@jasen - Rotations and reflections may be considered the same for the visual layout, but not for the assignment of values to specific points. There are twelve different solutions. If one considers rotations and reflections to be pertinent, so one should one allow the eversion of this particular figure due to the placement of the points. In such case there is but one solution.

Another consideration is the possibility that "Only one statement is true" is meant to refer to one statement of each of the pairs of statements. In such a case...

If each column has precisely 8 false coins and each column is composed of 8 coins, then...
..., or, perhaps
is intended to be interpreted as each pile has twenty-four coins, for a total of 72 coins in the three piles?

I am guessing I visually transposed one pair of rows in the spreadsheet. 6+6+6+2 does equal 20. Perhaps I was entranced by that twenty-year-old, and took my eyes off the mathematician's sister.

Request clarification to the definition intended for tangram:
The value of N aside, there is still a need to clarify what is meant by "that do not share the same (integer units) length of side." Does the qualification mean that each individual length of a side is a unique integer or that each ordered pair of integer sets [(length, width), such that length >= width] is unique?