{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "dbe7ced4",
   "metadata": {
    "tags": [
     "CSP",
     "easy",
     "AllDifferent",
     "Intension",
     "complex",
     "AllDifferentMatrix"
    ]
   },
   "source": [
    "# Problem *Sudoku*"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "28437889",
   "metadata": {},
   "source": [
    "This well-known problem is stated as follows: fill in a grid using digits ranging from 1 to 9 such that:\n",
    "- all digits occur on each row\n",
    "- all digits occur on each column\n",
    "- all digits occur in each $3 \\times 3$ block (starting at a position multiple of 3) "
   ]
  },
  {
   "attachments": {
    "sudoku.png": {
     "image/png": 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    }
   },
   "cell_type": "markdown",
   "id": "aac61629",
   "metadata": {},
   "source": [
    "Here is an illustration: \n",
    "<img src=\"attachment:sudoku.png\" alt=\"Sudoku\" width=\"800\" />"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ddb8e078",
   "metadata": {},
   "source": [
    "To build a CSP (Constraint Satisfaction Problem) model, we need first to import the library PyCSP$^3$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "e9ad9c6e",
   "metadata": {},
   "outputs": [],
   "source": [
    "from pycsp3 import *"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9db7f45e",
   "metadata": {},
   "source": [
    "In a first step, we shall build a Sudoku grid from no data/clues (i.e., from an empty grid)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c4b9334c",
   "metadata": {},
   "source": [
    "We start our CSP model with a two-dimensional array $x$ of $9 \\times 9$ variables, each variable having $\\{1,2,\\dots,9\\}$ as domain. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "a6a7d3de",
   "metadata": {},
   "outputs": [],
   "source": [
    "# x[i][j] is the value at row i and col j\n",
    "x = VarArray(size=[9, 9], dom=range(1, 10))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7d27a4c7",
   "metadata": {},
   "source": [
    "We can display the structure of the array, as well as the domain of the first variable (remember that all variables have the same domain)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "50dbf0a3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[\n",
      "  [x[0][0], x[0][1], x[0][2], x[0][3], x[0][4], x[0][5], x[0][6], x[0][7], x[0][8]]\n",
      "  [x[1][0], x[1][1], x[1][2], x[1][3], x[1][4], x[1][5], x[1][6], x[1][7], x[1][8]]\n",
      "  [x[2][0], x[2][1], x[2][2], x[2][3], x[2][4], x[2][5], x[2][6], x[2][7], x[2][8]]\n",
      "  [x[3][0], x[3][1], x[3][2], x[3][3], x[3][4], x[3][5], x[3][6], x[3][7], x[3][8]]\n",
      "  [x[4][0], x[4][1], x[4][2], x[4][3], x[4][4], x[4][5], x[4][6], x[4][7], x[4][8]]\n",
      "  [x[5][0], x[5][1], x[5][2], x[5][3], x[5][4], x[5][5], x[5][6], x[5][7], x[5][8]]\n",
      "  [x[6][0], x[6][1], x[6][2], x[6][3], x[6][4], x[6][5], x[6][6], x[6][7], x[6][8]]\n",
      "  [x[7][0], x[7][1], x[7][2], x[7][3], x[7][4], x[7][5], x[7][6], x[7][7], x[7][8]]\n",
      "  [x[8][0], x[8][1], x[8][2], x[8][3], x[8][4], x[8][5], x[8][6], x[8][7], x[8][8]]\n",
      "]\n",
      "Domain of any variable:  1..9\n"
     ]
    }
   ],
   "source": [
    "print(x)\n",
    "print(\"Domain of any variable: \", x[0][0].dom)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9c67a728",
   "metadata": {},
   "source": [
    "Interestingly, with a single constraint *AllDifferentMatrix*, we can impose that all digits are different on each row and column of $x$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "3369bd46",
   "metadata": {},
   "outputs": [],
   "source": [
    "satisfy(\n",
    "   # imposing distinct values on each row and each column\n",
    "   AllDifferent(x, matrix=True)\n",
    ");"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ded2198b",
   "metadata": {},
   "source": [
    "By calling the function *solve()*, we can check that the problem is satisfiable (SAT). We can also display the found solution. Here, we call the function *values()* that collects the values assigned to a specified list of variables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c268f917",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 2, 3, 4, 5, 6, 7, 8, 9]\n",
      "[2, 1, 4, 3, 6, 5, 8, 9, 7]\n",
      "[3, 4, 1, 2, 7, 8, 9, 5, 6]\n",
      "[4, 3, 2, 1, 8, 9, 6, 7, 5]\n",
      "[5, 6, 7, 8, 9, 1, 2, 3, 4]\n",
      "[6, 5, 8, 9, 1, 7, 3, 4, 2]\n",
      "[7, 8, 9, 5, 2, 3, 4, 6, 1]\n",
      "[8, 9, 6, 7, 4, 2, 5, 1, 3]\n",
      "[9, 7, 5, 6, 3, 4, 1, 2, 8]\n"
     ]
    }
   ],
   "source": [
    "if solve() is SAT:\n",
    "    for i in range(9):\n",
    "        print(values(x[i]))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eebe3754",
   "metadata": {},
   "source": [
    "Note that we get a structured string representation of multi-dimensional lists when printing them directly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "b760c35e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[\n",
      "  [1, 2, 3, 4, 5, 6, 7, 8, 9]\n",
      "  [2, 1, 4, 3, 6, 5, 8, 9, 7]\n",
      "  [3, 4, 1, 2, 7, 8, 9, 5, 6]\n",
      "  [4, 3, 2, 1, 8, 9, 6, 7, 5]\n",
      "  [5, 6, 7, 8, 9, 1, 2, 3, 4]\n",
      "  [6, 5, 8, 9, 1, 7, 3, 4, 2]\n",
      "  [7, 8, 9, 5, 2, 3, 4, 6, 1]\n",
      "  [8, 9, 6, 7, 4, 2, 5, 1, 3]\n",
      "  [9, 7, 5, 6, 3, 4, 1, 2, 8]\n",
      "]\n"
     ]
    }
   ],
   "source": [
    "if solve() is SAT:\n",
    "    print(values(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a74a9947",
   "metadata": {},
   "source": [
    "On can observe that some digits occur several times in the same block. This is why we add a list of constraints *AllDifferent*. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "a3376f3f",
   "metadata": {},
   "outputs": [],
   "source": [
    "satisfy(\n",
    "   # imposing distinct values on each block\n",
    "   AllDifferent(x[i:i + 3, j:j + 3]) for i in [0, 3, 6] for j in [0, 3, 6]\n",
    ");"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aa00992c",
   "metadata": {},
   "source": [
    "Note how the notation $x[i:i + 3, j:j + 3]$ extracts a list of variables corresponding to a block of size $3 \\times 3$ in $x$. This is similar to notations used in package NumPy. We can check this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "b1e87fbb",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[\n",
      "  [x[0][0], x[0][1], x[0][2]]\n",
      "  [x[1][0], x[1][1], x[1][2]]\n",
      "  [x[2][0], x[2][1], x[2][2]]\n",
      "]\n"
     ]
    }
   ],
   "source": [
    "print(x[0:3,0:3])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7f656a4d",
   "metadata": {},
   "source": [
    "We can control the scope (i.e., involved variables) of all posted constraints by displaying their internal representation.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "6a58887d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "allDifferent(matrix:x[][])\n",
      "allDifferent(list:[x[0][0], x[0][1], x[0][2], x[1][0], x[1][1], x[1][2], x[2][0], x[2][1], x[2][2]])\n",
      "allDifferent(list:[x[0][3], x[0][4], x[0][5], x[1][3], x[1][4], x[1][5], x[2][3], x[2][4], x[2][5]])\n",
      "allDifferent(list:[x[0][6], x[0][7], x[0][8], x[1][6], x[1][7], x[1][8], x[2][6], x[2][7], x[2][8]])\n",
      "allDifferent(list:[x[3][0], x[3][1], x[3][2], x[4][0], x[4][1], x[4][2], x[5][0], x[5][1], x[5][2]])\n",
      "allDifferent(list:[x[3][3], x[3][4], x[3][5], x[4][3], x[4][4], x[4][5], x[5][3], x[5][4], x[5][5]])\n",
      "allDifferent(list:[x[3][6], x[3][7], x[3][8], x[4][6], x[4][7], x[4][8], x[5][6], x[5][7], x[5][8]])\n",
      "allDifferent(list:[x[6][0], x[6][1], x[6][2], x[7][0], x[7][1], x[7][2], x[8][0], x[8][1], x[8][2]])\n",
      "allDifferent(list:[x[6][3], x[6][4], x[6][5], x[7][3], x[7][4], x[7][5], x[8][3], x[8][4], x[8][5]])\n",
      "allDifferent(list:[x[6][6], x[6][7], x[6][8], x[7][6], x[7][7], x[7][8], x[8][6], x[8][7], x[8][8]])\n"
     ]
    }
   ],
   "source": [
    "print(posted())"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a616e12c",
   "metadata": {},
   "source": [
    "Now, we can generate a valid Sudoku grid."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "078be930",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[\n",
      "  [1, 2, 3, 4, 5, 6, 7, 8, 9]\n",
      "  [4, 5, 6, 7, 8, 9, 1, 2, 3]\n",
      "  [7, 8, 9, 1, 2, 3, 4, 5, 6]\n",
      "  [2, 1, 4, 3, 6, 5, 8, 9, 7]\n",
      "  [3, 6, 5, 8, 9, 7, 2, 1, 4]\n",
      "  [8, 9, 7, 2, 1, 4, 3, 6, 5]\n",
      "  [5, 3, 1, 6, 4, 2, 9, 7, 8]\n",
      "  [6, 4, 2, 9, 7, 8, 5, 3, 1]\n",
      "  [9, 7, 8, 5, 3, 1, 6, 4, 2]\n",
      "]\n"
     ]
    }
   ],
   "source": [
    "if solve() is SAT:\n",
    "    print(values(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "de18c3ae",
   "metadata": {},
   "source": [
    "In a second step, we decide to take into account the clues as those in the illustration above. When 0 is present in a cell, it means that no clue is given for this cell."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "f1f06cda",
   "metadata": {},
   "outputs": [],
   "source": [
    " clues = [\n",
    "    [0, 2, 0, 5, 0, 1, 0, 9, 0],\n",
    "    [8, 0, 0, 2, 0, 3, 0, 0, 6],\n",
    "    [0, 3, 0, 0, 6, 0, 0, 7, 0],\n",
    "    [0, 0, 1, 0, 0, 0, 6, 0, 0],\n",
    "    [5, 4, 0, 0, 0, 0, 0, 1, 9],\n",
    "    [0, 0, 2, 0, 0, 0, 7, 0, 0],\n",
    "    [0, 9, 0, 0, 3, 0, 0, 8, 0],\n",
    "    [2, 0, 0, 8, 0, 4, 0, 0, 7],\n",
    "    [0, 1, 0, 9, 0, 7, 0, 6, 0]\n",
    "  ]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "27faca96",
   "metadata": {},
   "source": [
    "We can post an unary constraint *Intension* for any given clue. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "76874335",
   "metadata": {},
   "outputs": [],
   "source": [
    "satisfy (\n",
    "    # imposing clues\n",
    "    x[i][j] == clues[i][j] for i in range(9) for j in range(9) if clues[i][j] > 0\n",
    ");"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b63a399f",
   "metadata": {},
   "source": [
    "We can control that such instantiations are well defined by displaying, for example, the internal representation of the last posted constraint."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "c6027d04",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "intension(function:eq(x[8][7],6))\n"
     ]
    }
   ],
   "source": [
    "print(posted(-1,-1))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c53cfc8b",
   "metadata": {},
   "source": [
    "We can now find a solution for that grid."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "5a0cbf07",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[\n",
      "  [4, 2, 6, 5, 7, 1, 3, 9, 8]\n",
      "  [8, 5, 7, 2, 9, 3, 1, 4, 6]\n",
      "  [1, 3, 9, 4, 6, 8, 2, 7, 5]\n",
      "  [9, 7, 1, 3, 8, 5, 6, 2, 4]\n",
      "  [5, 4, 3, 7, 2, 6, 8, 1, 9]\n",
      "  [6, 8, 2, 1, 4, 9, 7, 5, 3]\n",
      "  [7, 9, 4, 6, 3, 2, 5, 8, 1]\n",
      "  [2, 6, 5, 8, 1, 4, 9, 3, 7]\n",
      "  [3, 1, 8, 9, 5, 7, 4, 6, 2]\n",
      "]\n"
     ]
    }
   ],
   "source": [
    "if solve() is SAT:\n",
    "    print(values(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6306d9ea",
   "metadata": {},
   "source": [
    "Most of the time, there is only one solution for a Sudoku puzzle. We can check this here:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "96debbdd",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Number of solutions:  1\n"
     ]
    }
   ],
   "source": [
    "if solve(sols=ALL) is SAT:\n",
    "    print(\"Number of solutions: \", n_solutions())"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "05b7ff70",
   "metadata": {},
   "source": [
    "Finally, we give below the model in one piece. Here the data is expected to be given by the user (in a command line). "
   ]
  },
  {
   "cell_type": "raw",
   "id": "4700d783",
   "metadata": {
    "raw_mimetype": "text/x-python"
   },
   "source": [
    "from pycsp3 import *\n",
    "\n",
    "clues = data  # if not 0, clues[i][j] is a value imposed at row i and col j\n",
    "\n",
    "# x[i][j] is the value at row i and col j\n",
    "x = VarArray(size=[9, 9], dom=range(1, 10))\n",
    "\n",
    "satisfy(\n",
    "   # imposing distinct values on each row and each column\n",
    "   AllDifferent(x, matrix=True),\n",
    "\n",
    "   # imposing distinct values on each block  tag(blocks)\n",
    "   [AllDifferent(x[i:i + 3, j:j + 3]) for i in [0, 3, 6] for j in [0, 3, 6]],\n",
    "\n",
    "   # imposing clues  tag(clues)\n",
    "   [x[i][j] == clues[i][j] for i in range(9) for j in range(9)\n",
    "     if clues and clues[i][j] > 0]\n",
    ")"
   ]
  }
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