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NinetyNine F# Problems  Problems 21  28  Lists again
These are F# solutions of NinetyNine Haskell Problems which are themselves translations of NinetyNine Lisp Problems and NinetyNine Prolog Problems. The solutions are hidden so you can try to solve them yourself.
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/// These are F# solutions of NinetyNine Haskell Problems
/// (http://www.haskell.org/haskellwiki/H99:_NinetyNine_Haskell_Problems),
/// which are themselves translations of NinetyNine Lisp Problems
/// (http://www.ic.unicamp.br/~meidanis/courses/mc336/2006s2/funcional/L99_NinetyNine_Lisp_Problems.html)
/// and NinetyNine Prolog Problems
/// (https://sites.google.com/site/prologsite/prologproblems).
///
/// If you would like to contribute a solution or fix any bugs, send
/// an email to paks at kitiara dot org with the subject "99 F# problems".
/// I'll try to update the problem as soon as possible.
///
/// The problems have different levels of difficulty. Those marked with a single asterisk (*)
/// are easy. If you have successfully solved the preceeding problems you should be able to
/// solve them within a few (say 15) minutes. Problems marked with two asterisks (**) are of
/// intermediate difficulty. If you are a skilled F# programmer it shouldn't take you more than
/// 3090 minutes to solve them. Problems marked with three asterisks (***) are more difficult.
/// You may need more time (i.e. a few hours or more) to find a good solution
///
/// Though the problems number from 1 to 99, there are some gaps and some additions marked with
/// letters. There are actually only 88 problems.

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/// Example:
/// * (insertat 'alfa '(a b c d) 2)
/// (A ALFA B C D)
///
/// Example in F#:
///
/// > insertAt 'X' (List.ofSeq "abcd") 2;;
/// val it : char list = ['a'; 'X'; 'b'; 'c'; 'd']
(Solution 1)
(Solution 2)

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/// Example:
/// * (range 4 9)
/// (4 5 6 7 8 9)
///
/// Example in F#:
///
/// > range 4 9;;
/// val it : int list = [4; 5; 6; 7; 8; 9]
(Solution)

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/// Example:
/// * (rndselect '(a b c d e f g h) 3)
/// (E D A)
///
/// Example in F#:
///
/// > rnd_select "abcdefgh" 3;;
/// val it : seq<char> = seq ['e'; 'a'; 'h']
(Solution)

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/// Example:
/// * (rndselect 6 49)
/// (23 1 17 33 21 37)
///
/// Example in F#:
///
/// > diff_select 6 49;;
/// val it : int list = [27; 20; 22; 9; 15; 29]
// using problem 23
(Solution)
(Solution)

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/// Example:
/// * (rndpermu '(a b c d e f))
/// (B A D C E F)
///
/// Example in F#:
///
/// > rnd_permu < List.ofSeq "abcdef";;
/// val it : char list = ['b'; 'c'; 'd'; 'f'; 'e'; 'a']
(Solution 1)
(Solution 2)

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/// In how many ways can a committee of 3 be chosen from a group of 12 people? We all know that
/// there are C(12,3) = 220 possibilities (C(N,K) denotes the wellknown binomial coefficients). For
/// pure mathematicians, this result may be great. But we want to really generate all the
/// possibilities in a list.
///
/// Example:
/// * (combinations 3 '(a b c d e f))
/// ((A B C) (A B D) (A B E) ... )
///
/// Example in F#:
///
/// > combinations 3 ['a' .. 'f'];;
/// val it : char list list =
/// [['a'; 'b'; 'c']; ['a'; 'b'; 'd']; ['a'; 'b'; 'e']; ['a'; 'b'; 'f'];
/// ['a'; 'c'; 'd']; ['a'; 'c'; 'e']; ['a'; 'c'; 'f']; ['a'; 'd'; 'e'];
/// ['a'; 'd'; 'f']; ['a'; 'e'; 'f']; ['b'; 'c'; 'd']; ['b'; 'c'; 'e'];
/// ['b'; 'c'; 'f']; ['b'; 'd'; 'e']; ['b'; 'd'; 'f']; ['b'; 'e'; 'f'];
/// ['c'; 'd'; 'e']; ['c'; 'd'; 'f']; ['c'; 'e'; 'f']; ['d'; 'e'; 'f']]
(Solution 1)
(Solution 2)

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/// a) In how many ways can a group of 9 people work in 3 disjoint subgroups of 2, 3
/// and 4 persons? Write a function that generates all the possibilities and returns them
/// in a list.
///
/// Example:
/// * (group3 '(aldo beat carla david evi flip gary hugo ida))
/// ( ( (ALDO BEAT) (CARLA DAVID EVI) (FLIP GARY HUGO IDA) )
/// ... )
///
/// b) Generalize the above predicate in a way that we can specify a list of group sizes
/// and the predicate will return a list of groups.
///
/// Example:
/// * (group '(aldo beat carla david evi flip gary hugo ida) '(2 2 5))
/// ( ( (ALDO BEAT) (CARLA DAVID) (EVI FLIP GARY HUGO IDA) )
/// ... )
///
/// Note that we do not want permutations of the group members; i.e. ((ALDO BEAT) ...)
/// is the same solution as ((BEAT ALDO) ...). However, we make a difference between
/// ((ALDO BEAT) (CARLA DAVID) ...) and ((CARLA DAVID) (ALDO BEAT) ...).
///
/// You may find more about this combinatorial problem in a good book on discrete
/// mathematics under the term "multinomial coefficients".
///
/// Example in F#:
///
/// > group [2;3;4] ["aldo";"beat";"carla";"david";"evi";"flip";"gary";"hugo";"ida"];;
/// val it : string list list list =
/// [[["aldo"; "beat"]; ["carla"; "david"; "evi"];
/// ["flip"; "gary"; "hugo"; "ida"]];...]
/// (altogether 1260 solutions)
///
/// > group [2;2;5] ["aldo";"beat";"carla";"david";"evi";"flip";"gary";"hugo";"ida"];;
/// val it : string list list list =
/// [[["aldo"; "beat"]; ["carla"; "david"];
/// ["evi"; "flip"; "gary"; "hugo"; "ida"]];...]
/// (altogether 756 solutions)
(Solution)

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/// a) We suppose that a list contains elements that are lists themselves. The objective
/// is to sort the elements of this list according to their length. E.g. short lists first,
/// longer lists later, or vice versa.
///
/// Example:
/// * (lsort '((a b c) (d e) (f g h) (d e) (i j k l) (m n) (o)))
/// ((O) (D E) (D E) (M N) (A B C) (F G H) (I J K L))
///
/// Example in F#:
///
/// > lsort ["abc";"de";"fgh";"de";"ijkl";"mn";"o"];;
/// val it : string list = ["o"; "de"; "de"; "mn"; "abc"; "fgh"; "ijkl"]
///
/// b) Again; we suppose that a list contains elements that are lists themselves. But this
/// time the objective is to sort the elements of this list according to their length
/// frequency; i.e.; in the default; where sorting is done ascendingly; lists with rare
/// lengths are placed first; others with a more frequent length come later.
///
/// Example:
/// * (lfsort '((a b c) (d e) (f g h) (d e) (i j k l) (m n) (o)))
/// ((i j k l) (o) (a b c) (f g h) (d e) (d e) (m n))
///
/// Example in F#:
///
/// > lfsort ["abc"; "de"; "fgh"; "de"; "ijkl"; "mn"; "o"];;
/// val it : string list = ["ijkl"; "o"; "abc"; "fgh"; "de"; "de"; "mn"]
(Solution)

let insertAt x xs n =
let rec insAt acc xs n =
match xs, n with
 [], 1 > List.rev (x::acc)
 [], _ > failwith "Empty list you fool!"
 xs, 1 > List.rev (x::acc) @ xs
 x::xs, n > insAt (x::acc) xs (n  1)
insAt [] xs n
let rec insertAt' x xs n =
match xs, n with
 [], 1 > [x]
 [], _ > failwith "Empty list you fool!"
 _, 1 > x::xs
 y::ys, n > y::insertAt' x ys (n  1)
let range a b = [a..b]
let rnd_select xs n =
let rndSeq = let r = new System.Random() in seq { while true do yield r.Next() }
xs > Seq.zip rndSeq > Seq.sortBy fst > Seq.map snd > Seq.take n
let diff_select n m = rnd_select (seq { 1 .. m }) n > List.ofSeq
let diff_select' n m =
let rndSeq = let r = new System.Random() in Seq.initInfinite(ignore >> r.Next )
seq { 1 .. m } > Seq.zip rndSeq > Seq.sortBy fst > Seq.map snd > Seq.take n > List.ofSeq
// using problem 23
let rnd_permu xs = List.length xs > rnd_select xs
let rnd_permu' xs =
let rec rndSeq =
let r = new System.Random()
seq { while true do yield r.Next() }
xs > Seq.zip rndSeq > Seq.sortBy fst > Seq.map snd > List.ofSeq
// as a bonus you get the powerset of xs with combinations 0 xs
let rec combinations n xs =
match xs, n with
 [],_ > [[]]
 xs, 1 > [for x in xs do yield [x]]
 x::xs, n >
[for ys in combinations (n1) xs do
yield x::ys
if List.length xs > n then
yield! combinations n xs
else
yield xs]
let rec combinations' n xs =
let rec tails = function
 [] > [[]]
 _::ys as xs > xs::tails ys
match xs, n with
 _, 0 > [[]]
 xs, n >
[ for tail in tails xs do
match tail with
 [] > ()
 y::xs' >
for ys in combinations' (n  1) xs' do
yield y::ys ]
let rec group ns xs =
let rec combination n xs =
match n,xs with
 0, xs > [([], xs)]
 _, [] > []
 n, x::xs >
let ts = [ for ys, zs in combination (n1) xs do yield (x::ys, zs)]
let ds = [ for ys, zs in combination n xs do yield (ys, x::zs)]
ts @ ds
match ns,xs with
 [], _ > [[]]
 n::ns, xs >
[ for g, rs in combination n xs do
for gs in group ns rs do
yield g::gs ]
let lsort xss = xss > List.sortBy Seq.length
let lfsort xss = xss > Seq.groupBy (Seq.length) > Seq.sortBy (snd >> Seq.length) > Seq.collect snd > List.ofSeq
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