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.
I still don't see how to get to level 4, let alone level 5. In order to go higher, more and more checkers are needed, and since they may only be placed on the first two rows, they stretch out over a wide area. The checkers on the edges ultimately need to jump toward the middle so they can meet and work together towards reaching higher levels. But whenever I set up lots of checkers and start jumping, the resulting checkers end up with an empty space between them, i.e. they can't jump each other.
If anyone can shed some light on this problem, I would appreciate it. Right now this puzzle seems impossible as stated.

Posted by Bryan
on 20030627 11:39:56 