{"id":6807,"date":"2022-03-23T06:00:21","date_gmt":"2022-03-23T04:00:21","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=6807"},"modified":"2022-04-02T11:35:20","modified_gmt":"2022-04-02T09:35:20","slug":"regiones-determinadas-por-n-rectas-del-plano","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/regiones-determinadas-por-n-rectas-del-plano\/","title":{"rendered":"Regiones determinadas por n rectas del plano"},"content":{"rendered":"<p>En los siguientes dibujos se observa que el n\u00famero m\u00e1ximo de regiones en el plano generadas con 1, 2 \u00f3 3 l\u00edneas son 2, 4 \u00f3 7, respectivamente.<\/p>\n<pre lang=\"text\">\n\n                      \\  |\n                       \\5|\n                        \\|\n                         \\\n                         |\\\n                         | \\\n               |         |  \\\n    1        1 | 3     1 | 3 \\  6\n   ------   ---|---   ---|----\\---\n    2        2 | 4     2 | 4   \\ 7\n               |         |      \\\n<\/pre>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   regiones :: Integer -> Integer\n<\/pre>\n<p>tal que <code>(regiones n)<\/code> es el n\u00famero m\u00e1ximo de regiones en el plano generadas con <code>n<\/code> l\u00edneas. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   regiones 1     ==  2\n   regiones 2     ==  4\n   regiones 3     ==  7\n   regiones 100   ==  5051\n   regiones 1000  ==  500501\n   regiones 10000 ==  50005001\n   length (show (regiones (10^(10^5)))) ==  200000\n   length (show (regiones (10^(10^6)))) ==  2000000\n   length (show (regiones (10^(10^6)))) ==  2000000\n   length (show (regiones (10^(10^7)))) ==  20000000\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (genericIndex)\nimport Test.QuickCheck\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nregiones1 :: Integer -> Integer\nregiones1 0 = 1\nregiones1 n = regiones1 (n-1) + n\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nregiones2 :: Integer -> Integer\nregiones2 n = 1 + sum [0..n]\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\nregiones3 :: Integer -> Integer\nregiones3 n = 1 + sumas `genericIndex` n\n\n-- (sumas n) es la suma 0 + 1 + 2 +...+ n. Por ejemplo,\n--    take 10 sumas  ==  [0,1,3,6,10,15,21,28,36,45]\nsumas :: [Integer]\nsumas = scanl1 (+) [0..]\n\n-- 4\u00aa soluci\u00f3n\n-- ===========\n\nregiones4 :: Integer -> Integer\nregiones4 n = 1 + n*(n+1) `div` 2\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_regiones :: Positive Integer -> Bool\nprop_regiones (Positive n) =\n  all (== regiones1 n)\n      [regiones2 n,\n       regiones3 n,\n       regiones4 n]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_regiones\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> regiones1 (4*10^6)\n--    8000002000001\n--    (2.20 secs, 938,105,888 bytes)\n--    \u03bb> regiones2 (4*10^6)\n--    8000002000001\n--    (0.77 secs, 645,391,624 bytes)\n--    \u03bb> regiones3 (4*10^6)\n--    8000002000001\n--    (1.22 secs, 1,381,375,296 bytes)\n--    \u03bb> regiones4 (4*10^6)\n--    8000002000001\n--    (0.01 secs, 484,552 bytes)\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium\/blob\/main\/src\/Regiones.hs\">GitHub<\/a>.<\/p>\n<p>La elaboraci\u00f3n de las soluciones se describe en el siguiente v\u00eddeo<\/p>\n<p><iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/lLl-jQ1tW-I\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\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>En los siguientes dibujos se observa que el n\u00famero m\u00e1ximo de regiones en el plano generadas con 1, 2 \u00f3 3 l\u00edneas son 2, 4 \u00f3 7, respectivamente. \\ | \\5| \\| \\ |\\ | \\ | | \\ 1 1 | 3 1 | 3 \\ 6 &#8212;&#8212; &#8212;|&#8212; &#8212;|&#8212;-\\&#8212; 2 2 | 4&#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":[8,498,523,11,6],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6807"}],"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=6807"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6807\/revisions"}],"predecessor-version":[{"id":6868,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6807\/revisions\/6868"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=6807"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=6807"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=6807"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}