A Crabby Word Problem

Date: 2/13/96 at 17:34:28
From: Anonymous
Subject: A crab trying to swim home...

Dr. Math,

  Our class had the following problem the other day:
  
  A crab was trying to swim home after lying on the beach.  Its 
home was 35 meters out in the ocean.  It took the crab one minute 
to swim ten meters.  Then a wave would wash it back five meters, 
where it would rest for one minute before trying again.  If the crab 
continued this way, how long did it take the crab to get home?

  I can see where 11, 13, 16 and 18 minutes could be possible
solutions to the problem. What might I be doing wrong? 

Date: 2/13/96 at 19:0:41
From: Doctor Ethan
Subject: Re: A crab trying to swim home...

Thanks for the very interesting problem!

Let's consider the crab after it has advanced 5 times, been washed 
back 5 times, and rested 5 times.  At this point, the crab is 25 meters 
out and about to start advancing again.  At its farthest up until this 
point, the crab has reached the 30 meter mark.  (It has just been 
washed back and rested.)  This procedure has taken 10 minutes.  

(From the problem description I am assuming that only the 
advancing and rest periods take any time - it seems that the wash 
back is instantaneous.)

So now we are at the 10-minute mark, and the crab begins 
advancing again.  After a minute it has gone from 25 to 35 meters 
out, thus reaching its home after 11 meters.  Admittedly, it may be 
washed back in moments, but it has, in fact, reached its home.  

Perhaps the crab has built its home well, and will be able to find 
shelter there from those pesky waves.

-Doctor Ethan,  The Math Forum