Math Puzzle: Find the Missing Piece

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A math puzzle is presented that focuses on finding a missing piece, described as more mathematical than a typical brain teaser. The original poster has shared additional insights and updates related to the puzzle, encouraging engagement from the community. Despite being posted in the Brain Teasers forum, there has been limited response to the challenge. The poster seeks further discussion and solutions from others interested in mathematical problems. The conversation highlights the need for more participation in solving the puzzle.
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I posted this in the Brain Teasers forum, but nobody's really taking a stab at it. It is more mathematical than a brain teaser. This is not homework; hope it's all right posting the link here.

https://www.physicsforums.com/showthread.php?p=609930
 
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I figured out some more stuff on this and posted it. (see link)
 

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