Quartominos

A Domino has two numbers, one at each end. A Triomino has three numbers, one in each corner of a triangle. A Quartomino has four numbers, one in each corner of a square.
  1. Find all 24 unique quartominos created by placing the numbers 0, 1 or 2 in each of the corners. A quartomino is unique if it cannot be rotated to make another.
  2. Place the 24 quartominos from part 1 into a 5x5 grid, duplicating only one, such that each quartomino matches its neighbors at both corners.
Source: Original.
Solutions were received from Denis Borris, Sandy Thompson, Kirk Bresniker, and Philippe Fondanaiche. Kirk says his computer was able to prove that a solution for part 2 can be found to duplicate any one of the 24 quartominos. Some representative solutions are below.

Sandy Thompson made an interactive Java applet for this problem. It can be found at: http://www.prevalence.com/games/quartominoes.

From Philippe Fondanaiche: 
Question 1:
a   b   c   d   e   f   g   h   i   j   k   l
00  00  00  00  00  00  00  01  01  01  01  01
00  10  20  11  21  12  22  10  11  21  12  22

m   n   o   p   q   r   s   t   u   v   w   x
02  02  02  02  02  02  11  11  11  12  12  22
10  20  11  21  12  22  11  21  22  21  22  22

Question 2:
  0  0  0  2  1  X
  0  0  1  1  0  1
  1  2  0  0  1  1
  0  0  2  2  1  1
  0  2  2  2  2  1
  2  1  0  1  1  2  

X is the NorthWest value of the redundant tile and can be 0 or 1 or 2
From Kirk Bresniker: 
       0---0     0---0     0---0     0---0     0---0     
       |Q00|     |Q00|     |Q01|     |Q03|     |Q05|
       |r=0|     |r=0|     |r=1|     |r=0|     |r=0|
       0---0     0---0     0---1     1---1     1---2     

       0---0     0---0     0---1     1---1     1---2     
       |Q02|     |Q04|     |Q07|     |Q10|     |Q15|
       |r=1|     |r=0|     |r=0|     |r=1|     |r=2|
       0---2     2---1     1---0     0---2     2---0     

       0---2     2---1     1---0     0---2     2---0     
       |Q06|     |Q22|     |Q11|     |Q08|     |Q13|
       |r=1|     |r=3|     |r=3|     |r=2|     |r=1|
       0---2     2---2     2---1     1---0     0---2     

       0---2     2---2     2---1     1---0     0---2     
       |Q17|     |Q23|     |Q20|     |Q09|     |Q14|
       |r=0|     |r=0|     |r=3|     |r=3|     |r=0|
       2---2     2---2     2---1     1---1     1---1     

       2---2     2---2     2---1     1---1     1---1     
       |Q12|     |Q16|     |Q21|     |Q19|     |Q18|
       |r=2|     |r=1|     |r=1|     |r=0|     |r=0|
       1---0     0---1     1---2     2---1     1---1     


the Q number is the number of the tile and the r is the
number of 90 counter clockwise rotations.
From Denis Borris: 
(I'll number the squares clockwise from top left corner)
  3331 3333 3213 2121 1222
  1323 3312*3131 1223 2222
  3232 2113 1311 3221 2232
  2321 3112 1111 1221 2332
  1211 2131 1133 1233*2333
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