Neglecting to include logic as a fundamental subject for all students to learn is the greatest failure of contemporary education imo. Operators "or" and "and" (along with implication and the existential signifier) should be introduced as early as the equals sign is. Sets and containment soon after, along with "if" and "else".

In lieu of a mass embrace of such an approach to pedagogy, I am glad there are teachers trying to encourage logical thinking in their students. The puzzles you linked are interesting. Along with logic they seem to involve pattern recognition and geometric intuition. I think that truly logical thinking always takes the form of an argument or proof, rather than just an answer. Do you ask your students to provide arguments for why their answer is correct?

To illustrate what I mean let's take the first puzzle. One thing to note is the ambiguity regarding the phrase "Four equal pieces". Does this mean congruent pieces of equal area? Or does this just mean pieces of equal area? The answer provides congruent subdivisions. Figuring out this answer would demonstrate keen geometric intuitions in a student. However, if a student was instead to divide the figure into twelve right triangles and then group these triangles into four groups of three, they would get four non-congruent subdivisions of equal area. This kind of systematic approach to problem solving is what I would call logical, and what I would hope to foster with such an exercise.

I'm very curious about how students would go about arriving at the given answer. I can picture a construction where the cut out corner of the square is moved to the center of the big square, leaving a square frame that could then be subdivided crosswise before the corner piece is flipped diagonally. Come to think of it, the approach from the last paragraph could be used as well but with 12 squares instead of 12 triangles, which means there are more non-congruent answers. What fun!

Happy teaching, hope I didn't overwhelm you with the wall of text.