There are actually mathematical proofs why the last puzzle has no solution.
In the last puzzle, the character must not touch the starting red square (called R) after starting, and must end on a square which is an even number far from R. Since the square under R (called B) and the square above R (called A) must be touched twice, and the character must not finally end on A or B, the squares next to either square must be touched 3 times (in->out->in->out).
Thus, the connected parts (including the start) next to square A and B must be touched at least altogether (2+1)*2=6 times! That's impossible! The three squares next to A and B can only suffice 2+2+1=5 passes! (R is only touched once in the very beginning!)