{"id":6021,"date":"2021-02-01T06:00:10","date_gmt":"2021-02-01T04:00:10","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=6021"},"modified":"2021-02-08T11:18:06","modified_gmt":"2021-02-08T09:18:06","slug":"con-algun-nueve","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/con-algun-nueve\/","title":{"rendered":"Con alg\u00fan nueve"},"content":{"rendered":"<p>Definir las funciones<\/p>\n<pre lang=\"text\">\n   numerosConNueve :: [Integer]\n   conNueve :: Integer -> Integer\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li>numerosConNueve es la lista de los n\u00fameros con alg\u00fan d\u00edgito igual a 9. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     \u03bb> take 20 numerosConNueve\n     [9,19,29,39,49,59,69,79,89,90,91,92,93,94,95,96,97,98,99,109]\n<\/pre>\n<ul>\n<li>(conNueve n) es la cantidad de n\u00fameros enteros no negativos menores o iguales que n con alg\u00fan d\u00edgito igual a 9. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     conNueve 1   ==  0\n     conNueve 10  ==  1\n     conNueve 90  ==  10\n     length (show (conNueve (10^3)))      ==  3\n     length (show (conNueve (10^30)))     ==  30\n     length (show (conNueve (10^300)))    ==  300\n     length (show (conNueve (10^3000)))   ==  3000\n     length (show (conNueve (10^30000)))  ==  30000\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (genericLength)\nimport Test.QuickCheck (Property, (==>), quickCheck)\n\nnumerosConNueve :: [Integer]\nnumerosConNueve = filter tieneNueve [9..]\n\n-- (tieneNueve n) se verifica si alg\u00fan d\u00edgito de n es igual a 9. Por\n-- ejemplo,\n--    tieneNueve 2919  ==  True\n--    tieneNueve 2717  ==  False\ntieneNueve :: Integer -> Bool\ntieneNueve n = '9' `elem` show n\n\n-- 1\u00aa definici\u00f3n de conNueve\n-- =========================\n\nconNueve :: Integer -> Integer\nconNueve n =\n  genericLength (takeWhile (<= n) numerosConNueve)\n\n-- 2\u00aa definici\u00f3n de conNueve\n-- =========================\n\nconNueve2 :: Integer -> Integer\nconNueve2 0 = 0\nconNueve2 n\n  | tieneNueve n = 1 + conNueve2 (n-1)\n  | otherwise   = conNueve2 (n-1)\n\n-- 3\u00aa definici\u00f3n de conNueve\n-- =========================\n\nconNueve3 :: Integer -> Integer\nconNueve3 n = n + 1 - sinNueve n\n\n-- (sinNueve n) es la cantidad de n\u00fameros enteros no negativos menores\n-- o iguales que n con ning\u00fan d\u00edgito igual a 9. Por ejemplo,\n--    sinNueve 1   ==  2\n--    sinNueve 9   ==  9\n--    sinNueve 90  ==  81\nsinNueve :: Integer -> Integer\nsinNueve n = aux (digitos n)\n  where aux []     = 0\n        aux [9]    = 9\n        aux [x]    = x+1\n        aux (9:xs) = 9^(1 + length xs)\n        aux (x:xs) = x*9^(length xs) + aux xs\n\n-- (digitos n) es la lista de los d\u00edgitod de n. Por ejemplo,\n--    digitos 2021  ==  [2,0,2,1]\ndigitos :: Integer -> [Integer]\ndigitos n = [read [c] | c <- show n]\n\n-- 4\u00aa definici\u00f3n de conNueve\n-- =========================\n\nconNueve4 :: Integer -> Integer\nconNueve4 n = n + 1 - sinNueve2 n\n\nsinNueve2 :: Integer -> Integer\nsinNueve2 n = aux ds (length ds)\n  where ds = digitos n\n        aux []     _ = 0\n        aux [9]    _ = 9\n        aux [x]    _ = x+1\n        aux (9:xs) m = 9^m\n        aux (x:xs) m = x*9^(m-1) + aux xs (m-1)\n\n-- Comprobacion de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_conNueve_equiv :: Integer -> Property\nprop_conNueve_equiv n =\n  n >= 0 ==>\n  all (== (conNueve n))\n      [conNueve2 n,\n       conNueve3 n,\n       conNueve4 n]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_conNueve_equiv\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> conNueve (10^7)\n--    5217031\n--    (5.21 secs, 5,215,047,968 bytes)\n--    \u03bb> conNueve2 (10^7)\n--    5217031\n--    (8.25 secs, 5,651,944,872 bytes)\n--    \u03bb> conNueve3 (10^7)\n--    5217031\n--    (0.02 secs, 144,000 bytes)\n--    \u03bb> conNueve4 (10^7)\n--    5217031\n--    (0.02 secs, 145,904 bytes)\n--\n--    \u03bb> length (show (conNueve3 (10^30000)))\n--    30000\n--    (3.24 secs, 776,076,072 bytes)\n--    \u03bb> length (show (conNueve4 (10^30000)))\n--    30000\n--    (2.00 secs, 786,452,856 bytes)\n<\/pre>\n<h4>Nuevas soluciones<\/h4>\n<ul>\n<li>En los comentarios se pueden escribir nuevas soluciones.\n<li>El c\u00f3digo se debe escribir entre una l\u00ednea con &#60;pre lang=&quot;haskell&quot;&#62; y otra con &#60;\/pre&#62;\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Definir las funciones numerosConNueve :: [Integer] conNueve :: Integer -> Integer tales que numerosConNueve es la lista de los n\u00fameros con alg\u00fan d\u00edgito igual a 9. Por ejemplo, \u03bb> take 20 numerosConNueve [9,19,29,39,49,59,69,79,89,90,91,92,93,94,95,96,97,98,99,109] (conNueve n) es la cantidad de n\u00fameros enteros no negativos menores o iguales que n con alg\u00fan d\u00edgito igual a 9. Por&#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":[4],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6021"}],"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=6021"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6021\/revisions"}],"predecessor-version":[{"id":6061,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6021\/revisions\/6061"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=6021"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=6021"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=6021"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}