A friend had a problem with a four color Dino cube in that all turned fine except the white side (corner). After looking into it a bit (five Dinos in the house at once!) I determined the problem was one white piece was half connected to the internal center.
So how to repair it?
First order of business was to get it apart. This was the source of a bit of anxiety. A Dino cube is the classic rarity and as it does not have springs, one must rely on old plastic bending to pop it open.
As carefully as I could I popped out a piece using my knife's flat screwdriver/nail file, and from there subsequent pieces were easier.

As you may notice, Dino cube pieces are molded hollow (a loop, it seems). This explains the odd mold line that gives the distinctive sticker line/bubble.

Sure enough the offending white piece was fused from its foot to the center. Try as I might, I was unable to get the piece dislodged from the internal center until !!! it broke. You can see the piece and foot are pretty weakly connected. Now I had the outer piece broken off but the foot still fused to the center.
To help get it apart, I noted the core can unscrew in halves:

Here are pictures with the fused foot to center:


Attempts to cut this away with my knife seemed destined to end up lodged in my finger, so I gave up after a few tries. One can't get much leverage on a twisting center.
In the end, cushioned pliers were used to break the piece away. It pains me to man-handle a rare cube such, but sometimes one must:

Once broken away, one can see the small scars in the plastic where it was fused and didn't break evenly, but it was not much work to shape those surfaces with a bit of cutting and polishing.

If you have ever tried super glue on old or unknown plastics, you may be familiar with disappointment. I was unconvinced the foot would hold to the piece. Early tests proved conclusive: It broke easily. So what to do?
Screws! I have numerous small #0-80 screws from my other puzzle work and these would just fit. It took careful drilling but luckily the angle was 90 degrees from the face edge so I could set the piece on the table and drill straight down through the foot into the piece. The real work was cutting away the inside of the foot to allow the screw head to be relatively flush so it wouldn't catch or scrape the core:


Here it is next to an unharmed piece:


Finally the repaired Dino back together. I think I just made a friend happy


Hopefully this answers the questions of anyone who wondered what a Dino looked like inside, and how one might repair similar problems if anyone else ever found a fused version.
Dave