Just for fun. You can choose freely whether to do 0, 1 or more of these. Don't expect us to spend time grading these however. There are no perfect, pre-defined answers here.

X. Perhaps if we picked the blank in a smarter way, the solve
function would go faster?
One idea is to always pick the blank spot where there are
Does your Do not forget to add appropriate properties that test your functions. |

Y. The solving method we have used in this lab assignment is very basic, and in some sense
naive. The best known methods to solve problems like Sudoku is to also include the notion of
propagation. This is the way most humans actually solve a Sudoku.
A simple variant of propagation is the following. Suppose we have Sudoku with a row with precisely one blank, such as the 3rd row in the example below: 36..712.. .5....18. ..92.47.. 596.13428 4..5.2..9 27.46.... ..53.89.. .83....6. ..769..43Our current solution would go and pick blanks, and start searching recursively, without making use of the fact that we already know the value of that blank (namely 7 in this case); all the other values have been used by the other cells in the row. Implement a function propagate :: Sudoku -> Sudokuthat, given a Sudoku, finds out which rows, columns, and 3x3 blocks only have one blank in them, and then fills those blanks with the only possibly remaining value. It repeats doing this until all rows, columns and 3x3 blocks are either completely filled up, or contain two holes or more. Now, add this function at the appropriate place in your solve function. Does it work faster now? For other, more powerful propagation, you can for example read the following webpage: Or come up with your own propagation rules!Do not forget to add appropriate properties that test your functions. |

Z. Write a function that produces interesting Sudoku puzzles. For example, one could have a function
createSudoku :: IO ()that every time we run it, would print a new, interesting Sudoku puzzle on the screen. One can discuss what an interesting Sudoku puzzle is. Here are three properties that an interesting Sudoku puzzle must have: Can you think of a way to define a function for generating an infinite supply of new Sudokus satisfying the above two properties? You should of course make use of the functions you already have.Do not forget to add appropriate properties that test your functions. |

Å. Generalize the Sudokus that are dealt with in this assignment to
other dimensions, for example 4x4 Sudokus, or 4x3 Sudokus. Do all dimensions
make sense? Make sure your solution works in general, for all possible dimensions that make sense.
For inspiration, look here: Monster Sudokus. Or here: xkcd comics :-) Do not forget to add appropriate properties that test your functions. |

Ö. We have stated the soundness of the solve function as a property;
every produced solution should be a real solution.
The dual of soundness is completeness. Completeness says that whenever there is
a solution, the solve function should also produce a solution. (Equivalently, if the
solve function says that there is no solution, then there really is no solution.)
If we define the following helper datatype: Then, implement a property prop_SolveComplete :: SolvableSudoku -> Bool prop_SolveComplete (Solvable sud) = ...that states that, for any solvale Sudoku, solve produces an answer.
Now, make the type SolvableSudoku an instance of Arbitrary: Here, you need to think about how to generate arbitrary Sudokus that are guaranteed to be solvable! One idea is to start with an arbitrary solved Sudoku, and randomly blank out some of the digits. Implement this! |