The Cantankerous Cube
2003
Here is a puzzle-solver who understands the simple math.
 three critical pieces three critical pieces
HOW TO MAKE THE PUZZLE
 This puzzle solver demonstrates, by the way he places those three little pieces, that he sees the fatal error that prevents so many people from solving this puzzle.  No two of those pieces may lie in the same plane.  There are 15 planes in the assembled cube, and each one of those 1X1X3 pieces will lie in five of those 15 planes.  If any of the 15 planes lacks one of the three small pieces, there will be no way that 25 unit cubes (an odd number) can be placed in that plane, because all the other pieces can add only an even number of unit cubes.  There are just enough odd numbers to go around with none to spare.

This is one way the three pieces might be arranged.
The upper left piece lies in the left three vertical planes that come out of the screen,
and also in the top-most horizontal plane and the foremost vertical plane that is parallel to the screen.

Seeing science is seeing mistakes we can make
that we never before realized were possible.

Dirty Tricks Department
(make the solution disappear!)

We may make the cube impossible to solve simply by replacing
two of the 1X1X3 pieces with a single piece, 1X2X3.

Replace these two...with this one.

Take a look at The Cantankerous Cube from Da Vinci Days, 2002,

and one of the lessons from the cube.

Scope -- Relevance -- Math -- Logic -- Tensor