Los n\u00fameros del 1 al 3 se pueden escribir de dos formas, con el signo m\u00e1s o menos entre ellos, tales que su suma sea 0:<\/p>\n
\n 1+2-3 = 0\n -1-2+3 = 0\n<\/pre>\nDefinir la funci\u00f3n<\/p>\n
\n ceros :: Int -> [[Int]]\n<\/pre>\ntal que (ceros n) son las posibles formas de obtener cero sumando los n\u00fameros del 1 al n, con el signo m\u00e1s o menos entre ellos. Por ejemplo,<\/p>\n
\n ceros 3 == [[1,2,-3],[-1,-2,3]]\n ceros 4 == [[1,-2,-3,4],[-1,2,3,-4]]\n ceros 5 == []\n length (ceros 7) == 8\n take 3 (ceros 7) == [[1,2,-3,4,-5,-6,7],[1,2,-3,-4,5,6,-7],[1,-2,3,4,-5,6,-7]]\n<\/pre>\nSoluciones<\/h4>\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nceros :: Int -> [[Int]]\nceros n = [xs | xs <- candidatos n, sum xs == 0]\n\n-- (candidatos n) es la lista de los n\u00fameros del 1 al n con los signos\n-- m\u00e1s o menos. Por ejemplo,\n-- \u03bb> candidatos 1\n-- [[1],[-1]]\n-- \u03bb> candidatos 2\n-- [[1,2],[1,-2],[-1,2],[-1,-2]]\n-- \u03bb> candidatos 3\n-- [[1,2,3],[1,2,-3],[1,-2,3],[1,-2,-3],[-1,2,3],[-1,2,-3],[-1,-2,3],[-1,-2,-3]]\ncandidatos :: Int -> [[Int]]\ncandidatos n = productoCartesiano [[i,-i] | i <- [1..n]]\n\n-- (productoCartesiano xss) es el producto cartesiano de los conjuntos\n-- xss. Por ejemplo, \n-- \u03bb> productoCartesiano [[1,3],[2,5],[6,4]]\n-- [[1,2,6],[1,2,4],[1,5,6],[1,5,4],[3,2,6],[3,2,4],[3,5,6],[3,5,4]]\nproductoCartesiano :: [[a]] -> [[a]]\nproductoCartesiano [] = [[]]\nproductoCartesiano (xs:xss) =\n [x:ys | x <- xs, ys <- productoCartesiano xss]\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nceros2 :: Int -> [[Int]]\nceros2 n = [xs | xs <- candidatos2 n, sum xs == 0]\n\ncandidatos2 :: Int -> [[Int]]\ncandidatos2 n = mapM (\\i -> [i,-i]) [1..n]\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\nceros3 :: Int -> [[Int]]\nceros3 n = [xs | xs <- mapM (\\i -> [i,-i]) [1..n], sum xs == 0]\n\n-- 4\u00aa soluci\u00f3n\n-- ===========\n\nceros4 :: Integer -> [[Integer]]\nceros4 = map fst\n . filter ((==0) . snd)\n . sumas\n\n-- (sumas n) es la lista de las candidatos con los n primeros n\u00fameros y\n-- sus sumas. Por ejemplo,\n-- \u03bb> sumas 2\n-- [([1,2],3),([-1,2],1),([1,-2],-1),([-1,-2],-3)]\n-- \u03bb> sumas 3\n-- [([1,2,3],6),([-1,2,3],4),([1,-2,3],2),([-1,-2,3],0),([1,2,-3],0),\n-- ([-1,2,-3],-2),([1,-2,-3],-4),([-1,-2,-3],-6)]\nsumas :: Integer -> [([Integer],Integer)]\nsumas = foldr aux [([], 0)]\n . enumFromTo 1\n where\n aux n = concatMap (\\(ns', s') -> [(n : ns', s' + n), (-n : ns', s' - n)])\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- \u03bb> length (ceros 18)\n-- 0\n-- (4.14 secs, 1,094,854,776 bytes)\n-- \u03bb> length (ceros2 18)\n-- 0\n-- (1.16 secs, 375,541,680 bytes)\n-- \u03bb> length (ceros3 18)\n-- 0\n-- (1.13 secs, 375,540,192 bytes)\n-- \u03bb> length (ceros4 18)\n-- 0\n-- (0.69 secs, 157,316,688 bytes)\n<\/pre>\n","protected":false},"excerpt":{"rendered":"Los n\u00fameros del 1 al 3 se pueden escribir de dos formas, con el signo m\u00e1s o menos entre ellos, tales que su suma sea 0: 1+2-3 = 0 -1-2+3 = 0 Definir la funci\u00f3n ceros :: Int -> [[Int]] tal que (ceros n) son las posibles formas de obtener cero sumando los n\u00fameros del…<\/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":[8,58,438,38,94,80,10,437,11,6,40],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3766"}],"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=3766"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3766\/revisions"}],"predecessor-version":[{"id":3830,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3766\/revisions\/3830"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=3766"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=3766"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=3766"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}