The Flexible Hexagon puzzle (also known as Betcha Can't) is a relative of the the Rubik's Magics, with the only difference that here we are dealing with regular hexagons, not squares. My experience with 6-tile magics (square version) that I have personally constructed, had helped me a great deal to understand this little wonder, as it also has (yeap you guessed it) six tiles! ;-)

It is definitely regarded as a super rare puzzle, and if you ever find one, make sure you treat it well, as it is much more fragile than its square-tiled cousins. Below, ALL four possible solutions (matching shapes) are presented. Each pair of photos shows both sides, the front side, and by flipping horizontally, the back side. Later, some exotic shapes that can be formed, are also presented!

Going from Solution 1 to Solution 2 and then to Solution 3 is very easy, and some simple moves are only required. However, to reach the Solution 4, many difficult moves are required. Notice that although the front parts of Solution 1 and Solution 4 seem identical, that is not true for their back sides! (this is shown below)
I find this connection between Solution 1 and Solution 4, very intriguing! :-)

(Note that, all solutions are presented such that the tile with copyright note of the front face, is always visible and readable)

The required sequences/generators (which are not unique!) to achieve all positions are two:

1. The "mirror" sequence, something like turning it inside out.
2. The "gear" sequence, that is, keeping two opposite (not adjacent) tiles invariant, while rotating the four other tiles (each by 120 degrees, two clockwise and two anticlockwise) - this is seen at the comparison of the back sides of solution 1 and solution 4... and it uses the mirror sequence as a composite!

I haven't done exhaustive research or wrote a proof, but logically speaking, by using the two above (restrictive) moves, it is impossible to have both the blue and red colors in a centre matching position at the same time. And since this is impossible, at least one blue or red part has to stay on the outside, which consequently means it has to be paired with a black or yellow part. Contradiction.

Meanwhile, if we leave all blue and red parts on the outside, then there will be matching problems with the black and yellow.