anywhere outside the water then the smiley face just goes to the closest point on the shore, rather than towards the cursor. Other than that, great game!

My record: First try, it took me a minute and several restarts to beat the game.

Second attempt, beat the game in 12 total seconds with one restart (level 4).

Sorry, I forgot to close close the quotes in the href=. Here's the correct link: http://smartflashgames.blogspot.com/2011/04/5-brain-teasers.html

BTW, 1 more fun fact. Have you ever seen such examples:
sqrt(3375) = 3sqrt(375) and sqrt(91125)=9sqrt(1125) ?
Can you find more? (Or prove that there aren't :) )

In base 10, those are basically the only examples, but in both of them you can multiply the number under the sqrt with 10^n for any n.

Sketch of proof:
We work in base b. Let p be a single digit, then the question becomes:
Find naturnal numbers n and q

If we square both sides of sides of the equation, it becomes a*b^n+q=p^2*q, isolate q to get q=b^n*(p/(p^2-1)).

It's easy to see that q can only be a natural number if the prime factorisation of p^2-1 only contain the same primes as b, and n is sufficiently larger.

For b=10, the only usable denominators occur for p=3 and p=9, with minimal n values of 3 and 4, this gives rises to the two given examples.

I'm trully dumb, I can't even bit level 1 =(

I beat the game :D

i got 413 points...beat all 5 levels. Where is the high scores page?

Sorry for the highscores - it was my first flash game and I didn't know how to implement it then. But later I made "5 Brain Teasers" game. It has highscore table and there are 5 different math puzzles (ogether with this one) in it.

beat lv 5! points 146, total points 324.

Can't I just beat the goblin with the oar? I mean, the math is beautiful, but if I can run faster, it's not as practical to sit in a boat for an hour or so to solve it.

519 points :) Beating the goblin with the oar could be implemented in a future version.

what about this variation?

An escaped prisoner finds himself in the middle of a square swimming pool. The guard that is chasing him is at one of the corners of the pool. The guard can run faster than the prisoner can swim. The prisoner can run faster than the guard can run. The guard does not swim. Which direction should the prisoner swim in in order to maximize the likelihood that he will get away?

Seems like the right side of the shore is easier for some reason. Perhaps because it is closer.

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