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(2): Thoughts... by DJ)
There are 8 checkers in the arrangement needed for level 3. Once you have created this arrangement, one unit higher, you will need only 7 moves and not 8 to reach level 4.
The number of moves will at most be one less than the number of checkers in the starting setup, since each move eliminates one checker and at least one checker has to survive in the end.
Since the arrangement needed for level 4 has 20 checkers, one will need 19 moves to reach level 4.

Posted by Sanjay
on 20030627 10:23:55 