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.
This arrangement of checkers can be used to reach level 4.
4 4 1 1 1 1 1
1 1 1 2 2
3 3 3 2 2 2
3 2
*************
First, use the checkers marked with 1's to reach level 3 (in the 5th column). Second, use the checkers marked with 2's to reach level 1 underneath the level 3 checker. Third, use the checkers marked with 3's to place a checker at level 0 one column to the left of the checker at level 3. Finally, use the checkers marked with 4's to place a checker at level 4.
20 checkers to level 4
