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 order to reach level 4 I think there must be one row more like Brian did it.
Some thoughts here...
Reaching the 3rd level:
*00
11*
The asterisks are the "ladder" to climb the levels. The checkers with numbers are the stepping stones needed to build that ladder. The lines are checkers which are simply used to move the stepping stones up two levels.
To get another checker up to 3 you need again 8 checkers, leaving 4 to get one up to level 2.
I can see no possible way to achieve that using only two rows of initials.
It seems the jumping around opens up another opportunity I can't grasp.
Btw: Is there an edit function on these boards?

Posted by abc
on 20030909 21:11:23 