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
Thoughts... by levik)
Getting to level 3 is pretty straight forward. It can be done with an arrangment like
o o o o o
o o o
The problem quickly becomes one of getting other checkers to level 2, then sideways so they get under the checker advanced to level 3. And the higher we go, the further sideways they have to go. If I saw how to get to level 4, I suspect I could determine how to get beyond it, or if it is impossible.

Posted by Bryan
on 20030624 16:08:05 