99 questions/90 to 94

These are Haskell translations of Ninety Nine Lisp Problems.

If you want to work on one of these, put your name in the block so we know someone's working on it. Then, change n in your block to the appropriate problem number, and fill in the <Problem description>,<example in lisp>,<example in Haskell>,<solution in haskell> and <description of implementation> fields.

Problem 90

Eight queens problem

This is a classical problem in computer science. The objective is to place eight queens on a chessboard so that no two queens are attacking each other; i.e., no two queens are in the same row, the same column, or on the same diagonal.

Hint: Represent the positions of the queens as a list of numbers 1..N. Example: [4,2,7,3,6,8,5,1] means that the queen in the first column is in row 4, the queen in the second column is in row 2, etc. Use the generate-and-test paradigm.

Example in Haskell:
> length queens
92
> take 1 queens
[[4,2,7,3,6,8,5,1]]

Solution:

queens = queens' 8

queens' 0     = [[]]
queens' (n+1) = [ try:alreadySet | alreadySet <- queens' n, try <- [1..8], isSafe try alreadySet]
    where isSafe try alreadySet       = not (sameRow try alreadySet || sameDiagonal try alreadySet)
          sameRow try alreadySet      = try `elem` alreadySet
          sameDiagonal try alreadySet = any (\(col,q) -> abs(try - q) == col) $ zip [1..] alreadySet

By definition/data representation no two queens can occupy the same column. "try `elem` alreadySet" checks for a queen in the same row, "abs(try - q) == col" checks for a queen in the same diagonal.

This is a modification of a function I wrote when I was just learning haskell, so there's certainly much to improve here! For one thing there is speedup potential in caching "blocked" rows, columns and diagonals.

Problem 91

<Problem description>

Example:
<example in lisp>

Example in Haskell:
<example in Haskell>

Solution:

<solution in haskell>

<description of implementation>

Problem 92

<Problem description>

Example:
<example in lisp>

Example in Haskell:
<example in Haskell>

Solution:

<solution in haskell>

<description of implementation>

Problem 93

An arithmetic puzzle

Given a list of integer numbers, find a correct way of inserting arithmetic signs (operators) such that the result is a correct equation. Example: With the list of numbers [2,3,5,7,11] we can form the equations 2-3+5+7 = 11 or 2 = (3*5+7)/11 (and ten others!).

Example in Haskell:
> puzzle [2,3,5,7,11]
["2+3-5=7/11", "2+3=5+7/11", "2+3=5-7/11", "2-3+5+7=11", "2/3/5=7/11", "2/3*5=7/11", "2/3=5/7/11", "2/3=5/7*11", "2=3-5-7+11"]

Solution:

import Text.ParserCombinators.Parsec
import Text.ParserCombinators.Parsec.Expr

puzzle :: [Integer] -> [String]
puzzle xs = filter ((\(a,b) -> calcExp a == calcExp b) . split) equations
    where equations = map concat . filter isValid . sequence . intersperse ops . map ((:[]) . show) $ xs
          split     = (\(a,b) -> (a, drop 1 b)) . break ('='==)
          ops       = ["+", "-", "/", "*", "="]
          isValid   = (1==) . length . filter ("="==)


expr    :: Parser Integer
expr    = buildExpressionParser optable factor

optable   = [[op "*" (*) AssocLeft, op "/" div AssocLeft]
            ,[op "+" (+) AssocLeft, op "-" (-) AssocLeft]
            ]          
                where op s f assoc = Infix (do{ string s; return f}) assoc

factor  = do{ char '('
            ; x <- expr
            ; char ')'
            ; return x 
            }
        <|> number

number  :: Parser Integer
number  = do{ ds <- many1 digit
            ; return (read ds)
            }

calcExp e = either (error . show) id (parse expr "source" e)

I chose using parsec to evaluate the arithmetic expressions. Parentheses are not yet supported. Integer division is used (i.e. 7/11 == 0).

Using Template Haskell or hs-plugins would likely yield a more concise solution.

Problem 94

<Problem description>

Example:
<example in lisp>

Example in Haskell:
<example in Haskell>

Solution:

<solution in haskell>

<description of implementation>

Problem 95

English number words

On financial documents, like cheques, numbers must sometimes be written in full words. Example: 175 must be written as one-seven-five. Write a predicate full-words/1 to print (non-negative) integer numbers in full words.

Example in Haskell:
> numbers 175
one-seven-five

Solution:

import Data.List
import Data.Maybe

numbers :: Integer -> String
numbers n = concat . intersperse "-" . map (fromJust . (`lookup` table)) $ show n
    where table = [('0',"zero"), ('1',"one"), ('2',"two"),   ('3',"three"), ('4',"four"), 
                   ('5',"five"), ('6',"six"), ('7',"seven"), ('8',"eight"), ('9',"nine")]

This solution does a simple table lookup after converting the positive integer into a string. Thus dividing into digits is much simplified.

Problem 96

<Problem description>

Example:
<example in lisp>

Example in Haskell:
<example in Haskell>

Solution:

<solution in haskell>

<description of implementation>

Problem 97

Sudoku

Sudoku puzzles go like this:

       Problem statement                 Solution

        .  .  4 | 8  .  . | .  1  7	     9  3  4 | 8  2  5 | 6  1  7	     
                |         |                          |         |
        6  7  . | 9  .  . | .  .  .	     6  7  2 | 9  1  4 | 8  5  3
                |         |                          |         |
        5  .  8 | .  3  . | .  .  4          5  1  8 | 6  3  7 | 9  2  4
        --------+---------+--------          --------+---------+--------
        3  .  . | 7  4  . | 1  .  .          3  2  5 | 7  4  8 | 1  6  9
                |         |                          |         |
        .  6  9 | .  .  . | 7  8  .          4  6  9 | 1  5  3 | 7  8  2
                |         |                          |         |
        .  .  1 | .  6  9 | .  .  5          7  8  1 | 2  6  9 | 4  3  5
        --------+---------+--------          --------+---------+--------
        1  .  . | .  8  . | 3  .  6	     1  9  7 | 5  8  2 | 3  4  6
                |         |                          |         |
        .  .  . | .  .  6 | .  9  1	     8  5  3 | 4  7  6 | 2  9  1
                |         |                          |         |
        2  4  . | .  .  1 | 5  .  .          2  4  6 | 3  9  1 | 5  7  8

Every spot in the puzzle belongs to a (horizontal) row and a (vertical) column, as well as to one single 3x3 square (which we call "square" for short). At the beginning, some of the spots carry a single-digit number between 1 and 9. The problem is to fill the missing spots with digits in such a way that every number between 1 and 9 appears exactly once in each row, in each column, and in each square.

see Sudoku

Problem 98

<Problem description>

Example:
<example in lisp>

Example in Haskell:
<example in Haskell>

Solution:

<solution in haskell>

<description of implementation>

Problem 99

<Problem description>

Example:
<example in lisp>

Example in Haskell:
<example in Haskell>

Solution:

<solution in haskell>

<description of implementation>