After Dinner question in seek of answer

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Geek’s post-dinner mind-bending question:  Given a nonogon (9 sided polygon) and all 126 quadrilaterals that can be made using groups of 4 points (vertices) of the nonogon, determine a method of choosing 10 to 20 of these quadrilaterals such that if they were all drawn on the nonogon at the same time the result would be symmetrical.

…but by the time I wrote this down he’s already got two instances of sets that have the desired property, though perhaps not a general method for selecting them (hard to tell since it’s all in his head still).  So I think he’s got what he was looking for, but I thought I’d write this down just to demonstrate what it’s like being Geek’s Dad (!).

Reason for answering this?  This forms the theoretical backbone of a system of magic for the world in which an upcoming Geek And Dad trading card game is set.

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