Solve Checkers Problem: 8x8 Board, 7 Moves, Many Paths

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The discussion revolves around solving a checkers problem on an 8x8 board, where a checker moves diagonally from a starting position to the top row in seven moves. The initial setup involves placing a "1" in the fourth square of the bottom row and "0s" in the others. Participants are encouraged to calculate the number of paths leading to each square in subsequent rows based on the previous row's values. After completing the calculations, the final count of distinct paths to the top row is determined to be 41, significantly lower than the initial estimate of 103. The problem emphasizes the importance of systematic counting and understanding movement patterns in combinatorial problems.
Chaotic Boredom
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All right...I've been at this all night, and any help whatsoever would be appreciated!

Problem-
An eight by eight square game board for checkers has a checker positioned in the fourth square of the bottom row. The checker is allowed to move one square at a time diagonally left or right to the row above. After seven moves the checker will be in the top row. How many different paths will lead to the top row?
 
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Start with the first row. Write in each square the number of paths that lead from the starting position to that square. (there will be seven zeroes and one one)

Use the numbers in the first row to figure out the number of paths from the starting square to each square in the second row.

Use the numbers in the second row to figure out the numbers in the third row.

Keep doing this until you've filled the last row, then just add up the numbers!
 
Danke! Very much! I owe you one! *runs off to solve problem*


EDIT: Final answer? 41 Wow...a lot smaller than what I was getting before...103...>_<
 
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