{"id":7065,"date":"2022-06-02T06:00:03","date_gmt":"2022-06-02T04:00:03","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=7065"},"modified":"2022-05-31T09:11:29","modified_gmt":"2022-05-31T07:11:29","slug":"suma-de-los-elementos-de-las-diagonales-matrices-espirales","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/suma-de-los-elementos-de-las-diagonales-matrices-espirales\/","title":{"rendered":"Suma de los elementos de las diagonales matrices espirales"},"content":{"rendered":"<p>Empezando con el n\u00famero 1 y movi\u00e9ndose en el sentido de las agujas del reloj se obtienen las matrices espirales<\/p>\n<pre lang=\"text\">\n   |1 2|   |7 8 9|   | 7  8  9 10|   |21 22 23 24 25|\n   |4 3|   |6 1 2|   | 6  1  2 11|   |20  7  8  9 10|\n           |5 4 3|   | 5  4  3 12|   |19  6  1  2 11|\n                     |16 15 14 13|   |18  5  4  3 12|\n                                     |17 16 15 14 13|\n<\/pre>\n<p>La suma los elementos de sus diagonales es<\/p>\n<pre lang=\"text\">\n   + en la 2x2: 1+3+2+4               =  10\n   + en la 3x3: 1+3+5+7+9             =  25\n   + en la 4x4: 1+2+3+4+7+10+13+16    =  56\n   + en la 5x5: 1+3+5+7+9+13+17+21+25 = 101\n<\/pre>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   sumaDiagonales :: Integer -> Integer\n<\/pre>\n<p>tal que (sumaDiagonales n) es la suma de los elementos en las diagonales de la matriz espiral de orden nxn. Por ejemplo.<\/p>\n<pre lang=\"text\">\n   sumaDiagonales 1         ==  1\n   sumaDiagonales 2         ==  10\n   sumaDiagonales 3         ==  25\n   sumaDiagonales 4         ==  56\n   sumaDiagonales 5         ==  101\n   sumaDiagonales (10^6)    ==  666667166668000000\n   sumaDiagonales (1+10^6)  ==  666669166671000001\n\n   sumaDiagonales (10^2)  ==         671800\n   sumaDiagonales (10^3)  ==        667168000\n   sumaDiagonales (10^4)  ==       666716680000\n   sumaDiagonales (10^5)  ==      666671666800000\n   sumaDiagonales (10^6)  ==     666667166668000000\n   sumaDiagonales (10^7)  ==    666666716666680000000\n   sumaDiagonales (10^8)  ==   666666671666666800000000\n   sumaDiagonales (10^9)  ==  666666667166666668000000000\n<\/pre>\n<p>Comprobar con QuickCheck que el \u00faltimo d\u00edgito de (sumaDiagonales n) es 0, 4 \u00f3 6 si n es par y es 1, 5 \u00f3 7 en caso contrario.<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Test.QuickCheck (Positive (Positive), quickCheck)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nsumaDiagonales1 :: Integer -> Integer\nsumaDiagonales1 = sum . elementosEnDiagonales\n\n-- (elementosEnDiagonales n) es la lista de los elementos en las\n-- diagonales de la matriz espiral de orden nxn. Por ejemplo,\n--    elementosEnDiagonales 1  ==  [1]\n--    elementosEnDiagonales 2  ==  [1,2,3,4]\n--    elementosEnDiagonales 3  ==  [1,3,5,7,9]\n--    elementosEnDiagonales 4  ==  [1,2,3,4,7,10,13,16]\n--    elementosEnDiagonales 5  ==  [1,3,5,7,9,13,17,21,25]\nelementosEnDiagonales :: Integer -> [Integer]\nelementosEnDiagonales n \n  | even n    = tail (scanl (+) 0 (concatMap (replicate 4) [1,3..n-1]))\n  | otherwise = scanl (+) 1 (concatMap (replicate 4) [2,4..n-1])\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nsumaDiagonales2 :: Integer -> Integer\nsumaDiagonales2 n\n  | even n    = (-1) + n `div` 2 + sum [2*k^2-k+1 | k <- [0..n]]\n  | otherwise = 1 + sum [4*k^2-6*k+6 | k <- [3,5..n]]\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\nsumaDiagonales3 :: Integer -> Integer\nsumaDiagonales3 n\n  | even n    = n * (4*n^2 + 3*n + 8) `div` 6\n  | otherwise = (4*n^3 + 3*n^2 + 8*n - 9) `div` 6\n\n-- Equivalencia de las definiciones\n-- ================================\n\n-- La propiedad es\nprop_sumaDiagonales :: Positive Integer -> Bool\nprop_sumaDiagonales (Positive n) =\n  all (== sumaDiagonales1 n)\n      [sumaDiagonales2 n,\n       sumaDiagonales3 n]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_sumaDiagonales_equiv\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> sumaDiagonales (2*10^6)\n--    5333335333336000000\n--    (2.30 secs, 1,521,955,848 bytes)\n--    \u03bb> sumaDiagonales2 (2*10^6)\n--    5333335333336000000\n--    (2.77 secs, 1,971,411,440 bytes)\n--    \u03bb> sumaDiagonales3 (2*10^6)\n--    5333335333336000000\n--    (0.01 secs, 139,520 bytes)\n\n-- Propiedad\n-- =========\n\n-- La propiedad es\nprop_sumaDiagonales2 :: Positive Integer -> Bool\nprop_sumaDiagonales2 (Positive n) \n  | even n    = x `elem` [0,4,6] \n  | otherwise = x `elem` [1,5,7] \n  where x = sumaDiagonales1 n `mod` 10\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_sumaDiagonales2\n--    +++ OK, passed 100 tests.\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium\/blob\/main\/src\/Suma_de_los_elementos_de_las_diagonales_matrices_espirales.hs\">GitHub<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Empezando con el n\u00famero 1 y movi\u00e9ndose en el sentido de las agujas del reloj se obtienen las matrices espirales |1 2| |7 8 9| | 7 8 9 10| |21 22 23 24 25| |4 3| |6 1 2| | 6 1 2 11| |20 7 8 9 10| |5 4 3| | 5&#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":[2],"tags":[569],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7065"}],"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=7065"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7065\/revisions"}],"predecessor-version":[{"id":7066,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7065\/revisions\/7066"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=7065"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=7065"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=7065"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}