Números con dígitos 1 y 2

PorJosé A. Alonso 28-enero-201926-marzo-2022

Definir las funciones

   numerosCon1y2               :: Int -> [Int]
   restosNumerosCon1y2         :: Int -> [Int]
   grafica_restosNumerosCon1y2 :: Int -> IO ()

numerosCon1y2 :: Int -> [Int]

restosNumerosCon1y2 :: Int -> [Int]

grafica_restosNumerosCon1y2 :: Int -> IO ()

tales que

(numerosCon1y2 n) es la lista ordenada de números de n dígitos que se pueden formar con los dígitos 1 y 2. Por ejemplo,

     numerosCon1y2 2  ==  [11,12,21,22] 
     numerosCon1y2 3  ==  [111,112,121,122,211,212,221,222]

1 2	numerosCon1y2 2 == [11,12,21,22] numerosCon1y2 3 == [111,112,121,122,211,212,221,222]

(restosNumerosCon1y2 n) es la lista de los restos de dividir los elementos de (restosNumerosCon1y2 n) entre 2^n. Por ejemplo,

     restosNumerosCon1y2 2 == [3,0,1,2]
     restosNumerosCon1y2 3 == [7,0,1,2,3,4,5,6]
     restosNumerosCon1y2 4 == [7,8,1,2,11,12,5,6,15,0,9,10,3,4,13,14]

restosNumerosCon1y2 2 == [3,0,1,2]

restosNumerosCon1y2 3 == [7,0,1,2,3,4,5,6]

restosNumerosCon1y2 4 == [7,8,1,2,11,12,5,6,15,0,9,10,3,4,13,14]

(graficaRestosNumerosCon1y2 n) dibuja la gráfica de los restos de dividir los elementos de (restosNumerosCon1y2 n) entre 2^n. Por ejemplo, (graficaRestosNumerosCon1y2 3) dibuja

(graficaRestosNumerosCon1y2 4) dibuja

y (graficaRestosNumerosCon1y2 5) dibuja

Nota: En la definición usar la función plotListStyle y como su segundo argumento (el PloStyle) usar

     (defaultStyle {plotType = LinesPoints,
                    lineSpec = CustomStyle [PointType 7]})

1 2	(defaultStyle {plotType = LinesPoints, lineSpec = CustomStyle [PointType 7]})

Comprobar con QuickCheck que todos los elementos de (restosNumerosCon1y2 n) son distintos.

Soluciones

import Test.QuickCheck
import Graphics.Gnuplot.Simple

-- Definición de numerosCon1y2
-- ===========================

numerosCon1y2 :: Int -> [Int]
numerosCon1y2 = map digitosAnumero . digitos

-- (dígitos n) es la lista ordenada de de listas de n elementos que
-- se pueden formar con los dígitos 1 y 2. Por ejemplo,
--    λ> digitos 2
--    [[1,1],[1,2],[2,1],[2,2]]
--    λ> digitos 3
--    [[1,1,1],[1,1,2],[1,2,1],[1,2,2],[2,1,1],[2,1,2],[2,2,1],[2,2,2]]
digitos :: Int -> [[Int]]
digitos 0 = [[]]
digitos n = map (1:) xss ++ map (2:) xss
  where xss = digitos (n-1)

-- (digitosAnumero ds) es el número cuyos dígitos son ds. Por ejemplo,
--    digitosAnumero [2,0,1,9]  ==  2019
digitosAnumero :: [Int] -> Int
digitosAnumero = read . concatMap show

-- Definición de restosNumerosCon1y2
-- =================================

restosNumerosCon1y2 :: Int -> [Int]
restosNumerosCon1y2 n =
  [x `mod` m | x <- numerosCon1y2 n]
  where m = 2^n

-- Definición de graficaRestosNumerosCon1y2
-- =========================================

graficaRestosNumerosCon1y2 :: Int -> IO ()
graficaRestosNumerosCon1y2 n =
  plotListStyle
    [ Key Nothing
    , PNG ("Numeros_con_digitos_1_y_2_" ++ show n ++ ".png")
    ]
    (defaultStyle {plotType = LinesPoints,
                   lineSpec = CustomStyle [PointType 7]})
    (restosNumerosCon1y2 n)

-- Propiedad de restosNumerosCon1y2
-- ================================

-- La propiedad
prop_restosNumerosCon1y2 :: Positive Int -> Bool
prop_restosNumerosCon1y2 (Positive n) =
  todosDistintos (restosNumerosCon1y2 n)
  where todosDistintos xs = xs == nub xs

-- La comprobación es
--    λ> quickCheckWith (stdArgs {maxSize=12}) prop_restosNumerosCon1y2
--    +++ OK, passed 100 tests.

import Test.QuickCheck

import Graphics.Gnuplot.Simple

-- Definición de numerosCon1y2

-- ===========================

numerosCon1y2 :: Int -> [Int]

numerosCon1y2 = map digitosAnumero . digitos

-- (dígitos n) es la lista ordenada de de listas de n elementos que

-- se pueden formar con los dígitos 1 y 2. Por ejemplo,

-- λ> digitos 2

-- [[1,1],[1,2],[2,1],[2,2]]

-- λ> digitos 3

-- [[1,1,1],[1,1,2],[1,2,1],[1,2,2],[2,1,1],[2,1,2],[2,2,1],[2,2,2]]

digitos :: Int -> [[Int]]

digitos 0 = [[]]

digitos n = map (1:) xss ++ map (2:) xss

where xss = digitos (n-1)

-- (digitosAnumero ds) es el número cuyos dígitos son ds. Por ejemplo,

-- digitosAnumero [2,0,1,9] == 2019

digitosAnumero :: [Int] -> Int

digitosAnumero = read . concatMap show

-- Definición de restosNumerosCon1y2

-- =================================

restosNumerosCon1y2 :: Int -> [Int]

restosNumerosCon1y2 n =

[x `mod` m | x <- numerosCon1y2 n]

where m = 2^n

-- Definición de graficaRestosNumerosCon1y2

-- =========================================

graficaRestosNumerosCon1y2 :: Int -> IO ()

graficaRestosNumerosCon1y2 n =

plotListStyle

[ Key Nothing

, PNG ("Numeros_con_digitos_1_y_2_" ++ show n ++ ".png")

]

(defaultStyle {plotType = LinesPoints,

lineSpec = CustomStyle [PointType 7]})

(restosNumerosCon1y2 n)

-- Propiedad de restosNumerosCon1y2

-- ================================

-- La propiedad

prop_restosNumerosCon1y2 :: Positive Int -> Bool

prop_restosNumerosCon1y2 (Positive n) =

todosDistintos (restosNumerosCon1y2 n)

where todosDistintos xs = xs == nub xs

-- La comprobación es

-- λ> quickCheckWith (stdArgs {maxSize=12}) prop_restosNumerosCon1y2

-- +++ OK, passed 100 tests.

Pensamiento

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a los surcos del azar? …
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como Jesús, sobre el mar.

Antonio Machado

4 Comentarios

import Control.Monad (replicateM)
import Data.List (nub)
import Graphics.Gnuplot.Simple
import Test.QuickCheck

numerosCon1y2 :: Int -> [Int]
numerosCon1y2 n
  | n <= 0    = []
  | otherwise = map aEntero (replicateM n [1,2])
 
aEntero :: [Integer] -> Int
aEntero xs =
  sum [fromIntegral (a*10^b) | (a,b) <- zip (reverse xs) [0..]]
 
restosNumerosCon1y2 :: Int -> [Int]
restosNumerosCon1y2 n =
  map (\x -> mod x (2^n)) (numerosCon1y2 n)
 
grafica_numerosCon1y2 :: Int -> IO ()
grafica_numerosCon1y2 n =
  plotListStyle [ Key Nothing
                , PNG "restos_Numeros_Con_1_y_2.png"]
                (defaultStyle { plotType = LinesPoints
                              , lineSpec = CustomStyle [PointType 7]})
  (numerosCon1y2 n)
 
prop_restosNumerosCon1y2 :: Int -> Property
prop_restosNumerosCon1y2 n = n >= 0 ==> nub zs == zs
   where zs = restosNumerosCon1y2 n

-- La comprobación es
--    λ> quickCheckWith (stdArgs {maxSize=12}) prop_restosNumerosCon1y2
--    +++ OK, passed 100 tests.

import Control.Monad (replicateM)

import Data.List (nub)

import Graphics.Gnuplot.Simple

import Test.QuickCheck

numerosCon1y2 :: Int -> [Int]

numerosCon1y2 n

| n <= 0 = []

| otherwise = map aEntero (replicateM n [1,2])

aEntero :: [Integer] -> Int

aEntero xs =

sum [fromIntegral (a*10^b) | (a,b) <- zip (reverse xs) [0..]]

restosNumerosCon1y2 :: Int -> [Int]

restosNumerosCon1y2 n =

map (\x -> mod x (2^n)) (numerosCon1y2 n)

grafica_numerosCon1y2 :: Int -> IO ()

grafica_numerosCon1y2 n =

plotListStyle [ Key Nothing

, PNG "restos_Numeros_Con_1_y_2.png"]

(defaultStyle { plotType = LinesPoints

, lineSpec = CustomStyle [PointType 7]})

(numerosCon1y2 n)

prop_restosNumerosCon1y2 :: Int -> Property

prop_restosNumerosCon1y2 n = n >= 0 ==> nub zs == zs

where zs = restosNumerosCon1y2 n

-- La comprobación es

-- λ> quickCheckWith (stdArgs {maxSize=12}) prop_restosNumerosCon1y2

-- +++ OK, passed 100 tests.

Responder

import Data.List (nub, permutations, sort)
import Graphics.Gnuplot.Simple

numerosCon1y2 :: Int -> [Int]
numerosCon1y2 n = sort (nub (map listaNumero (aux 0)))
  where
    aux i
      | i < n =     permutations (take i [1,1..] ++ take (n-i) [2,2..]) ++ aux (i+1)
      | otherwise = permutations (take n [1,1..])  
 
listaNumero :: [Int] -> Int
listaNumero [x]    = x
listaNumero (x:xs) = x + 10 * listaNumero xs
 
restosNumerosCon1y2 :: Int -> [Int]
restosNumerosCon1y2 n = map (`rem` (2^n)) (numerosCon1y2 n)
 
grafica_restosNumerosCon1y2 :: Int -> IO ()
grafica_restosNumerosCon1y2 n =
  plotListStyle [ Key Nothing
                , PNG "restos_Numeros_Con_1_y_2.png"]
                (defaultStyle { plotType = LinesPoints
                              , lineSpec = CustomStyle [PointType 7]})
                (restosNumerosCon1y2 n)

import Data.List (nub, permutations, sort)

import Graphics.Gnuplot.Simple

numerosCon1y2 :: Int -> [Int]

numerosCon1y2 n = sort (nub (map listaNumero (aux 0)))

where

aux i

| i < n = permutations (take i [1,1..] ++ take (n-i) [2,2..]) ++ aux (i+1)

| otherwise = permutations (take n [1,1..])

listaNumero :: [Int] -> Int

listaNumero [x] = x

listaNumero (x:xs) = x + 10 * listaNumero xs

restosNumerosCon1y2 :: Int -> [Int]

restosNumerosCon1y2 n = map (`rem` (2^n)) (numerosCon1y2 n)

grafica_restosNumerosCon1y2 :: Int -> IO ()

grafica_restosNumerosCon1y2 n =

plotListStyle [ Key Nothing

, PNG "restos_Numeros_Con_1_y_2.png"]

(defaultStyle { plotType = LinesPoints

, lineSpec = CustomStyle [PointType 7]})

(restosNumerosCon1y2 n)

Responder

import Graphics.Gnuplot.Simple
import Test.QuickCheck
import Data.List (nub)

numerosCon1y2 :: Int -> [Int]
numerosCon1y2 0 = []
numerosCon1y2 n = [x | x <- [10^(n-1)..10^n], aux x]
  where aux x = and [elem y "12" | y <- show x ]

restosNumerosCon1y2 :: Int -> [Int]
restosNumerosCon1y2 n =
  [mod x (2^n) | x <- numerosCon1y2 n]

grafica_restosNumerosCon1y2 :: Int -> IO ()
grafica_restosNumerosCon1y2 n =
  plotListStyle [ Key Nothing
                , PNG "restos_Numeros_Con_1_y_2.png"]
                (defaultStyle { plotType = LinesPoints
                              , lineSpec = CustomStyle [PointType 7]})
                (restosNumerosCon1y2 n )
  
prop_numerosCon1y2 :: Int -> Property  
prop_numerosCon1y2 n =
  n >= 1 ==> nub (numerosCon1y2 n) == numerosCon1y2 n

import Graphics.Gnuplot.Simple

import Test.QuickCheck

import Data.List (nub)

numerosCon1y2 :: Int -> [Int]

numerosCon1y2 0 = []

numerosCon1y2 n = [x | x <- [10^(n-1)..10^n], aux x]

where aux x = and [elem y "12" | y <- show x ]

restosNumerosCon1y2 :: Int -> [Int]

restosNumerosCon1y2 n =

[mod x (2^n) | x <- numerosCon1y2 n]

grafica_restosNumerosCon1y2 :: Int -> IO ()

grafica_restosNumerosCon1y2 n =

plotListStyle [ Key Nothing

, PNG "restos_Numeros_Con_1_y_2.png"]

(defaultStyle { plotType = LinesPoints

, lineSpec = CustomStyle [PointType 7]})

(restosNumerosCon1y2 n )

prop_numerosCon1y2 :: Int -> Property

prop_numerosCon1y2 n =

n >= 1 ==> nub (numerosCon1y2 n) == numerosCon1y2 n

Responder

import Test.QuickCheck
import Graphics.Gnuplot.Simple
 
numerosCon1y2 :: Int -> [Int]
numerosCon1y2 n =
  [x | x <- filter compuesto [1..10^n]
     , length (digitos x) == n]
 
digitos :: Int -> [Int]
digitos x = [read [y] |y <- show x ]
 
compuesto :: Int -> Bool
compuesto x = all (`elem`[1,2]) (digitos x)
 
restosNumerosCon1y2 :: Int -> [Int]
restosNumerosCon1y2 n =
  map (`mod` (2^n)) (numerosCon1y2 n)
 
prop :: [Int] -> Bool
prop []     = True
prop (x:xs) = all (/=x) xs && prop xs
 
comprobacion :: Int -> Property
comprobacion n = n >= 1 ==> prop (numerosCon1y2 n)
 
grafica_restosNumerosCon1y2 :: Int -> IO ()
grafica_restosNumerosCon1y2 n =
  plotListStyle [ Key Nothing
                , PNG "restos_Numeros_Con_1_y_2.png"]
                (defaultStyle { plotType = LinesPoints
                              , lineSpec = CustomStyle [PointType 7]})
                (restosNumerosCon1y2 n)

import Test.QuickCheck

import Graphics.Gnuplot.Simple

numerosCon1y2 :: Int -> [Int]

numerosCon1y2 n =

[x | x <- filter compuesto [1..10^n]

, length (digitos x) == n]

digitos :: Int -> [Int]

digitos x = [read [y] |y <- show x ]

compuesto :: Int -> Bool

compuesto x = all (`elem`[1,2]) (digitos x)

restosNumerosCon1y2 :: Int -> [Int]

restosNumerosCon1y2 n =

map (`mod` (2^n)) (numerosCon1y2 n)

prop :: [Int] -> Bool

prop [] = True

prop (x:xs) = all (/=x) xs && prop xs

comprobacion :: Int -> Property

comprobacion n = n >= 1 ==> prop (numerosCon1y2 n)

grafica_restosNumerosCon1y2 :: Int -> IO ()

grafica_restosNumerosCon1y2 n =

plotListStyle [ Key Nothing

, PNG "restos_Numeros_Con_1_y_2.png"]

(defaultStyle { plotType = LinesPoints

, lineSpec = CustomStyle [PointType 7]})

(restosNumerosCon1y2 n)

Responder

Números con dígitos 1 y 2

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