Assume you have a checker board with 7 rows and infinite columns. You can place checkers on only the first 2 rows initially (number these 1 and 0). Then you may jump other checkers up, down, right, and left but not diagonally. The goal is to get as high a row as possible. For example you can get to the second level with four checkers like this:
Level Setup Turn 1 Turn 2 Turn 3
    
2 ····· ····· ····· ···a·
1 ····· ···d· ···d· ·····
0 ·abc· ·ab·· ···a· ·····
1 ···d· ····· ····· ·····
It turns out you need at least 2 checkers to get to level 1, 4 to get to level 2, 8 to get to level 3, and 20 to get to level 4.
Prove the least number of jumps it would take to get to level 5, and how you would do it.
Note: You may place the initial checkers anywhere you wish, as necessary.
(In reply to
re: Impossible? by Yoseph)
Actually, Yoseph, I've been trying to show how to get to level 4 in twenty moves myself. I have the mathematical proofs that it is possible, but I want to put up the actual method to level 4  that lack is the only reason I haven't posted a solution. So, if you could post even the start of your method, I'll type it all out and post a full solution. By the way, good job for figuring it out! =)

Posted by DJ
on 20030820 11:14:54 