Wednesday, January 27, 2010

Lions and Lambs

There are 17 perfectly rational lions and 1 lamb on a magical island.  Lions only eat lambs; they will not eat each other.  The magical thing about this island is that if a lion eats the lamb, it becomes the lamb by magical transformation.  The lions are very intelligent and live by two rules:
  1. Don't get eaten
  2. Eat a lamb only if it doesn't result in a violation of rule number 1.  
 A biologist observes these 17 lions and 1 lamb for a little while and then leaves for several years.  When the biologist returns, how many lions and how many lambs will remain?

To understand what happens with 17 lions and 1 lamb, one must first consider the case where there is only 1 lion and 1 lamb.  The lion eats the lamb.  Then consider the case of 2 lions and 1 lamb.  If either lion eats the lamb, he will be a lamb in the 1 lion, 1 lamb scenario.  Knowing this, neither lion will eat the lamb.  Next we consider 3 lions, 1 lamb.  Let's put it in a chart:

1 lion, 1 lamb: lion eats lamb
2 lions, 1 lamb: lions don't eat    
3 lions, 1 lamb: lion eats lamb
4 lions, 1 lamb: lions don't eat

A pattern emerges.  In situations with an even number of lions, a lamb will get eaten.  In situations with an odd number, the lamb will not get eaten.  Applying this rule to the case of 17 lions and 1 lamb, we see that the lamb will get eaten, leaving 16 lions and 1 lamb.  At that point, the situation will remain stable.