{"id":5947,"date":"2020-06-03T07:27:01","date_gmt":"2020-06-03T05:27:01","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=5947"},"modified":"2021-01-02T22:31:07","modified_gmt":"2021-01-02T20:31:07","slug":"numero-como-suma-de-sus-digitos","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/numero-como-suma-de-sus-digitos\/","title":{"rendered":"N\u00famero como suma de sus d\u00edgitos"},"content":{"rendered":"<p>El n\u00famero 23 se puede escribir de 4 formas como suma de sus d\u00edgitos<\/p>\n<pre lang=\"text\"> \n   2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 3\n   2 + 2 + 2 + 2 + 2 + 2 + 2 + 3 + 3 + 3\n   2 + 2 + 2 + 2 + 3 + 3 + 3 + 3 + 3\n   2 + 3 + 3 + 3 + 3 + 3 + 3 + 3\n<\/pre>\n<p>La de menor n\u00famero de sumando es la \u00faltima, que tiene 8 sumandos.<\/p>\n<p>Definir las funciones<\/p>\n<pre lang=\"text\"> \n   minimoSumandosDigitos        :: Integer -> Integer\n   graficaMinimoSumandosDigitos :: Integer -> IO ()\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li>(minimoSumandosDigitos n) es el menor n\u00famero de d\u00edgitos de n cuya suma es n. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">   \n     minimoSumandosDigitos 23    ==  8\n     minimoSumandosDigitos 232   ==  78\n     minimoSumandosDigitos 2323  ==  775\n     map minimoSumandosDigitos [10..20] == [10,11,6,5,5,3,6,5,4,3,10]\n<\/pre>\n<ul>\n<li>(graficaMinimoSumandosDigitos n) dibuja la gr\u00e1fica de (minimoSumandosDigitos k) par los k primeros n\u00fameros naturales. Por ejemplo, (graficaMinimoSumandosDigitos 300) dibuja<\/li>\n<\/ul>\n<p><a href=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2019\/03\/Numero_como_suma_de_sus_digitos.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2019\/03\/Numero_como_suma_de_sus_digitos.png?resize=640%2C480\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-4795\" srcset=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2019\/03\/Numero_como_suma_de_sus_digitos.png?w=640&amp;ssl=1 640w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2019\/03\/Numero_como_suma_de_sus_digitos.png?resize=300%2C225&amp;ssl=1 300w\" sizes=\"(max-width: 640px) 100vw, 640px\" data-recalc-dims=\"1\" \/><\/a><\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Test.QuickCheck\nimport Graphics.Gnuplot.Simple\nimport Data.List (nub, genericLength, sort)\nimport Data.Array (array, (!))\n\nminimoSumandosDigitos :: Integer -> Integer\nminimoSumandosDigitos n =\n  minimoSumandos (digitos n) n\n\n-- (digitos n) es el conjunto de los d\u00edgitos no nulos de n. Por ejemplo,\n--    digitos 2032  ==  [2,3]\ndigitos :: Integer -> [Integer]\ndigitos n =\n  nub [read [c] | c <- show n, c \/= '0']\n\n-- (minimoSumandos xs n) es el menor n\u00famero de elementos de la lista de\n-- enteros positivos xs (con posibles repeticiones) cuya suma es n. Por\n-- ejemplo, \n--    minimoSumandos [7,2,4] 11  ==  2\nminimoSumandos :: [Integer] -> Integer -> Integer\nminimoSumandos xs n =\n  minimum (map genericLength (sumas xs n))\n\n-- (sumas xs n) es la lista de elementos de la lista de enteros\n-- positivos xs (con posibles repeticiones) cuya suma es n. Por ejemplo,  \n--    sumas [7,2,4] 11  ==  [[7,2,2],[7,4]]\nsumas :: [Integer] -> Integer -> [[Integer]]\nsumas [] 0 = [[]]\nsumas [] _ = []\nsumas (x:xs) n\n  | x <= n    = map (x:) (sumas (x:xs) (n-x)) ++ sumas xs n\n  | otherwise = sumas xs n\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nminimoSumandosDigitos2 :: Integer -> Integer\nminimoSumandosDigitos2 n = aux n \n  where\n    aux 0 = 0\n    aux k = 1 + minimo [aux (k - x) | x <- ds,  k >= x]\n    ds    = digitos n\n    infinito = 10^100\n    minimo xs | null xs   = infinito\n              | otherwise = minimum xs\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\nminimoSumandosDigitos3 :: Integer -> Integer\nminimoSumandosDigitos3 n = v ! n\n  where\n    v   = array (0,n) [(i,f i) | i <- [0..n]]\n    f 0 = 0\n    f k = 1 + minimo [v ! (k - x) | x <- ds, k >= x]\n    ds       = digitos n\n    infinito = 10^100\n    minimo xs | null xs   = infinito\n              | otherwise = minimum xs\n        \n-- Equivalencia de las definiciones\n-- ================================\n\n-- La propiedad es\nprop_minimoSumandosDigitos :: Positive Integer -> Bool\nprop_minimoSumandosDigitos (Positive n) =\n  r1 == r2 && r2 == r3\n  where\n    r1 = minimoSumandosDigitos n\n    r2 = minimoSumandosDigitos n\n    r3 = minimoSumandosDigitos n\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheckWith (stdArgs {maxSize=9}) prop_minimoSumandosDigitos\n--    +++ OK, passed 100 tests.\n\n-- Definici\u00f3n de graficaMinimoSumandosDigitos\n-- ==========================================\n\ngraficaMinimoSumandosDigitos :: Integer -> IO ()\ngraficaMinimoSumandosDigitos n =\n  plotList [ Key Nothing\n           -- , PNG \"Numero_como_suma_de_sus_digitos.png\"\n           ]\n           [minimoSumandosDigitos k | k <- [0..n-1]]\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>El n\u00famero 23 se puede escribir de 4 formas como suma de sus d\u00edgitos 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 3 2 + 2 + 2 + 2 + 2 + 2 + 2 + 3 + 3 + 3 2&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[5],"tags":[8,258,10,340,24,11,95,6,32,33,14],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5947"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/comments?post=5947"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5947\/revisions"}],"predecessor-version":[{"id":5966,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5947\/revisions\/5966"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=5947"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=5947"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=5947"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}