Problem Description
AmazingCaddy likes Sudoku very much. One day, he got some Sudoku problems and he wondered whether these problems are well designed. He needs your help now.
Sudoku is a logic-based, combinatorial number-placement puzzle. The objective is to fill a 9 × 9 grid with digits so that each column, each row, and each of the nine 3 × 3 regions contain all of the digits from 1 to 9.
-- Wikipedia
In this problem, the grid has three different sizes, with the formal (N^2) × (N^2), and N will be one in 2, 3 and 4. The rule is the same. The objective is to fill the (N^2) × (N^2) grid with characters so that each column, each row, and each of the N^2 regions (each size is N × N) contains all of the characters from 1 to 4(N = 2), 1 to 9(N = 3) or 1 to G (N = 4).
You task is that when you got a grid, you should tell the grid whether a puzzle or not. If it`s a puzzle, you should tell whether it is a minimal puzzle; else you should tell it has no solution or has multiple solutions.
A puzzle is a partially completed grid (the size is (N^2) × (N^2), N = 2, 3, 4), and has one and only one solution. If remove any character from a puzzle, it will lead to multiple solutions, then it is called a minimal puzzle.
Input
The input contains several cases. For each case, the first line of the input is the N (2<= N <=4). The next N^2 lines will contain the grid. Each line will have N^2 characters, represent the grid. The empty cell will be represented as ’.’. All input data is legal, and you can get more information from the sample input.
Output
For each case:
If the grid is not a puzzle, and has no solution, print “No Solution”, or has multiple solutions print “Multiple Solutions”.
If the grid is a puzzle, but not a minimal, print “Not Minimal”, or print N lines represent the answer in the same format in the input, and ‘.’ should be replaced with the right characters. You can get more information from the sample output.
Sample Input
2
4312
12.4
....
...1
3
5...9.31.
71.8...9.
32...6...
........3
.8.5.3...
4....1.5.
8..9...4.
...1..9..
.....7...
4
...86.....D.A...
.A.C.G.49....65E
.......3B61C..DG
3...89.D7....24.
....4..G.9F2BD..
49...C5.....7...
1.C..8.B6.......
.6A.2.D...4.89.5
8F3BG.E24.A...7.
...6.3.A.1......
D.........5.1E2.
.G...D.9........
......F6.7.....3
6D.9.7..EG...B..
51B.A..8........
........F.C..71.
Sample Output
Not Minimal
Multiple Solutions
9BG86F253ED4A1C7
7ADC1GB4928F365E
F245EA73B61C98DG
3E6189CD7AG5F24B
E873461G59F2BDAC
492D3C5FA8EB7G61
15CF98AB6D7G43E2
B6AG2ED7C34189F5
8F3BG1E24CAD6579
C756F38A219EG4BD
D49A7B6CGF531E28
2G1E5D498B67CF3A
GCE4D5F617B82A93
6DF9C731EG2A5B84
51B7A298D436ECGF
A382B4GEF5C9D716
