Peons
November 5th, 2007
This is a cute puzzle. Consider an infinite checkerboard divided in two with an infinite line lying along the x-axis, as depicted below:
You have several peons at your disposal. In each turn you can move the peons by making one of them jump over the other, thereby killing it (removing it from the board). The peon movement is demonstrated here:
When the game starts, all the peons are below the line. Your goal is to place a peon above the line. The example shows how to accomplish this starting with two peons. Indeed it is obvious that in order to make a peon cross the line, two peons are needed (clearly one is not enough as it cannot move at all).
How many peons are needed in order to not only cross the line, but place a peon two squares above it? As is demonstrated below, four peons are sufficient, and indeed this is the minimum (check that you see why 2 and 3 peons cannot do it).
Now for the riddle. How many peons are needed (again, starting below the line) in order to place a peon 5 squares above the line?
Pages: 1 2
November 7th, 2007 at 7:25 pm
If n is the “height” of a single peon above the line (n=1 being the first square above the red line), then the first few terms are
n=0 –> 0 (no peons are needed)
n=1 –> 2 (the first example)
n=2 –> 4 (the second example)
n=3 –> 8 (since you first need a n=2 peon and need to reconstruct the “cannon” below the line)
It would of course be esthetical to have n=4 –> 16 but this is clearly not the case.
November 7th, 2007 at 10:32 pm
Hi Seb,
Your numbers are right
Now, although your comment is not a solution, (it is at most a very slim hint), in the future, please post such comments on the second page of riddles, so as not to spoil the riddle for people that haven’t solved it yet. That said, this type of comments is excellent, and please keep posting such insights.
On article posts, you are welcome to comment anywhere (i.e. you cannot “spoil” an article).
Thanks!
November 7th, 2007 at 11:25 pm
Sorry about that. I thought all comments went automatically to the second page. My mistake – but since it’s far from a solution, I guess I won’t have spoiled the fun.