Can you make a puzzle that looks correct until the very last piece?

Question

Lately I've been enjoying a little game called Flow Fit: Sudoku.

It's basically sudoku meets jigsaw. You need to place the pieces into predetermined slots, and numbers cannot repeat in the same rows/columns. If you make a mistake, the game alerts you immediately:

And I've been wondering - is it possible to make a mistake, but not realize it until the very last piece? I cannot come up with such an example, but neither can I come up with a reason why it's impossible.

So, here's my challenge - make such a puzzle! Or prove that it's impossible. The rules:

• The grid can be of any size
• Pieces cannot be rotated/flipped by the player (but you can present them with pre-rotated pieces like in the screenshots above)
• The allowed pieces are: rectangles 1x2, 1x3, 2x2; "Z" (4 squares); "T" (4 squares), "L" (3 or 4 squares). [Clarification: you can rotate/flip these shapes every possible way when creating the puzzle, but the user who will be solving your puzzle won't be able to rotate/flip them]
• The grid must have one and only one correct solution
• It must be possible to place the pieces in such a way, that all but one piece are placed in the grid without conflicts, but putting in the last piece would result in a conflict.
• Optional rule: no two pieces are the same. At least, so far I haven't seen a board in the game that would include two identical pieces. However I'll also accept answers that have identical pieces.

The game also has boards with additional tweaks like holes or hints or wildcard pieces. I'm not looking for a board with them, but if that's all you can come up with, you'll get an upvote from me too (no checkmark though).

Answer 1

There is a simple, almost trivial way in which the conditions can be satisfied. The smallest example of this works on a 3x3 grid.

The board looks like this:

 AAB
 CCB
 DDD

with the pieces

1  123  31  23
2 

Result:

 Solution    Dead end
 231         31.  1
 312         23.  2
 123         123

You can use a similar domino arrangement as part of a larger board. It is of course very easy to avoid the dead end solution, cause the vertical domino would normally be placed before you do the others.