Why is the math behind finger landing on a certain square related to parity?

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The discussion centers on the mathematical concept of parity in relation to finger placement on a game board. It establishes that certain squares can only be reached in an even number of moves, while others can only be accessed in an odd number of moves. Specifically, starting from the "Start" square, one can only return to it after an even number of moves, thus eliminating it as a possible landing square after the first move. This parity principle allows for strategic removal of squares based on the required number of moves.

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http://technology.todaysbigthing.com/2009/10/08

Can anyone prove that why your finger must be landed on that square?
 
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It's a matter of "parity". Starting from a given square, you can reach some of the squares only in an even number of moves and the others only in an odd number of moves. For example, starting on the "Start" square you could only get back to it by an even number of moves. Since he requires you to make an odd number of moves first, he knows you cannot be on the "Start" square after the first step and so can remove that one. By forcing you to make an even or odd number of moves (the exact number is not relevant) he knows which squares he can remove and so reduces you possible moves.
 

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