# 99 questions/90 to 94

### From HaskellWiki

m (minor change in solution 2, prob. 91) |
(moved solutions to subpages of 99 questions/Solutions. Still need solution for P94) |
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__NOTOC__ |
__NOTOC__ |
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− | These are Haskell translations of [http://www.ic.unicamp.br/~meidanis/courses/mc336/2006s2/funcional/L-99_Ninety-Nine_Lisp_Problems.html Ninety Nine Lisp Problems], |
+ | This is part of [[H-99:_Ninety-Nine_Haskell_Problems|Ninety-Nine Haskell Problems]], based on [https://prof.ti.bfh.ch/hew1/informatik3/prolog/p-99/ Ninety-Nine Prolog Problems]. |

− | which are themselves translations of [http://www.hta-bi.bfh.ch/~hew/informatik3/prolog/p-99/ Ninety-Nine Prolog Problems]. |
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− | 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. |
+ | 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 Haskell>,<solution in haskell> and <description of implementation> fields. |

== Miscellaneous problems == |
== Miscellaneous problems == |
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Example in Haskell: |
Example in Haskell: |
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− | <pre> |
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− | > length queens |
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− | 92 |
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− | > take 1 queens |
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− | [[4,2,7,3,6,8,5,1]] |
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− | </pre> |
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− | Solution: |
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<haskell> |
<haskell> |
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− | queens = queens' 8 |
+ | > length (queens 8) |

− | where queens' 0 = [[]] |
+ | 92 |

− | queens' n = [ try:qs | qs <- queens' (n-1), try <- [1..8], isSafe try qs] |
+ | > head (queens 8) |

− | isSafe try qs = not (try `elem` qs || sameDiag try qs) |
+ | [1,5,8,6,3,7,2,4] |

− | sameDiag try qs = any (\(colDist,q) -> abs(try - q) == colDist) $ zip [1..] qs |
+ | </haskell> |

− | </haskell> |
+ | [[99 questions/Solutions/90 | Solutions]] |

− | 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. |
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− | 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. |
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− | |||

== Problem 91 == |
== Problem 91 == |
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(**) Knight's tour |
(**) Knight's tour |
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− | Another famous problem is this one: How can a knight jump on an NxN chessboard in such a way that it visits every square exactly once? |
+ | Another famous problem is this one: How can a knight jump on an NxN chessboard in such a way that it visits every square exactly once? A set of solutions is given on the [[The_Knights_Tour]] page. |

Hints: Represent the squares by pairs of their coordinates of the form X/Y, where both X and Y are integers between 1 and N. (Note that '/' is just a convenient functor, not division!) Define the relation jump(N,X/Y,U/V) to express the fact that a knight can jump from X/Y to U/V on a NxN chessboard. And finally, represent the solution of our problem as a list of N*N knight positions (the knight's tour). |
Hints: Represent the squares by pairs of their coordinates of the form X/Y, where both X and Y are integers between 1 and N. (Note that '/' is just a convenient functor, not division!) Define the relation jump(N,X/Y,U/V) to express the fact that a knight can jump from X/Y to U/V on a NxN chessboard. And finally, represent the solution of our problem as a list of N*N knight positions (the knight's tour). |
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Example in Haskell: |
Example in Haskell: |
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− | <pre> |
+ | |

+ | <haskell> |
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Knights> head $ knightsTo 8 (1,1) |
Knights> head $ knightsTo 8 (1,1) |
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[(2,7),(3,5),(5,6),(4,8),(3,6),(4,4),(6,5),(4,6), |
[(2,7),(3,5),(5,6),(4,8),(3,6),(4,4),(6,5),(4,6), |
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(6,2),(8,1),(7,3),(5,2),(7,1),(8,3),(7,5),(8,7), |
(6,2),(8,1),(7,3),(5,2),(7,1),(8,3),(7,5),(8,7), |
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(6,8),(4,7),(2,8),(1,6),(2,4),(1,2),(3,1),(2,3)] |
(6,8),(4,7),(2,8),(1,6),(2,4),(1,2),(3,1),(2,3)] |
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− | </pre> |
+ | </haskell> |

− | Solution: |
+ | [[99 questions/Solutions/91 | Solutions]] |

− | <haskell> |
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− | module Knights where |
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− | import Data.List |
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− | type Square = (Int, Int) |
+ | == Problem 92 == |

− | -- Possible knight moves from a given square on an nxn board |
+ | (***) Von Koch's conjecture |

− | knightMoves :: Int -> Square -> [Square] |
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− | knightMoves n (x, y) = filter (onBoard n) |
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− | [(x+2, y+1), (x+2, y-1), (x+1, y+2), (x+1, y-2), |
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− | (x-1, y+2), (x-1, y-2), (x-2, y+1), (x-2, y-1)] |
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− | -- Is the square within an nxn board? |
+ | Several years ago I met a mathematician who was intrigued by a problem for which he didn't know a solution. His name was Von Koch, and I don't know whether the problem has been solved since. |

− | onBoard :: Int -> Square -> Bool |
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− | onBoard n (x, y) = 1 <= x && x <= n && 1 <= y && y <= n |
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− | -- Knight's tours on an nxn board ending at the given square |
+ | https://prof.ti.bfh.ch/hew1/informatik3/prolog/p-99/p92a.gif |

− | knightsTo :: Int -> Square -> [[Square]] |
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− | knightsTo n finish = [pos:path | (pos, path) <- tour (n*n)] |
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− | where tour 1 = [(finish, [])] |
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− | tour k = [(pos', pos:path) | |
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− | (pos, path) <- tour (k-1), |
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− | pos' <- sortImage (entrances path) |
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− | (filter (`notElem` path) (knightMoves n pos))] |
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− | entrances path pos = |
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− | length (filter (`notElem` path) (knightMoves n pos)) |
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− | -- Closed knight's tours on an nxn board |
+ | Anyway the puzzle goes like this: Given a tree with N nodes (and hence N-1 edges). Find a way to enumerate the nodes from 1 to N and, accordingly, the edges from 1 to N-1 in such a way, that for each edge K the difference of its node numbers equals to K. The conjecture is that this is always possible. |

− | closedKnights :: Int -> [[Square]] |
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− | closedKnights n = [pos:path | (pos, path) <- tour (n*n), pos == start] |
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− | where tour 1 = [(finish, [])] |
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− | tour k = [(pos', pos:path) | |
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− | (pos, path) <- tour (k-1), |
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− | pos' <- sortImage (entrances path) |
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− | (filter (`notElem` path) (knightMoves n pos))] |
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− | entrances path pos |
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− | | pos == start = 100 -- don't visit start until there are no others |
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− | | otherwise = length (filter (`notElem` path) (knightMoves n pos)) |
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− | start = (1,1) |
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− | finish = (2,3) |
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− | -- Sort by comparing the image of list elements under a function f. |
+ | For small trees the problem is easy to solve by hand. However, for larger trees, and 14 is already very large, it is extremely difficult to find a solution. And remember, we don't know for sure whether there is always a solution! |

− | -- These images are saved to avoid recomputation. |
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− | sortImage :: Ord b => (a -> b) -> [a] -> [a] |
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− | sortImage f xs = map snd (sortBy cmpFst [(f x, x) | x <- xs]) |
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− | where cmpFst x y = compare (fst x) (fst y) |
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− | </haskell> |
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− | This has a similar structure to the 8 Queens problem, except that we apply a heuristic invented by Warnsdorff: when considering next possible moves, we prefer squares with fewer open entrances. |
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− | This speeds things up enormously, and finds the first solution to boards smaller than 76x76 without backtracking. |
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+ | Write a predicate that calculates a numbering scheme for a given tree. What is the solution for the larger tree pictured below? |
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− | Solution 2: |
+ | https://prof.ti.bfh.ch/hew1/informatik3/prolog/p-99/p92b.gif |

− | |||

− | <haskell> |
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− | knights :: Int -> [[(Int,Int)]] |
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− | knights n = loop (n*n) [[(1,1)]] |
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− | where loop 1 = map reverse . id |
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− | loop i = loop (i-1) . concatMap nextMoves |
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− | |||

− | nextMoves already@(x:xs) = [next:already | next <- possible] |
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− | where possible = filter (\x -> on_board x && not (x `elem` already)) $ jumps x |
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− | |||

− | jumps (x,y) = [(x+a, y+b) | (a,b) <- [(1,2), (2,1), (2,-1), (1,-2), (-1,-2), (-2,-1), (-2,1), (-1,2)]] |
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− | on_board (x,y) = (x >= 1) && (x <= n) && (y >= 1) && (y <= n) |
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− | </haskell> |
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− | |||

− | This is just the naive backtracking approach. I tried a speedup using Data.Map, but the code got too verbose to post. |
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− | |||

− | == Problem 92 == |
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− | |||

− | <Problem description> |
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− | |||

− | Example: |
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− | <pre> |
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− | <example in lisp> |
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− | </pre> |
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Example in Haskell: |
Example in Haskell: |
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− | <pre> |
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− | <example in Haskell> |
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− | </pre> |
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− | Solution: |
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<haskell> |
<haskell> |
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− | <solution in haskell> |
+ | > head $ vonKoch [(1,6),(2,6),(3,6),(4,6),(5,6),(5,7),(5,8),(8,9),(5,10),(10,11),(11,12),(11,13),(13,14)] |

+ | [6,7,8,9,3,4,10,11,5,12,2,13,14,1] |
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</haskell> |
</haskell> |
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− | <description of implementation> |
+ | [[99 questions/Solutions/92 | Solutions]] |

+ | |||

== Problem 93 == |
== Problem 93 == |
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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!). |
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!). |
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+ | |||

+ | Division should be interpreted as operating on rationals, and division by zero should be avoided. |
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Example in Haskell: |
Example in Haskell: |
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− | <pre> |
+ | |

− | P93> putStr $ unlines $ puzzle [2,3,5,7,11] |
+ | <haskell> |

+ | P93> mapM_ putStrLn $ puzzle [2,3,5,7,11] |
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2 = 3-(5+7-11) |
2 = 3-(5+7-11) |
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2 = 3-5-(7-11) |
2 = 3-5-(7-11) |
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2-(3-5)+7 = 11 |
2-(3-5)+7 = 11 |
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2-3+5+7 = 11 |
2-3+5+7 = 11 |
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− | </pre> |
+ | </haskell> |

+ | |||

The other two solutions alluded to in the problem description are dropped by the Haskell solution as trivial variants: |
The other two solutions alluded to in the problem description are dropped by the Haskell solution as trivial variants: |
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+ | |||

<pre> |
<pre> |
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2 = 3-(5+(7-11)) |
2 = 3-(5+(7-11)) |
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Line 128: | Line 117: | ||

</pre> |
</pre> |
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− | Solution: |
+ | [[99 questions/Solutions/93 | Solutions]] |

− | <haskell> |
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− | module P93 where |
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− | import Control.Monad |
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− | import Data.List |
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− | import Data.Maybe |
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− | |||

− | type Equation = (Expr, Expr) |
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− | -- cache value in binary expression |
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− | data Expr = Const Integer | Binary Value Expr Op Expr |
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− | deriving (Eq, Show) |
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− | data Op = Plus | Minus | Multiply | Divide |
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− | deriving (Bounded, Eq, Enum, Show) |
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− | type Value = Rational |
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− | |||

− | -- top-level function: all correct equations generated from the list of |
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− | -- numbers, as pretty strings. |
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− | puzzle :: [Integer] -> [String] |
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− | puzzle ns = map (flip showsEquation "") (equations ns) |
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− | |||

− | -- generate all correct equations from the list of numbers |
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− | equations :: [Integer] -> [Equation] |
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− | equations [] = error "empty list of numbers" |
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− | equations [n] = error "only one number" |
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− | equations ns = [(e1, e2) | |
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− | (ns1, ns2) <- splits ns, |
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− | e1 <- exprs ns1, e2 <- exprs ns2, |
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− | value e1 == value e2] |
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− | |||

− | -- generate all expressions from the numbers, except those containing |
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− | -- a division by zero, or redundant right-associativity. |
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− | exprs :: [Integer] -> [Expr] |
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− | exprs [n] = [Const n] |
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− | exprs ns = [Binary n e1 op e2 | (ns1, ns2) <- splits ns, |
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− | e1 <- exprs ns1, e2 <- exprs ns2, op <- [minBound..maxBound], |
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− | not (right_associative op e2), |
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− | n <- maybeToList (apply op (value e1) (value e2))] |
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− | |||

− | -- splittings of a list into two non-empty lists |
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− | splits :: [a] -> [([a],[a])] |
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− | splits xs = tail (init (zip (inits xs) (tails xs))) |
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− | |||

− | -- the cached values of an expression |
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− | value :: Expr -> Value |
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− | value (Const n) = fromInteger n |
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− | value (Binary n _ _ _) = n |
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− | |||

− | -- applying an operator to arguments may fail (division by zero) |
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− | apply :: Op -> Value -> Value -> Maybe Value |
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− | apply Plus x y = Just (x + y) |
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− | apply Minus x y = Just (x - y) |
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− | apply Multiply x y = Just (x * y) |
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− | apply Divide x 0 = Nothing |
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− | apply Divide x y = Just (x / y) |
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− | |||

− | -- e1 op (e2 op' e3) == (e1 op e2) op' e3 |
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− | right_associative :: Op -> Expr -> Bool |
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− | right_associative Plus (Binary _ _ Plus _) = True |
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− | right_associative Plus (Binary _ _ Minus _) = True |
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− | right_associative Multiply (Binary _ _ Multiply _) = True |
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− | right_associative Multiply (Binary _ _ Divide _) = True |
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− | right_associative _ _ = False |
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− | |||

− | -- Printing of equations and expressions |
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− | |||

− | showsEquation :: Equation -> ShowS |
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− | showsEquation (l, r) = showsExprPrec 0 l . showString " = " . showsExprPrec 0 r |
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− | |||

− | -- all operations are left associative |
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− | showsExprPrec :: Int -> Expr -> ShowS |
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− | showsExprPrec _ (Const n) = shows n |
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− | showsExprPrec p (Binary _ e1 op e2) = showParen (p > op_prec) $ |
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− | showsExprPrec op_prec e1 . showString (opName op) . |
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− | showsExprPrec (op_prec+1) e2 |
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− | where op_prec = precedence op |
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− | |||

− | precedence :: Op -> Int |
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− | precedence Plus = 6 |
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− | precedence Minus = 6 |
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− | precedence Multiply = 7 |
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− | precedence Divide = 7 |
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− | |||

− | opName :: Op -> String |
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− | opName Plus = "+" |
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− | opName Minus = "-" |
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− | opName Multiply = "*" |
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− | opName Divide = "/" |
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− | </haskell> |
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− | I've used rational division, because that's what the Prolog solution does. |
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− | Unlike the Prolog solution, I've eliminated solutions like |
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− | <tt>"1+(2+3) = 6"</tt> as a trivial variant of <tt>"1+2+3 = 6"</tt> (cf the function <tt>right_associative</tt>). |
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− | Apart from that, the Prolog solution is shorter because it uses built-in evaluation and printing of expressions. |
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== Problem 94 == |
== Problem 94 == |
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− | <Problem description> |
+ | (***) Generate K-regular simple graphs with N nodes |

− | Example: |
+ | In a K-regular graph all nodes have a degree of K; i.e. the number of edges incident in each node is K. How many (non-isomorphic!) 3-regular graphs with 6 nodes are there? |

− | <pre> |
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− | <example in lisp> |
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− | </pre> |
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− | Example in Haskell: |
+ | [https://prof.ti.bfh.ch/hew1/informatik3/prolog/p-99/p94.txt Sample results] |

− | <pre> |
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− | <example in Haskell> |
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− | </pre> |
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Solution: |
Solution: |
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<description of implementation> |
<description of implementation> |
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− | |||

− | == Problem 95 == |
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− | |||

− | (**) English number words |
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− | |||

− | 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. |
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− | |||

− | Example in Haskell: |
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− | <pre> |
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− | > numbers 175 |
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− | one-seven-five |
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− | </pre> |
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− | |||

− | Solution: |
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− | <haskell> |
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− | import Data.List |
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− | import Data.Maybe |
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− | |||

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

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

− | == Problem 96 == |
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− | |||

− | (**) Syntax checker |
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− | |||

− | In a certain programming language (Ada) identifiers are defined by the [http://www.hta-bi.bfh.ch/~hew/informatik3/prolog/p-99/p96.gif syntax diagram]. Transform the syntax diagram into a system of syntax diagrams which do not contain loops; i.e. which are purely recursive. Using these modified diagrams, write a predicate identifier/1 that can check whether or not a given string is a legal identifier. |
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− | |||

− | Example in Prolog: |
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− | <pre> |
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− | % identifier(Str) :- Str is a legal identifier |
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− | </pre> |
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− | |||

− | Example in Haskell: |
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− | <pre> |
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− | > identifier "this-is-a-long-identifier" |
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− | True |
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− | > identifier "this-ends-in-" |
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− | False |
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− | > identifier "two--hyphens" |
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− | False |
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− | </pre> |
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− | |||

− | Solution: |
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− | <haskell> |
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− | identifier [] = False |
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− | identifier (x:xs) = letter x && loop xs |
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− | where loop [] = True |
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− | loop (x':xs') | x' == '-' = (not . null) xs' && (letter (head xs') || digit (head xs')) && loop (tail xs') |
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− | | letter x' = loop xs' |
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− | | digit x' = loop xs' |
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− | | otherwise = False |
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− | letter x = x `elem` (['a'..'z'] ++ ['A'..'Z']) |
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− | digit x = x `elem` ['0'..'9'] |
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− | </haskell> |
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− | |||

− | Simple functional transcription of the diagram. |
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− | |||

− | Another direct transcription of the diagram: |
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− | <haskell> |
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− | identifier :: String -> Bool |
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− | identifier (c:cs) = isLetter c && hyphen cs |
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− | where hyphen [] = True |
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− | hyphen ('-':cs) = alphas cs |
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− | hyphen cs = alphas cs |
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− | alphas [] = False |
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− | alphas (c:cs) = isAlpha c && hyphen cs |
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− | </haskell> |
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− | The functions <tt>hyphen</tt> and <tt>alphas</tt> correspond to states in the automaton at the start of the loop and before a compulsory alphanumeric, respectively. |
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− | == Problem 97 == |
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− | |||

− | (**) Sudoku |
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− | |||

− | Sudoku puzzles go like this: |
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− | |||

− | <pre> |
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− | Problem statement Solution |
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− | |||

− | . . 4 | 8 . . | . 1 7 9 3 4 | 8 2 5 | 6 1 7 |
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− | | | | | |
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− | 6 7 . | 9 . . | . . . 6 7 2 | 9 1 4 | 8 5 3 |
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− | | | | | |
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− | 5 . 8 | . 3 . | . . 4 5 1 8 | 6 3 7 | 9 2 4 |
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− | --------+---------+-------- --------+---------+-------- |
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− | 3 . . | 7 4 . | 1 . . 3 2 5 | 7 4 8 | 1 6 9 |
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− | | | | | |
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− | . 6 9 | . . . | 7 8 . 4 6 9 | 1 5 3 | 7 8 2 |
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− | | | | | |
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− | . . 1 | . 6 9 | . . 5 7 8 1 | 2 6 9 | 4 3 5 |
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− | --------+---------+-------- --------+---------+-------- |
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− | 1 . . | . 8 . | 3 . 6 1 9 7 | 5 8 2 | 3 4 6 |
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− | | | | | |
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− | . . . | . . 6 | . 9 1 8 5 3 | 4 7 6 | 2 9 1 |
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− | | | | | |
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− | 2 4 . | . . 1 | 5 . . 2 4 6 | 3 9 1 | 5 7 8 |
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− | </pre> |
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− | |||

− | 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. |
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− | |||

− | Solutions: see [[Sudoku]] |
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− | |||

− | == Problem 98 == |
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− | |||

− | <Problem description> |
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− | |||

− | Example: |
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− | <pre> |
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− | <example in lisp> |
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− | </pre> |
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− | |||

− | Example in Haskell: |
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− | <pre> |
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− | <example in Haskell> |
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− | </pre> |
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− | |||

− | Solution: |
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− | <haskell> |
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− | <solution in haskell> |
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− | </haskell> |
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− | |||

− | <description of implementation> |
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− | |||

− | == Problem 99 == |
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− | |||

− | (***) Crossword puzzle |
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− | |||

− | Given an empty (or almost empty) framework of a crossword puzzle and a set of words. The problem is to place the words into the framework. |
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− | The particular crossword puzzle is specified in a text file which first lists the words (one word per line) in an arbitrary order. Then, after an empty line, the crossword framework is defined. In this framework specification, an empty character location is represented by a dot (.). In order to make the solution easier, character locations can also contain predefined character values. The puzzle opposite is defined in the file [http://www.hta-bi.bfh.ch/~hew/informatik3/prolog/p-99/p99a.dat p99a.dat], other examples are [http://www.hta-bi.bfh.ch/~hew/informatik3/prolog/p-99/p99b.dat p99b.dat] and [http://www.hta-bi.bfh.ch/~hew/informatik3/prolog/p-99/p99d.dat p99d.dat]. There is also an example of a puzzle ([http://www.hta-bi.bfh.ch/~hew/informatik3/prolog/p-99/p99c.dat p99c.dat]) which does not have a solution. |
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− | Words are strings (character lists) of at least two characters. A horizontal or vertical sequence of character places in the crossword puzzle framework is called a site. Our problem is to find a compatible way of placing words onto sites. |
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− | Hints: (1) The problem is not easy. You will need some time to thoroughly understand it. So, don't give up too early! And remember that the objective is a clean solution, not just a quick-and-dirty hack! |
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− | (2) Reading the data file is a tricky problem for which a solution is provided in the file [http://www.hta-bi.bfh.ch/~hew/informatik3/prolog/p-99/p99-readfile.pl p99-readfile.pl]. See the predicate read_lines/2. |
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− | (3) For efficiency reasons it is important, at least for larger puzzles, to sort the words and the sites in a particular order. For this part of the problem, the solution of P28 may be very helpful. |
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− | Example in Haskell: |
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− | <pre> |
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− | </pre> |
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− | Solution: |
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− | <haskell> |
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− | </haskell> |
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[[Category:Tutorials]] |
[[Category:Tutorials]] |

## Revision as of 17:08, 15 July 2010

This is part of Ninety-Nine Haskell Problems, based on Ninety-Nine Prolog 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 Haskell>,<solution in haskell> and <description of implementation> fields.

## 1 Miscellaneous problems

## 2 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 8) 92 > head (queens 8) [1,5,8,6,3,7,2,4]

## 3 Problem 91

(**) Knight's tour

Another famous problem is this one: How can a knight jump on an NxN chessboard in such a way that it visits every square exactly once? A set of solutions is given on the The_Knights_Tour page.

Hints: Represent the squares by pairs of their coordinates of the form X/Y, where both X and Y are integers between 1 and N. (Note that '/' is just a convenient functor, not division!) Define the relation jump(N,X/Y,U/V) to express the fact that a knight can jump from X/Y to U/V on a NxN chessboard. And finally, represent the solution of our problem as a list of N*N knight positions (the knight's tour).

There are two variants of this problem:

- find a tour ending at a particular square
- find a circular tour, ending a knight's jump from the start (clearly it doesn't matter where you start, so choose (1,1))

Example in Haskell:

Knights> head $ knightsTo 8 (1,1) [(2,7),(3,5),(5,6),(4,8),(3,6),(4,4),(6,5),(4,6), (5,4),(7,5),(6,3),(5,5),(4,3),(2,4),(1,6),(2,8), (4,7),(6,8),(8,7),(6,6),(4,5),(6,4),(5,2),(7,1), (8,3),(6,2),(8,1),(7,3),(8,5),(7,7),(5,8),(3,7), (1,8),(2,6),(3,4),(1,5),(2,3),(3,1),(1,2),(3,3), (1,4),(2,2),(4,1),(5,3),(7,4),(8,2),(6,1),(4,2), (2,1),(1,3),(2,5),(1,7),(3,8),(5,7),(7,8),(8,6), (6,7),(8,8),(7,6),(8,4),(7,2),(5,1),(3,2),(1,1)] Knights> head $ closedKnights 8 [(1,1),(3,2),(1,3),(2,1),(3,3),(5,4),(6,6),(4,5), (2,6),(1,8),(3,7),(5,8),(4,6),(2,5),(4,4),(5,6), (6,4),(8,5),(7,7),(6,5),(5,3),(6,1),(4,2),(6,3), (8,2),(7,4),(5,5),(3,4),(1,5),(2,7),(4,8),(3,6), (1,7),(3,8),(5,7),(7,8),(8,6),(6,7),(8,8),(7,6), (8,4),(7,2),(5,1),(4,3),(3,5),(1,4),(2,2),(4,1), (6,2),(8,1),(7,3),(5,2),(7,1),(8,3),(7,5),(8,7), (6,8),(4,7),(2,8),(1,6),(2,4),(1,2),(3,1),(2,3)]

## 4 Problem 92

(***) Von Koch's conjecture

Several years ago I met a mathematician who was intrigued by a problem for which he didn't know a solution. His name was Von Koch, and I don't know whether the problem has been solved since.

Anyway the puzzle goes like this: Given a tree with N nodes (and hence N-1 edges). Find a way to enumerate the nodes from 1 to N and, accordingly, the edges from 1 to N-1 in such a way, that for each edge K the difference of its node numbers equals to K. The conjecture is that this is always possible.

For small trees the problem is easy to solve by hand. However, for larger trees, and 14 is already very large, it is extremely difficult to find a solution. And remember, we don't know for sure whether there is always a solution!

Write a predicate that calculates a numbering scheme for a given tree. What is the solution for the larger tree pictured below?

Example in Haskell:

> head $ vonKoch [(1,6),(2,6),(3,6),(4,6),(5,6),(5,7),(5,8),(8,9),(5,10),(10,11),(11,12),(11,13),(13,14)] [6,7,8,9,3,4,10,11,5,12,2,13,14,1]

## 5 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!).

Division should be interpreted as operating on rationals, and division by zero should be avoided.

Example in Haskell:

P93> mapM_ putStrLn $ 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 2-3+5+7 = 11

The other two solutions alluded to in the problem description are dropped by the Haskell solution as trivial variants:

2 = 3-(5+(7-11)) 2-3+(5+7) = 11

## 6 Problem 94

(***) Generate K-regular simple graphs with N nodes

In a K-regular graph all nodes have a degree of K; i.e. the number of edges incident in each node is K. How many (non-isomorphic!) 3-regular graphs with 6 nodes are there?

Solution:

<solution in haskell>

<description of implementation>