I'm all that into brainteasers like this...
... but it was enjoyable for a short play. I don't have a lot to add, other than a minor grouch about the overabundance of blue coloring. A little variety would be good to break up the monochromatics. I realize you probably had an intended muting effect of some kind, what with the varying shades of plue, but imho, it's a tad overdone.
Good and a possable fix
First I would like to say that it is a fun puzzle concept. The problem though lies in your connectedness algorithm, which in level 9 is unable to detect solutions.
There are ways to make a checker independent of the full map matching an answer key. An example would be to count each path segment as a separate path (the L tile fore example is made up of 2). If you keep a reference as to how many path segments there are and for each segment a color, R (used) or B(unused). You could have the checker start a pointer at a random path and all pats set to B. Now every time a pointer goes over a segment, it switches its color to R and adds one to a check counter. Every time it encounters a fork, it starts a new pointer and pointers end when they encounter a R path. If the counter's value is equal to the total number of paths, then it is a solution to the cube. This should be a viable solution and if it is not, a slight modification to it should fix it. Good luck
its a good concept, though it needs more levels, I got through the first 9 levels in less than 15 minutes.
also, when I completed level 9 it didn't let me move on to the next level.
I love it, but.....
this is a very neat design for 3D puzzle solvers to enjoy. I loved it, however, once I got to level 9, it became buggy. I have solved the puzzle twice, but apparently it wasn't the solution it needed. so it is possible to connect all the lines without moving to level 10, which is annoying, because it is solved. it was hard though. my congratulations for making a fun game, however, you should test it enough to realize that if there are multiple solutions, they should all finish the level.
did the 1st 8 lvl without a prob but lvl 9 is hard. i dont even wanna see 10