The coin toss example given here seems tautological. Why would my unfair coin have to abide by some norm of probability? If I were designing such a coin, it would obviously include a mechanism for tracking state such that if the last toss was heads the next would always be tails. For example, it could literally redraw the upwards-facing side to the desired outcome an instant before striking the table.

I've never had a great head for mathematics because it feels like most impossibilities arise from arbitrary constraints and definitions. Reading such proofs feels like: 1) I've created an arbitrary universe with made up rules 2) This condition does/doesn't break one of those rules I made up 3) QED

For example, why on earth can't a straight line in a rectangle have a 180 degree angle at its center making that rectangle an octagon too? It's not clear why one system is epistemologically valid and the other is not. The proof of impossibility feels like a "no true scotsman"/"no true octagon" fallacy disguised by a bit of added notation.