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

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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.