{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "7b33ad92",
   "metadata": {},
   "source": [
    "# Septième exercice en Python (Niveau Lycée)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7f4e9974",
   "metadata": {},
   "source": [
    "<img src=\"https://blog.univ-angers.fr/mathsinfo/files/2022/06/image-6.png\">"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "60fae855",
   "metadata": {},
   "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."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "755fa541",
   "metadata": {},
   "source": [
    "## Version classique"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7d7bf9bc",
   "metadata": {},
   "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."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "7190588b",
   "metadata": {},
   "outputs": [],
   "source": [
    "def mul(n):\n",
    "  chaine = str(n)            # Transformation nombre en chaine\n",
    "  liste = list(chaine)       # puis en liste\n",
    "  produit = 1                # Initialisation du résultat du produit\n",
    "  for c in liste:\n",
    "    produit *= int(c)        # int = convertir chaine en entier\n",
    "  return produit  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "ab2f3607",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "729"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mul(999)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "04952584",
   "metadata": {},
   "source": [
    "On peut également utiliser une librairie comme `operator` ou `numpy` :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "c7596ea6",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "def mul(n):\n",
    "  liste = list(str(n))                      # Nombre en liste\n",
    "  return np.prod([int(v) for v in liste])   # \"prod\" pour produit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "7a163b64",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "729"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mul(999)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7a7aef7f",
   "metadata": {},
   "source": [
    "Ce qui donne ce programme pour la version classique :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "ab411e7f",
   "metadata": {},
   "outputs": [],
   "source": [
    "def persistence(n):\n",
    "  compteur = 0\n",
    "  while n > 9:\n",
    "    compteur += 1\n",
    "    n = np.prod([int(v) for v in list(str(n))])\n",
    "  return compteur "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "3ddb3512",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    " persistence(999)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ebf47515",
   "metadata": {},
   "source": [
    "## Version récursive"
   ]
  },
  {
   "cell_type": "markdown",
   "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>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7b8ad345",
   "metadata": {},
   "source": [
    "Programme final version récursive :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "a5aaf28e",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "def persistence(n):\n",
    "    if n < 10: return 0\n",
    "    p = np.prod([int(v) for v in list(str(n))])\n",
    "    return 1 + persistence(p) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "f8b3c126",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "persistence(999)"
   ]
  }
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