Some notes:
Good luck!
Lab Assignment 3: Sudoku
In this Lab Assignment, you will design a Haskell program that will be able
to solve Sudokus, a popular logical puzzle originating from Japan.
Assignments and Deadlines
There are 6 regular assignments as part of this Lab: A, B, C, D, E, and F. The
lab consists (again) of two parts.
For submission, assignments A, B, C and D are called Lab 3A.
Assignments E and F are called Lab 3B.
Deadlines for each of these parts are given on
the home page.
There are also extra assignments. You can choose freely
whether to do one of these. Those are just for fun.
Hints
Some assignments have hints. Often, these involve particular standard Haskell
functions that you could use. Some of these functions are defined in modules
that you have to import yourself explicitly. You can use the following
resources to find more information about those functions:
We encourage you to actually go and find information about the functions that
are mentioned in the hints!
Sudokus
Sudoku is a logic puzzle originating in Japan and has
caught on in popularity also in the West in recent years. Most
newspapers now publish a daily Sudoku puzzle for the readers to solve.
A Sudoku puzzle consists of a 9x9 grid. Some of the cells in the grid have
digits (from 1 to 9), others are blank. The objective of the puzzle is to fill
in the blank cells with digits from 1 to 9, in such a way that every row, every
column and every 3x3 block has exactly one occurrence of each digit 1 to 9.
Here is an example of a Sudoku puzzle
3 | 6 | | | 7 | 1 | 2 | |
|
| 5 | | | | | 1 | 8 |
|
| | 9 | 2 | | 4 | 7 | |
|
| | | | 1 | 3 | | 2 | 8
|
4 | | | 5 | | 2 | | | 9
|
2 | 7 | | 4 | 6 | | | |
|
| | 5 | 3 | | 8 | 9 | |
|
| 8 | 3 | | | | | 6 |
|
| | 7 | 6 | 9 | | | 4 | 3
|
Here is the solution to the Sudoku puzzle
3 | 6 | 4 | 8 | 7 | 1 | 2 | 9 | 5
|
---|
7 | 5 | 2 | 9 | 3 | 6 | 1 | 8 | 4
|
---|
8 | 1 | 9 | 2 | 5 | 4 | 7 | 3 | 6
|
---|
5 | 9 | 6 | 7 | 1 | 3 | 4 | 2 | 8
|
---|
4 | 3 | 1 | 5 | 8 | 2 | 6 | 7 | 9
|
---|
2 | 7 | 8 | 4 | 6 | 9 | 3 | 5 | 1
|
---|
6 | 4 | 5 | 3 | 2 | 8 | 9 | 1 | 7
|
---|
9 | 8 | 3 | 1 | 4 | 7 | 5 | 6 | 2
|
---|
1 | 2 | 7 | 6 | 9 | 5 | 8 | 4 | 3
|
---|
In this lab assignment, you will write a Haskell program that can read in a
Sudoku puzzle and solve it.
More Information
If you want to read more about Sudokus, here are a few links:
Modelling Sudokus
To implement a Sudoku-solving program, we need to come up with a way of
modelling Sudokus. A Sudoku is a matrix of digits or blanks. The natural way of
modelling a matrix is as a list of lists. The outer list represents all the
rows, and the elements of the list are the elements of each row. Digits or
blanks can be represented by using the Haskell Maybe
type. Digits are
simply represented by Int
.
Summing up, a natural way to represent Sudokus is using the following Haskell
datatype:
data Sudoku = Sudoku [[Maybe Int]]
Since it is convenient to have a function that extracts the actual rows from the
Sudoku, we actually use the following equivalent datatype definition:
data Sudoku = Sudoku { rows :: [[Maybe Int]] }
For example, the above Sudoku puzzle has the following representation in Haskell:
example :: Sudoku
example =
Sudoku
[ [Just 3, Just 6, Nothing,Nothing,Just 7, Just 1, Just 2, Nothing,Nothing]
, [Nothing,Just 5, Nothing,Nothing,Nothing,Nothing,Just 1, Just 8, Nothing]
, [Nothing,Nothing,Just 9, Just 2, Nothing,Just 4, Just 7, Nothing,Nothing]
, [Nothing,Nothing,Nothing,Nothing,Just 1, Just 3, Nothing,Just 2, Just 8]
, [Just 4, Nothing,Nothing,Just 5, Nothing,Just 2, Nothing,Nothing,Just 9]
, [Just 2, Just 7, Nothing,Just 4, Just 6, Nothing,Nothing,Nothing,Nothing]
, [Nothing,Nothing,Just 5, Just 3, Nothing,Just 8, Just 9, Nothing,Nothing]
, [Nothing,Just 8, Just 3, Nothing,Nothing,Nothing,Nothing,Just 6, Nothing]
, [Nothing,Nothing,Just 7, Just 6, Just 9, Nothing,Nothing,Just 4, Just 3]
]
or, written in a more compact way:
example :: Sudoku
example =
Sudoku
[ [j 3,j 6,n ,n ,j 7,j 1,j 2,n ,n ]
, [n ,j 5,n ,n ,n ,n ,j 1,j 8,n ]
, [n ,n ,j 9,j 2,n ,j 4,j 7,n ,n ]
, [n ,n ,n ,n ,j 1,j 3,n ,j 2,j 8]
, [j 4,n ,n ,j 5,n ,j 2,n ,n ,j 9]
, [j 2,j 7,n ,j 4,j 6,n ,n ,n ,n ]
, [n ,n ,j 5,j 3,n ,j 8,j 9,n ,n ]
, [n ,j 8,j 3,n ,n ,n ,n ,j 6,n ]
, [n ,n ,j 7,j 6,j 9,n ,n ,j 4,j 3]
]
where
n = Nothing
j = Just
Now, a number of assignments follows, which will lead you step-by-step towards
an implementation of a Sudoku-solver.
Some Basic Sudoku Functions
To warm up, we start with a number of basic functions on Sudukos.
Assignment A
A1. Implement a function
allBlankSudoku :: Sudoku
that represents a Sudoku that only contains blank cells (this means that no
digits are present).
Do not use copy-and-paste programming here! Your definition does not need to be longer than
a few short lines.
A2. The Sudoku type we have defined allows for more things than Sudokus.
For example, there is nothing in the type definition that says that a
Sudoku has 9 rows and 9 columns,
or that digits need to lie between 1 and 9. Implement a function
isSudoku :: Sudoku -> Bool
that checks if all such extra conditions are met by the given Sudoku.
Examples:
isSudoku (Sudoku [])
False
isSudoku allBlankSudoku
True
isSudoku example
True
isSudoku (Sudoku (tail (rows example)))
False
A3. Our job is to solve Sudokus. So, it would be handy to know
when a Sudoku is solved or not. We say that a Sudoku is solved if there
are no blank cells left to be filled in anymore. Implement the following
function:
isSolved :: Sudoku -> Bool
Note that we do not check here if the Sudoku is a
valid solution; we
will do this later. This means that any Sudoku without blanks (even Sudokus with
the same digit appearing twice in a row) is considered
solved by this function!
Hints
To implement the above, use list comprehensions! Also, the following standard
Haskell functions might come in handy:
replicate :: Int -> a -> [a]
length :: [a] -> Int
and :: [Bool] -> Bool
To help you get started, here is a file that you can use:
- Sudoku.hs, with some definitions that help you
get going with Assignments A, B and C.
Reading and Printing Sudokus
Next, we need to have a way of representing Sudokus in a file. In that way, our
program can read Sudokus from a file, and it is easy for us to create and store
several Sudoku puzzles.
The following is an example text-representation that we will use in this
assignment. It actually represents the example above.
36..712..
.5....18.
..92.47..
....13.28
4..5.2..9
27.46....
..53.89..
.83....6.
..769..43
There are 9 lines of text in this representation, each corresponding to a row.
Each line contains 9 characters. A digit 1 -- 9 represents a filled cell, and a
period (.) represents a blank cell.
Assignment B
B1. Implement a function:
printSudoku :: Sudoku -> IO ()
that, given a Sudoku, creates instructions to print the Sudoku on the screen,
using the format shown above.
Example:
printSudoku allBlankSudoku
.........
.........
.........
.........
.........
.........
.........
.........
.........
printSudoku example
36..712..
.5....18.
..92.47..
....13.28
4..5.2..9
27.46....
..53.89..
.83....6.
..769..43
B2. Implement a function:
readSudoku :: FilePath -> IO Sudoku
that, given a filename, creates instructions that read the Sudoku from the
file, and deliver it as the result of the instructions. You may decide yourself
what to do when the file does not contain a representation of a Sudoku.
Examples:
sud <- readSudoku "example.sud"
printSudoku sud
36..712..
.5....18.
..92.47..
....13.28
4..5.2..9
27.46....
..53.89..
.83....6.
..769..43
readSudoku "Sudoku.hs"
Program error: Not a Sudoku!
Hints
To implement the above, you will need to be able to convert between characters
(type Char
) and digits/integers (type Int
). The standard functions
show
and digitToInt
(import the module Data.Char
) will come in handy
here.
Here are some more functions that might come in handy:
digitToInt :: Char -> Int
putStr :: String -> IO ()
putStrLn :: String -> IO ()
readFile :: FilePath -> IO String
lines :: String -> [String]
unlines :: [String] -> String
Here are some example Sudoku-files that you can download and use:
- example.sud, containing the above example.
- sudokus.zip, a ZIPped collection of sudokus,
both easy and hard ones. The easy ones should all be solvable by your final program
within minutes; the hard ones will probably take a very long time (unless you do
extra Assignment X and/or Y)!.
Generating Sudokus as Test Data
Finally, we need to be able to test properties about the functions related to
our Sudokus. In order to do so, QuickCheck needs to be able to generate
arbitrary Sudokus.
Let us split this problem into a number of smaller problems. First, we need to
know how to generate arbitrary cell values (of type Maybe Int
). Then,
we need to know how to generate 81 such cells, and compose them all into a
Sudoku.
Assignment C
C1. Implement a function:
cell :: Gen (Maybe Int)
that, contains instructions for generating a Sudoku cell. You have to think
about the following:
- Cells either contain a digit between 1 and 9 (for example
Just 3
) or
are empty (Nothing
),
- We would like our generated Sudokus to resemble realistic Sudoku puzzles.
Therefore, the distribution should be around 10% probability non-empty cells
vs. 90% probability for empty cells. (This is not something strict; you can play
around with this if you like.)
Example:
sample cell
Just 3
Nothing
Nothing
Just 7
Nothing
C2. Make Sudokus an instance of the class Arbitrary.
instance Arbitrary Sudoku where
...
We have already done this for you in the file
Sudoku.hs.
C3. Define a property
prop_Sudoku :: Sudoku -> Bool
that expresses that each generated Sudoku actually is a Sudoku according to
Assignment A2. Also use QuickCheck to check that the property
actually holds for all Sudokus that are generated.
Hints
Here are some functions that might come in handy:
sample :: Show a => Gen a -> IO ()
choose :: Random a => (a,a) -> Gen a
frequency :: [(Int,Gen a)] -> Gen a
sequence :: [Gen a] -> Gen [a]
You might want to take a look at the lecture notes and example code on test data
generation.
Rows, Columns, Blocks
Now, we are going to think about what actually constitutes a valid solution of a
Sudoku. There are three constraints that a valid solution has to forfill:
- No row can contain the same digit twice;
- No column can contain the same digit twice;
- No 3x3 block can contain the same digit twice.
This leads us to the definition of a block; a block is either a row or
a column or a 3x3 block. A block therefore contains 9 cells:
type Block = [Maybe Int]
We are going to define a function that checks if a Sudoku is not violating any
of the above constraints, by checking that none of the blocks violate those
constraints.
Assignment D
D1. Implement a function:
isOkayBlock :: Block -> Bool
that, given a block, checks if that block does not contain the same digit twice.
Examples:
isOkayBlock [Just 1, Just 7, Nothing, Nothing, Just 3, Nothing, Nothing, Nothing, Just 2]
True
isOkayBlock [Just 1, Just 7, Nothing, Just 7, Just 3, Nothing, Nothing, Nothing, Just 2]
False
D2. Implement a function:
blocks :: Sudoku -> [Block]
that, given a Sudoku, creates a list of all blocks of that Sudoku. This means:
- 9 rows,
- 9 columns,
- 9 3x3 blocks.
Also add a property that states that, for each Sudoku, there are 3*9 blocks,
and each block has exactly 9 cells.
D3. Now, implement a function:
isOkay :: Sudoku -> Bool
that, given a Soduko, checks that all rows, colums and 3x3 blocks do not contain
the same digit twice.
Examples:
isOkay allBlankSudoku
True
sud <- readSudoku "example.sud"
isOkay sud
True
Hints
Here are some functions that might come in handy:
nub :: Eq a => [a] -> [a]
transpose :: [[a]] -> [[a]]
take :: Int -> [a] -> [a]
drop :: Int -> [a] -> [a]
Note that some of the above functions only appear when you import
Data.List
.
You might want to take a look at the exercises and answers on lists and list
comprehensions.
Positions and Finding Blanks
We are getting closer to the final solving function. Let us start thinking about
how such a function would work.
Given a Sudoku, if there are no blanks left in the Sudoku, we are done.
Otherwise, there is at least one blank cell that needs to be filled in somehow.
We are going to write functions to find and manipulate blank cells.
It is quite natural to start to talk about positions. A position is a
coordinate that identifies a cell in the Sudoku. Here is a way of modelling
coordinates:
type Pos = (Int,Int)
We use positions as indicating first the row and then the column.
For example, the position (3,5) denotes the 5th cell in the 3rd row.
Note: It is common in programming languages to start counting at 0! Therefore,
the position that indicates the upper left corner is (0,0), and the position
indicating the lower right corner is (8,8).
Assignment E
E1. Implement a function:
blanks :: Sudoku -> [Pos]
that, given a Sudoku returns a list of the positions in the
Sudoku that are still blank. You may decide on the order in which the
positions appear.
Examples:
length (blanks allBlankSudoku) == 9*9
True
blanks example
[(0,2),(0,3),(0,7),(0,8),(1,0),(1,2),(1,3),(1,4),(1,5),(1,8),(2,0),(2,1),
(2,4),(2,7),(2,8),(3,0),(3,1),(3,2),(3,3),(3,6),(4,1),(4,2),(4,4),(4,6),
(4,7),(5,2),(5,5),(5,6),(5,7),(5,8),(6,0),(6,1),(6,4),(6,7),(6,8),(7,0),
(7,3),(7,4),(7,5),(7,6),(7,8),(8,0),(8,1),(8,5),(8,6)]
In addition, write a property that states that all cells in the blanks list
are actually blank.
E2. Implement a function:
(!!=) :: [a] -> (Int,a) -> [a]
that, given a list, and a tuple containing an index in the list and a new value,
updates the given list with the new value at the given index.
Examples:
["a","b","c","d"] !!= (1,"apa")
["a","apa","c","d"]
["p","qq","rrr"] !!= (0,"bepa")
["bepa","qq","rrr"]
Also write (a) propert(y/ies) that state(s) the expected properties of this
function. Think about what can go wrong!
E3. Implement a function:
update :: Sudoku -> Pos -> Maybe Int -> Sudoku
that, given a Sudoku, a position, and a new cell value, updates the given Sudoku
at the given position with the new value.
Example:
printSudoku (update allBlankSudoku (1,3) (Just 5))
.........
...5.....
.........
.........
.........
.........
.........
.........
.........
Also write a property that checks that the updated position really has gotten
the new value.
E4. Implement a function:
candidates :: Sudoku -> Pos -> [Int]
that, given a Sudoku, and a blank position, determines which numbers could be
legally written into that position.
Example:
candidates example (0,2)
[4,8]
candidates allBlankSudoku (8,8)
[1,2,3,4,5,6,7,8,9]
In addition, write a property that relates the function candidates with the functions update, isSudoku, and isOkay. (This property can be very useful to understand how to solve Sudokus!)
Hints
There is a standard function (!!)
in Haskell for getting a specific
element from a list. It starts indexing at 0, so for example to get the
first element from a list xs, you can use xs !! 0.
We usually use the standard function zip
to pair up elements in a list
with their corresponding index. Example:
["apa","bepa","cepa"] `zip` [1..3]
[("apa",1),("bepa",2),("cepa",3)]
This, in combination with list comprehensions, should be very useful for this
assignment!
When testing a property that is polymorphic (meaning that it has type variables
in its type), you need to add a type signature that picks an arbitrary type. For
example, when testing properties for the function (!!=), which works for lists
of any type, you have to fix the type when testing, for example lists of
Integers. Do this by adding a type signature to your properties.
Here are some more useful functions:
head :: [a] -> a
(!!) :: [a] -> Int -> a
zip :: [a] -> [b] -> [(a,b)]
Solving Sudokus
Finally, we have all the bits in place to attack our main problem: Solving a
given Sudoku.
Our objective is to define a Haskell function
solve :: Sudoku -> Maybe Sudoku
The basic idea is as follows. Function solve must first check that its argument
is not already a bad Sudoku. This means that (1) it represents a 9x9 sudoku, (2) it has no blocks
(rows, columns, 3x3 blocks) that contain the same digit twice.
We will only do this check once. If the argument is bad then
solve must return Nothing
Now if we have such a Sudoku sud
that we would like to solve, we give it to a recursive helper function solve'
.
The solve'
function must
consider all the blanks in sud
. If this list is empty then
by (1) and (2) above we are done, and the answer of solve'
(and hence solve
)
must be Just sud
.
Otherwise there is at least one blank position. We choose one of them.
For this blank position we
we try to recursively solve sud
, once for each possible candidate; in each recursive case we update the
blank cell with a candidate. The first recursive attempt that
does not give Nothing
provides our solution. But if none of the recursive attempts succeed, we return
Nothing
.
This method of problem solving is called
backtracking.
Assignment F
F1. Implement a function:
solve :: Sudoku -> Maybe Sudoku
using the above idea.
Examples:
printSudoku (fromJust (solve allBlankSudoku))
123456789
456789123
789123456
214365897
365897214
897214365
531642978
642978531
978531642
sud <- readSudoku "example.sud"
printSudoku (fromJust (solve sud))
364871295
752936184
819254736
596713428
431582679
278469351
645328917
983147562
127695843
*Sudoku> sud <- readSudoku "impossible.sud"
solve sud
Nothing
(In the above examples, we use the standard function fromJust
from the library
Data.Maybe
.)
F2. For your own convenience, define a function:
readAndSolve :: FilePath -> IO ()
that produces instructions for reading the Sudoku from the given file, solving it, and printing the answer.
Examples:
readAndSolve "example.sud"
364871295
752936184
819254736
596713428
431582679
278469351
645328917
983147562
127695843
readAndSolve "impossible.sud"
(no solution)
F3. Implement a function:
isSolutionOf :: Sudoku -> Sudoku -> Bool
that checks, given two Sudokus, whether the first one is a solution (i.e. all blocks
are okay, there are no blanks), and also whether the first one is a solution of the
second one (i.e. all digits in the second sudoku are maintained in the first one).
Examples:
fromJust (solve allBlankSudoku) `isSolutionOf` allBlankSudoku
True
allBlankSudoku `isSolutionOf` allBlankSudoku
False
fromJust (solve allBlankSudoku) `isSolutionOf` example
False
F4. Define a property:
prop_SolveSound :: Sudoku -> Property
that says that the function
solve
is
sound. Soundness means that every
supposed solution
produced by
solve
actually is a valid solution of the original problem.
Hints
All the work we did in the assignments A -- E
should be used in order to implement the function solve
.
QuickChecking the property prop_SolveSound
will probably take a long time. Be patient!
Alternatively, there are a number of things you can do about this.
You can test on fewer examples (using the QuickCheck function
quickCheckWith). You can for example define:
fewerChecks prop = quickCheckWith stdArgs{ maxSuccess = 30 } prop
and then write fewerChecks prop_SolveSound
when you want to QuickCheck the property.
You can also generate Sudokus with a different probability distribution. Try increasing
the amount of digits in an arbitrary Sudoku by fiddling with the frequencies in the cell
function
from Assignment C1 and see what happens.
You can use a compiler, such as GHC.
It is okay if you do not find a completely satisfactory solution to this issue.
Here are some useful functions:
fromJust :: Maybe a -> a
listToMaybe :: [a] -> Maybe a
catMaybes :: [Maybe a] -> [a]
Here is an example of an impossible Sudoku:
Extra Assignments (for fun)
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.
Submission
Submit your solutions using the Fire system.
Your submission should consist of the following file:
- Sudoku.hs, containing your solution.
It should contain enough comments to understand what is going
on.
Before you submit your code, Clean It Up! Remember, submitting clean code
is Really Important, and simply the polite thing to do.
After you feel you are done,
spend some time on cleaning your code; make it simpler,
remove unneccessary things, etc.
We will reject your solution if it is not clean. Clean code:
- Does not have long lines (< 78 characters)
- Has a consistent layout
- Has type signatures for all top-level functions
- Has good comments
- Has no junk (junk is unused code, commented code, unneccessary comments)
- Has no overly complicated function definitions
- Does not contain any repetitive code (copy-and-paste programming)
To the Fire System
Good Luck!