Exercice7JS.ipynb

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   "source": [
    "# Septième exercice en JavaScript"
   ]
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  {
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   "id": "7f4e9974",
   "metadata": {},
   "source": [
    "<img src=\"https://blog.univ-angers.fr/mathsinfo/files/2022/06/image-6.png\">"
   ]
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   "source": [
    "*Résumé en français* : Ecrire une fonction qui admet en paramètre un nombre entier positif et qui retourne le nombre de fois où vous devez multiplier ses chiffres pour obtenir un seul chiffre."
   ]
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   "source": [
    "## Version classique"
   ]
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   "source": [
    "On doit compter (variable `compteur`) combien de fois il faut multiplier les chiffres entre eux jusqu'à obtenir un seul chiffre. Ce nombre d'itérations étant inconnu, nous allons naturellement utiliser une boucle **Tant Que** (`while`). Maintenant, comment faire la multiplication des chiffres d'un nombre ? On peut par exemple **transformer** ce nombre en **chaine** puis en **liste** et multiplier les éléments de cette liste grâce à une boucle."
   ]
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   "execution_count": 1,
   "id": "7190588b",
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   "outputs": [],
   "source": [
    "var mul = n => \n",
    "{\n",
    "     liste = [...''+n] ;    // Conversion nombre > chaine > liste\n",
    "     produit = 1 ;\n",
    "     for (c of liste) produit *= +c ;\n",
    "     return produit\n",
    "}"
   ]
  },
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   "cell_type": "code",
   "execution_count": 2,
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   "outputs": [
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      "text/plain": [
       "729"
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     "execution_count": 2,
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   "source": [
    "mul(999)"
   ]
  },
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   "source": [
    "Ce qui donne ce programme pour la version classique :"
   ]
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   "execution_count": 3,
   "id": "ab411e7f",
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   "source": [
    "var persistence = n =>\n",
    "{\n",
    "   var compteur = 0;                  // Notre compteur de tours\n",
    "   while (n > 9) {\n",
    "     compteur ++;\n",
    "     produit = 1;                     // Calcul du produit des chiffres\n",
    "     liste = [...''+n];               // Conversion du nombre en liste\n",
    "     for (c of liste) produit *= +c;  // +c --> entier\n",
    "     n = produit  \n",
    "   }\n",
    "   \n",
    "   return compteur;\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "3ddb3512",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
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     "execution_count": 4,
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   "source": [
    " persistence(999)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ebf47515",
   "metadata": {},
   "source": [
    "## Version récursive"
   ]
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  {
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   "id": "85a4d639",
   "metadata": {},
   "source": [
    "Si un nombre est plus petit que 10, sa **persistence** est **0**. Sinon, sa **persistence** est **1 +** la **persistence** du **produit de ses chiffres**. Par exemple :\n",
    "\n",
    "<pre>persistence(39) = 1 + persistence(3 * 9)\n",
    "                = 1 + persistence(27)\n",
    "                = 1 + (1 + persistence(2 * 7)) \n",
    "                = 1 + 1 + persistence(14)\n",
    "                = 1 + 1 + (1 + persistence(1 * 4))\n",
    "                = 1 + 1 + 1 + persistence(4)              \n",
    "                = 1 + 1 + 1 + 0\n",
    "                = 3</pre>"
   ]
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  {
   "cell_type": "markdown",
   "id": "7b8ad345",
   "metadata": {},
   "source": [
    "Programme final version récursive :"
   ]
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  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "a5aaf28e",
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   "outputs": [],
   "source": [
    "var persistence = n => {\n",
    "    if (n < 10) return 0;\n",
    "     produit = 1;\n",
    "     liste = [...''+n];\n",
    "     for (c of liste) produit *= +c;  \n",
    "    return 1 + persistence(produit) \n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "f8b3c126",
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
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     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "persistence(999)"
   ]
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   "cell_type": "markdown",
   "id": "fe674992",
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   "source": [
    "La multiplication des chiffres peut se faire par `reduce` plutôt que par une boucle `for` classique :"
   ]
  },
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   "cell_type": "code",
   "execution_count": 7,
   "id": "d0975eb5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "729"
      ]
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     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "[...'999'].reduce((a,c) => a * (+c), 1)   // avec parenthèses"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "dbf904f4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "729"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[...'999'].reduce((a,c) => a * +c, 1)    // ou sans parenthèse !"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "b296c2a1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "729"
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     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[...'999'].reduce((a,c) => a * +c)      // ou sans initialisation !!"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8cdb4402",
   "metadata": {},
   "source": [
    "On obtient finalement :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "bc7b69d2",
   "metadata": {},
   "outputs": [],
   "source": [
    "var persistence = num => \n",
    "    num < 10 ? \n",
    "    0 :\n",
    "    1 + persistence([...'' + num].reduce((a, c) => a * +c))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
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     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "persistence(999)"
   ]
  }
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