Question

I am trying to create my own normal 9×9 sudoku puzzle.

I divided the problem into two parts –

1. creating a fully filled sudoku, and
2. removing unnecessary numbers from
   the grid

Right now, I am stuck with the first part.

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This is the algorithm I use in brief:

a) first of all I choose a number (say 1), generate a random cell position, and place it there if

• the cell is not already occupied, and
• if the row does not already have the number, and
• if the column does not already have the number, and
• if the 3×3 box does not already have the number

b) now I check for a situation in which in a row, or a column or a box, only one place is empty and I fill that

c) I check that if there is a number that in not present in a box but is present in the boxes in the same row and the same column (i am talking about 3×3 boxes here), the number’s place is fixed and I fill it.

d) I repeat the above steps until every number appears nine times on the grid.

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The problem I am facing is that, more than often I am getting an intermediate situation like this:

 0  1  0 | 0  0  3 | 0[4/2]0 
 0 [2] 0 | 0 [4] 1 | 3  0  0 
 3  0 [4]|[2] 0  0 | 0  0  1 
---------+---------+---------
 2  0  3 | 0  5  4 | 0  1  0 
 0  0  1 | 3  0  2 |[4] 0  0 
 0  4  0 | 0  1  0 |[2] 3  0 
---------+---------+---------
 1  0  2 | 0  3  0 | 0  0 [4] 
 4  3  0 | 1  0  0 | 0  0 [2] 
 5  0  0 | 4  2  0 | 1  0  3

See the place with [4/2] written? that is the place of 2 as well as 4 because of the boxes marked [].

What can I do to avoid getting in this situation (because this situation is a deadlock – I cannot move further)

Answer 1

There’s another way to generate sudoku puzzles: Start with a known good grid – any one will do – then randomly ‘shuffle’ it by applying operations that don’t destroy the invariants. Valid operations include:

• Swapping rows within a block
• Swapping columns within a block
• Swapping entire rows of blocks (eg, first 3, middle 3, last 3 rows)
• Swapping entire columns of blocks
• Swapping all instances of one number with another
• Reflecting the board
• Rotating the board

With these operations, you can generate a very large range of possible boards. You need to be careful about how you apply the operations, however – much like the naive shuffle, it’s easy to write an algorithm that makes some boards more likely than others. A technique similar to the Knuth shuffle may help here.

Edit: It has been pointed out in the comments that these operations alone aren’t sufficient to create every possible grid.