{"id":7400,"date":"2022-09-29T06:00:33","date_gmt":"2022-09-29T04:00:33","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=7400"},"modified":"2022-12-14T14:20:20","modified_gmt":"2022-12-14T12:20:20","slug":"suma-de-cuadrados-menos-cuadrado-de-la-suma","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/suma-de-cuadrados-menos-cuadrado-de-la-suma\/","title":{"rendered":"Suma de cuadrados menos cuadrado de la suma"},"content":{"rendered":"

Definir la funci\u00f3n<\/p>\n

\n   euler6 :: Integer -> Integer\n<\/pre>\n
tal que euler6 n<\/code> es la diferencia entre el cuadrado de la suma de los n<\/code> primeros n\u00fameros y la suma de los cuadrados de los n<\/code>primeros n\u00fameros. Por ejemplo,<\/p>\n
\n   euler6 10       ==  2640\n   euler6 (10^10)  ==  2500000000166666666641666666665000000000\n<\/pre>\n
Nota:<\/strong> Este ejercicio est\u00e1 basado en el problema 6 del proyecto Euler<\/a>.<\/p>\n
Soluciones<\/b><\/p>\n
A continuaci\u00f3n se muestran las soluciones en Haskell<\/a> y las soluciones en Python<\/a>.<\/p>\n
<\/a>
\nSoluciones en Haskell<\/b><\/p>\n
\nimport Data.List (foldl')\nimport Test.QuickCheck\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\neuler6a :: Integer -> Integer\neuler6a n = (suma1 n)^2 - sumaDeCuadrados1 n\n\n-- (suma n) es la suma de los n primeros n\u00fameros. Por ejemplo,\n--    suma 3  ==  6\nsuma1 :: Integer -> Integer\nsuma1 n = sum [1..n]\n\n-- (sumaDeCuadrados n) es la suma de los cuadrados de los\n-- primeros n n\u00fameros; es decir, 1^2 + 2^2 + ... + n^2. Por ejemplo,\n--    sumaDeCuadrados 3    ==  14\n--    sumaDeCuadrados 100  ==  338350\n\nsumaDeCuadrados1 :: Integer -> Integer\nsumaDeCuadrados1 n = sum [x^2 | x <- [1..n]]\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\neuler6b :: Integer -> Integer\neuler6b n = (suma2 n)^ 2 - sumaDeCuadrados2 n\n\nsuma2 :: Integer -> Integer\nsuma2 n = (1+n)*n `div` 2\n\nsumaDeCuadrados2 :: Integer -> Integer\nsumaDeCuadrados2 n = n*(n+1)*(2*n+1) `div` 6\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\neuler6c :: Integer -> Integer\neuler6c n = (suma3 n)^ 2 - sumaDeCuadrados3 n\n\nsuma3 :: Integer -> Integer\nsuma3 1 = 1\nsuma3 n = n + suma3 (n-1)\n\nsumaDeCuadrados3 :: Integer -> Integer\nsumaDeCuadrados3 1 = 1\nsumaDeCuadrados3 n = n^2 + sumaDeCuadrados3 (n-1)\n\n-- 4\u00aa soluci\u00f3n\n-- ===========\n\neuler6d :: Integer -> Integer\neuler6d n = (suma4 n)^ 2 - sumaDeCuadrados4 n\n\nsuma4 :: Integer -> Integer\nsuma4 n = foldl (+) 0 [0..n]\n\nsumaDeCuadrados4 :: Integer -> Integer\nsumaDeCuadrados4 n = foldl (+) 0 (map (^2) [0..n])\n\n-- 5\u00aa soluci\u00f3n\n-- ===========\n\neuler6e :: Integer -> Integer\neuler6e n = (suma5 n)^ 2 - sumaDeCuadrados5 n\n\nsuma5 :: Integer -> Integer\nsuma5 n = foldl' (+) 0 [0..n]\n\nsumaDeCuadrados5 :: Integer -> Integer\nsumaDeCuadrados5 n = foldl' (+) 0 (map (^2) [0..n])\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_euler6 :: Positive Integer -> Bool\nprop_euler6 (Positive n) =\n  all (== euler6a n)\n      [euler6b n,\n       euler6c n,\n       euler6d n,\n       euler6e n]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_euler6\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> euler6a (3*10^6)\n--    20250004499997749999500000\n--    (3.32 secs, 2,577,174,640 bytes)\n--    \u03bb> euler6b (3*10^6)\n--    20250004499997749999500000\n--    (0.01 secs, 569,288 bytes)\n--    \u03bb> euler6c (3*10^6)\n--    20250004499997749999500000\n--    (5.60 secs, 2,849,479,288 bytes)\n--    \u03bb> euler6d (3*10^6)\n--    20250004499997749999500000\n--    (2.52 secs, 2,457,175,248 bytes)\n--    \u03bb> euler6e (3*10^6)\n--    20250004499997749999500000\n--    (1.08 secs, 2,016,569,472 bytes)\n--\n--    \u03bb> euler6a (10^7)\n--    2500000166666641666665000000\n--    (11.14 secs, 8,917,796,648 bytes)\n--    \u03bb> euler6b (10^7)\n--    2500000166666641666665000000\n--    (0.01 secs, 570,752 bytes)\n--    \u03bb> euler6c (10^7)\n--    *** Exception: stack overflow\n--    \u03bb> euler6d (10^7)\n--    2500000166666641666665000000\n--    (9.47 secs, 8,517,796,760 bytes)\n--    \u03bb> euler6e (10^7)\n--    2500000166666641666665000000\n--    (3.78 secs, 7,049,100,104 bytes)\n<\/pre>\n
El c\u00f3digo se encuentra en GitHub<\/a>.<\/p>\n
<\/a>
\nSoluciones en Python<\/b><\/p>\n
\nfrom functools import reduce\nfrom operator import add\nfrom sys import setrecursionlimit\nfrom timeit import Timer, default_timer\nfrom hypothesis import given, strategies as st\nsetrecursionlimit(10**6)\n\n# 1\u00aa soluci\u00f3n\n# ===========\n\ndef euler6a(n: int) -> int:\n    return suma1(n)**2 - sumaDeCuadrados1(n)\n\n# suma(n) es la suma de los n primeros n\u00fameros. Por ejemplo,\n#    suma(3)  ==  6\ndef suma1(n: int) -> int:\n    return sum(range(1, n + 1))\n\n# sumaDeCuadrados(n) es la suma de los cuadrados de los\n# primeros n n\u00fameros; es decir, 1^2 + 2^2 + ... + n^2. Por ejemplo,\n#    sumaDeCuadrados(3)    ==  14\n#    sumaDeCuadrados(100)  ==  338350\ndef sumaDeCuadrados1(n: int) -> int:\n    return sum(x**2 for x in range(1, n + 1))\n\n# 2\u00aa soluci\u00f3n\n# ===========\n\ndef euler6b(n: int) -> int:\n    return suma2(n)**2 - sumaDeCuadrados2(n)\n\ndef suma2(n: int) -> int:\n    return (1 + n) * n \/\/ 2\n\ndef sumaDeCuadrados2(n: int) -> int:\n    return n * (n + 1) * (2 * n + 1) \/\/ 6\n\n# 3\u00aa soluci\u00f3n\n# ===========\n\ndef euler6c(n: int) -> int:\n    return suma3(n)**2 - sumaDeCuadrados3(n)\n\ndef suma3(n: int) -> int:\n    if n == 1:\n        return 1\n    return n + suma3(n - 1)\n\ndef sumaDeCuadrados3(n: int) -> int:\n    if n == 1:\n        return 1\n    return n**2 + sumaDeCuadrados3(n - 1)\n\n# 4\u00aa soluci\u00f3n\n# ===========\n\ndef euler6d(n: int) -> int:\n    return suma4(n)**2 - sumaDeCuadrados4(n)\n\ndef suma4(n: int) -> int:\n    return reduce(add, range(1, n + 1))\n\ndef sumaDeCuadrados4(n: int) -> int:\n    return reduce(add, (x**2 for x in range(1, n + 1)))\n\n# 5\u00aa soluci\u00f3n\n# ===========\n\ndef euler6e(n: int) -> int:\n    return suma5(n)**2 - sumaDeCuadrados5(n)\n\ndef suma5(n: int) -> int:\n    x, r = 1, 0\n    while x <= n:\n        r = r + x\n        x = x + 1\n    return r\n\ndef sumaDeCuadrados5(n: int) -> int:\n    x, r = 1, 0\n    while x <= n:\n        r = r + x**2\n        x = x + 1\n    return r\n\n# 6\u00aa soluci\u00f3n\n# ===========\n\ndef euler6f(n: int) -> int:\n    return suma6(n)**2 - sumaDeCuadrados6(n)\n\ndef suma6(n: int) -> int:\n    r = 0\n    for x in range(1, n + 1):\n        r = r + x\n    return r\n\ndef sumaDeCuadrados6(n: int) -> int:\n    r = 0\n    for x in range(1, n + 1):\n        r = r + x**2\n    return r\n\n# Comprobaci\u00f3n de equivalencia\n# ============================\n\n# La propiedad es\n@given(st.integers(min_value=1, max_value=1000))\ndef test_euler6(n):\n    r = euler6a(n)\n    assert euler6b(n) == r\n    assert euler6c(n) == r\n    assert euler6d(n) == r\n    assert euler6e(n) == r\n    assert euler6f(n) == r\n\n# La comprobaci\u00f3n es\n#    src> poetry run pytest -q suma_de_cuadrados_menos_cuadrado_de_la_suma.py\n#    1 passed in 0.21s\n\n# Comparaci\u00f3n de eficiencia\n# =========================\n\ndef tiempo(e):\n    \"\"\"Tiempo (en segundos) de evaluar la expresi\u00f3n e.\"\"\"\n    t = Timer(e, \"\", default_timer, globals()).timeit(1)\n    print(f\"{t:0.2f} segundos\")\n\n# La comparaci\u00f3n es\n#    >>> tiempo('euler6a(20000)')\n#    0.02 segundos\n#    >>> tiempo('euler6b(20000)')\n#    0.00 segundos\n#    >>> tiempo('euler6c(20000)')\n#    0.02 segundos\n#    >>> tiempo('euler6d(20000)')\n#    0.01 segundos\n#    >>> tiempo('euler6e(20000)')\n#    0.01 segundos\n#    >>> tiempo('euler6f(20000)')\n#    0.01 segundos\n#\n#    >>> tiempo('euler6a(10**7)')\n#    2.26 segundos\n#    >>> tiempo('euler6b(10**7)')\n#    0.00 segundos\n#    >>> tiempo('euler6d(10**7)')\n#    2.58 segundos\n#    >>> tiempo('euler6e(10**7)')\n#    2.89 segundos\n#    >>> tiempo('euler6f(10**7)')\n#    2.45 segundos\n<\/pre>\n
El c\u00f3digo se encuentra en GitHub<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"
Definir la funci\u00f3n euler6 :: Integer -> Integer tal que euler6 n es la diferencia entre el cuadrado de la suma de los n primeros n\u00fameros y la suma de los cuadrados de los nprimeros n\u00fameros. Por ejemplo, euler6 10 == 2640 euler6 (10^10) == 2500000000166666666641666666665000000000 Nota: Este ejercicio est\u00e1 basado en el problema 6…<\/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":[581],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7400"}],"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=7400"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7400\/revisions"}],"predecessor-version":[{"id":7681,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7400\/revisions\/7681"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=7400"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=7400"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=7400"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}