435 lines
33 KiB
Plaintext
435 lines
33 KiB
Plaintext
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"<a href=\"https://colab.research.google.com/github/udlbook/udlbook/blob/main/Trees/SAT_Sudoku_Answers.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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\">"
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],
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"metadata": {
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"id": "QjHXD27ieTS-"
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}
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},
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{
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"cell_type": "markdown",
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"source": [
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"# Graph coloring\n",
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"\n",
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"The purpose of this Python notebook is to use investigate using SAT to solve Sudoku problems. \n",
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"\n",
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"You should have completed the notebook on SAT constructions before attempting this notebook.\n",
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"\n",
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"You can save a local copy of this notebook in your Google account and work through it in Colab (recommended) or you can download the notebook and run it locally using Jupyter notebook or similar.\n",
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"\n",
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"Contact me at iclimbtreesmail@gmail.com if you find any mistakes or have any suggestions."
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],
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"metadata": {
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"id": "jtMs90veeZIn"
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}
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},
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{
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"cell_type": "code",
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"source": [
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"# Install relevant packages\n",
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"!pip install z3-solver\n",
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"from z3 import *\n",
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"import numpy as np\n",
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"import geopandas as gpd\n",
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"import matplotlib.pyplot as plt"
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],
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"metadata": {
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"id": "mF6ngqCses3n",
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"colab": {
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"base_uri": "https://localhost:8080/"
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},
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"outputId": "fad1115b-b146-46bc-b298-7c0885fe75ac"
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},
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"execution_count": 1,
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"outputs": [
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{
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"output_type": "stream",
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"name": "stdout",
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"text": [
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"Collecting z3-solver\n",
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" Downloading z3_solver-4.14.1.0-py3-none-manylinux_2_17_x86_64.manylinux2014_x86_64.whl.metadata (602 bytes)\n",
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"Downloading z3_solver-4.14.1.0-py3-none-manylinux_2_17_x86_64.manylinux2014_x86_64.whl (29.5 MB)\n",
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"\u001b[2K \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m29.5/29.5 MB\u001b[0m \u001b[31m17.6 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
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"\u001b[?25hInstalling collected packages: z3-solver\n",
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"Successfully installed z3-solver-4.14.1.0\n"
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]
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}
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]
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},
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{
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"cell_type": "markdown",
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"source": [
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"Now let's define a Sudoku problem. We'll make a 9x9 matrix use zeros to represent unknown values."
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],
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"metadata": {
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"id": "3A5_7mByYrur"
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}
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},
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{
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"cell_type": "code",
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"source": [
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"puzzle = [[0,0,9,4,2,0,0,6,0],\n",
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" [0,7,0,9,0,5,3,0,2],\n",
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" [5,0,0,0,0,3,0,9,0],\n",
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" [0,0,0,8,0,1,0,2,0],\n",
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" [2,6,0,0,0,0,0,5,1],\n",
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" [0,1,8,2,0,0,4,0,0],\n",
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" [3,8,0,0,0,4,0,1,9],\n",
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" [0,9,4,0,3,0,6,8,5],\n",
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" [0,2,1,0,0,8,0,3,0]]"
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],
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"metadata": {
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"id": "cvGNbKkf-Qix"
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},
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"execution_count": 2,
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"outputs": []
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},
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{
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"cell_type": "code",
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"source": [
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"def expandLine(line):\n",
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" return line[0]+line[5:9].join([line[1:5]*2]*3)+line[9:13]\n",
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"\n",
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"def drawPuzzle(puzzle):\n",
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" line0 = expandLine(\"╔═══╤═══╦═══╗\")\n",
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" line1 = expandLine(\"║ . │ . ║ . ║\")\n",
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" line2 = expandLine(\"╟───┼───╫───╢\")\n",
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" line3 = expandLine(\"╠═══╪═══╬═══╣\")\n",
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" line4 = expandLine(\"╚═══╧═══╩═══╝\")\n",
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"\n",
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" symbol = \" 123456789\"\n",
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" nums = [ [\"\"]+[symbol[n] for n in row] for row in puzzle ]\n",
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" print(line0)\n",
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" for r in range(1,9+1):\n",
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" print( \"\".join(n+s for n,s in zip(nums[r-1],line1.split(\".\"))) )\n",
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" print([line2,line3,line4][(r%9==0)+(r%3==0)])"
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],
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"metadata": {
|
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"id": "0EkPFukj_-qc"
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},
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"execution_count": 3,
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"outputs": []
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},
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{
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"cell_type": "code",
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"source": [
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"drawPuzzle(puzzle)"
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],
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"metadata": {
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"colab": {
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"base_uri": "https://localhost:8080/"
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},
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"id": "CB0QFnLODWTu",
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"outputId": "3698caa2-e0a2-466a-a542-5e141d7b6bce"
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},
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"execution_count": 4,
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"outputs": [
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{
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"output_type": "stream",
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"name": "stdout",
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"text": [
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"╔═══╤═══╤═══╦═══╤═══╤═══╦═══╤═══╤═══╗\n",
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"║ │ │ 9 ║ 4 │ 2 │ ║ │ 6 │ ║\n",
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"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
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"║ │ 7 │ ║ 9 │ │ 5 ║ 3 │ │ 2 ║\n",
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"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
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"║ 5 │ │ ║ │ │ 3 ║ │ 9 │ ║\n",
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"╠═══╪═══╪═══╬═══╪═══╪═══╬═══╪═══╪═══╣\n",
|
|
"║ │ │ ║ 8 │ │ 1 ║ │ 2 │ ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ 2 │ 6 │ ║ │ │ ║ │ 5 │ 1 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ │ 1 │ 8 ║ 2 │ │ ║ 4 │ │ ║\n",
|
|
"╠═══╪═══╪═══╬═══╪═══╪═══╬═══╪═══╪═══╣\n",
|
|
"║ 3 │ 8 │ ║ │ │ 4 ║ │ 1 │ 9 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ │ 9 │ 4 ║ │ 3 │ ║ 6 │ 8 │ 5 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ │ 2 │ 1 ║ │ │ 8 ║ │ 3 │ ║\n",
|
|
"╚═══╧═══╧═══╩═══╧═══╧═══╩═══╧═══╧═══╝\n"
|
|
]
|
|
}
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"source": [
|
|
"# Takes a list of z3.Bool variables and returns constraints\n",
|
|
"# ensuring that there is exactly one true\n",
|
|
"def exactly_one(x):\n",
|
|
" return PbEq([(i,1) for i in x],1)\n",
|
|
"\n",
|
|
"def solve_soduku(puzzle):\n",
|
|
" # Create a 9 x 9 x 9 structure containing Boolean variables\n",
|
|
" # If position (i,j,k) is true, this means position (i,j) in the puzzle matrix\n",
|
|
" # takes color k\n",
|
|
" x = [[[ z3.Bool(\"x_{%d,%d,%d}\"%((i,j,k))) for k in range(0,9)] for j in range(0,9) ] for i in range(0,9) ]\n",
|
|
" # Set up the sat solver\n",
|
|
" s = Solver()\n",
|
|
"\n",
|
|
" # Add constraint one: each position where the entry is already known\n",
|
|
" # should be set to true.\n",
|
|
" for i in range(0,9):\n",
|
|
" for j in range(0,9):\n",
|
|
" if puzzle[i][j] > 0:\n",
|
|
" s.add(x[i][j][puzzle[i][j]-1] == True)\n",
|
|
"\n",
|
|
"\n",
|
|
" # Add constraint two: each position in the matrix can have only one color\n",
|
|
" # For fixed i and j, only one value of k can be true\n",
|
|
" for i in range(0,9):\n",
|
|
" for j in range(0,9):\n",
|
|
" s.add(exactly_one([x[i][j][k] for k in range(0,9)]))\n",
|
|
"\n",
|
|
" # Add constraint three: each row of the table can only have each value once\n",
|
|
" for i in range(0,9):\n",
|
|
" for k in range(0,9):\n",
|
|
" s.add(exactly_one([x[i][j][k] for j in range(0,9)]))\n",
|
|
"\n",
|
|
" # Add constraint four: each column of the table can only have each value once\n",
|
|
" for j in range(0,9):\n",
|
|
" for k in range(0,9):\n",
|
|
" s.add(exactly_one([x[i][j][k] for i in range(0,9)]))\n",
|
|
"\n",
|
|
" # Add constraint five: each 3x3 region can only have each value once\n",
|
|
" for i in range(0,9,3):\n",
|
|
" for j in range(0,9,3):\n",
|
|
" for k in range(0,9):\n",
|
|
" s.add(exactly_one([x[i+a][j+b][k] for a in range(0,3) for b in range(0,3)]))\n",
|
|
"\n",
|
|
" # Check if it's SAT (creates the model)\n",
|
|
" sat_result = s.check()\n",
|
|
" print(sat_result)\n",
|
|
"\n",
|
|
" # If it isn't then return\n",
|
|
" if sat_result == z3.sat:\n",
|
|
" result = s.model()\n",
|
|
" x_vals = np.array([[[int(bool(result[z3.Bool(\"x_{%d,%d,%d}\" % (i, j, k))])) for k in range(9)] for j in range(9)] for i in range(9)] )\n",
|
|
" solution = np.argmax(x_vals, axis=2) + 1\n",
|
|
" drawPuzzle(solution)\n",
|
|
" else:\n",
|
|
" print(\"No solution\")\n",
|
|
"\n"
|
|
],
|
|
"metadata": {
|
|
"id": "dRmdNKGdZRks"
|
|
},
|
|
"execution_count": 23,
|
|
"outputs": []
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"source": [
|
|
"# Call the routine to solve the Soduku puzzle\n",
|
|
"solve_soduku(puzzle)"
|
|
],
|
|
"metadata": {
|
|
"id": "0HeFo62NZfBp",
|
|
"colab": {
|
|
"base_uri": "https://localhost:8080/"
|
|
},
|
|
"outputId": "8840695c-8ce0-4ce1-93ef-d68d1c7098ed"
|
|
},
|
|
"execution_count": 24,
|
|
"outputs": [
|
|
{
|
|
"output_type": "stream",
|
|
"name": "stdout",
|
|
"text": [
|
|
"sat\n",
|
|
"╔═══╤═══╤═══╦═══╤═══╤═══╦═══╤═══╤═══╗\n",
|
|
"║ 1 │ 3 │ 9 ║ 4 │ 2 │ 7 ║ 5 │ 6 │ 8 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ 8 │ 7 │ 6 ║ 9 │ 1 │ 5 ║ 3 │ 4 │ 2 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ 5 │ 4 │ 2 ║ 6 │ 8 │ 3 ║ 1 │ 9 │ 7 ║\n",
|
|
"╠═══╪═══╪═══╬═══╪═══╪═══╬═══╪═══╪═══╣\n",
|
|
"║ 4 │ 5 │ 3 ║ 8 │ 7 │ 1 ║ 9 │ 2 │ 6 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ 2 │ 6 │ 7 ║ 3 │ 4 │ 9 ║ 8 │ 5 │ 1 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ 9 │ 1 │ 8 ║ 2 │ 5 │ 6 ║ 4 │ 7 │ 3 ║\n",
|
|
"╠═══╪═══╪═══╬═══╪═══╪═══╬═══╪═══╪═══╣\n",
|
|
"║ 3 │ 8 │ 5 ║ 7 │ 6 │ 4 ║ 2 │ 1 │ 9 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ 7 │ 9 │ 4 ║ 1 │ 3 │ 2 ║ 6 │ 8 │ 5 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ 6 │ 2 │ 1 ║ 5 │ 9 │ 8 ║ 7 │ 3 │ 4 ║\n",
|
|
"╚═══╧═══╧═══╩═══╧═══╧═══╩═══╧═══╧═══╝\n"
|
|
]
|
|
}
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"source": [
|
|
"Now let's try it with a more difficult example. Try to solve this yourself and you'll see that it's pretty hard."
|
|
],
|
|
"metadata": {
|
|
"id": "vL-sDmF7tnxa"
|
|
}
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"source": [
|
|
"puzzle2 = [[0,5,0,0,0,9,0,8,0],\n",
|
|
" [7,0,0,0,0,3,6,0,5],\n",
|
|
" [0,0,0,0,6,0,0,1,0],\n",
|
|
" [0,0,0,0,0,0,0,6,0],\n",
|
|
" [0,1,0,0,3,0,4,0,9],\n",
|
|
" [0,0,2,0,0,7,0,0,0],\n",
|
|
" [0,9,0,0,4,0,5,0,3],\n",
|
|
" [1,0,0,9,0,0,0,0,0],\n",
|
|
" [0,0,0,0,0,0,0,0,8]]\n",
|
|
"drawPuzzle(puzzle2)\n",
|
|
"solve_soduku(puzzle2)"
|
|
],
|
|
"metadata": {
|
|
"colab": {
|
|
"base_uri": "https://localhost:8080/"
|
|
},
|
|
"id": "3SWS2QUqtlNI",
|
|
"outputId": "8c7419ff-70b2-4322-81c0-2f8d8727de72"
|
|
},
|
|
"execution_count": 32,
|
|
"outputs": [
|
|
{
|
|
"output_type": "stream",
|
|
"name": "stdout",
|
|
"text": [
|
|
"╔═══╤═══╤═══╦═══╤═══╤═══╦═══╤═══╤═══╗\n",
|
|
"║ │ 5 │ ║ │ │ 9 ║ │ 8 │ ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ 7 │ │ ║ │ │ 3 ║ 6 │ │ 5 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ │ │ ║ │ 6 │ ║ │ 1 │ ║\n",
|
|
"╠═══╪═══╪═══╬═══╪═══╪═══╬═══╪═══╪═══╣\n",
|
|
"║ │ │ ║ │ │ ║ │ 6 │ ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ │ 1 │ ║ │ 3 │ ║ 4 │ │ 9 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ │ │ 2 ║ │ │ 7 ║ │ │ ║\n",
|
|
"╠═══╪═══╪═══╬═══╪═══╪═══╬═══╪═══╪═══╣\n",
|
|
"║ │ 9 │ ║ │ 4 │ ║ 5 │ │ 3 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ 1 │ │ ║ 9 │ │ ║ │ │ ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ │ │ ║ │ │ ║ │ │ 8 ║\n",
|
|
"╚═══╧═══╧═══╩═══╧═══╧═══╩═══╧═══╧═══╝\n",
|
|
"sat\n",
|
|
"╔═══╤═══╤═══╦═══╤═══╤═══╦═══╤═══╤═══╗\n",
|
|
"║ 6 │ 5 │ 1 ║ 7 │ 2 │ 9 ║ 3 │ 8 │ 4 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ 7 │ 8 │ 9 ║ 4 │ 1 │ 3 ║ 6 │ 2 │ 5 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ 3 │ 2 │ 4 ║ 8 │ 6 │ 5 ║ 9 │ 1 │ 7 ║\n",
|
|
"╠═══╪═══╪═══╬═══╪═══╪═══╬═══╪═══╪═══╣\n",
|
|
"║ 9 │ 3 │ 5 ║ 1 │ 8 │ 4 ║ 7 │ 6 │ 2 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ 8 │ 1 │ 7 ║ 2 │ 3 │ 6 ║ 4 │ 5 │ 9 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ 4 │ 6 │ 2 ║ 5 │ 9 │ 7 ║ 8 │ 3 │ 1 ║\n",
|
|
"╠═══╪═══╪═══╬═══╪═══╪═══╬═══╪═══╪═══╣\n",
|
|
"║ 2 │ 9 │ 8 ║ 6 │ 4 │ 1 ║ 5 │ 7 │ 3 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ 1 │ 7 │ 3 ║ 9 │ 5 │ 8 ║ 2 │ 4 │ 6 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ 5 │ 4 │ 6 ║ 3 │ 7 │ 2 ║ 1 │ 9 │ 8 ║\n",
|
|
"╚═══╧═══╧═══╩═══╧═══╧═══╩═══╧═══╧═══╝\n"
|
|
]
|
|
}
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"source": [
|
|
"Finally, let's modify the last puzzle so it is (non-obviously) impossible to solve. I've just added a 3 to position (3,3) on the grid. It looks innoccuous, but actually makes the puzzle unsolvable."
|
|
],
|
|
"metadata": {
|
|
"id": "opc4hxI2ugU7"
|
|
}
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"source": [
|
|
"puzzle3 = [[0,5,0,0,0,9,0,8,0],\n",
|
|
" [7,0,0,0,0,3,6,0,5],\n",
|
|
" [0,0,3,0,6,0,0,1,0],\n",
|
|
" [0,0,0,0,0,0,0,6,0],\n",
|
|
" [0,1,0,0,3,0,4,0,9],\n",
|
|
" [0,0,2,0,0,7,0,0,0],\n",
|
|
" [0,9,0,0,4,0,5,0,3],\n",
|
|
" [1,0,0,9,0,0,0,0,0],\n",
|
|
" [0,0,0,0,0,0,0,0,8]]\n",
|
|
"drawPuzzle(puzzle3)\n",
|
|
"solve_soduku(puzzle3)"
|
|
],
|
|
"metadata": {
|
|
"colab": {
|
|
"base_uri": "https://localhost:8080/"
|
|
},
|
|
"id": "0lfUKX66ubj0",
|
|
"outputId": "e74941c8-addd-46fd-d908-34b0d975fd71"
|
|
},
|
|
"execution_count": 34,
|
|
"outputs": [
|
|
{
|
|
"output_type": "stream",
|
|
"name": "stdout",
|
|
"text": [
|
|
"╔═══╤═══╤═══╦═══╤═══╤═══╦═══╤═══╤═══╗\n",
|
|
"║ │ 5 │ ║ │ │ 9 ║ │ 8 │ ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ 7 │ │ ║ │ │ 3 ║ 6 │ │ 5 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ │ │ 3 ║ │ 6 │ ║ │ 1 │ ║\n",
|
|
"╠═══╪═══╪═══╬═══╪═══╪═══╬═══╪═══╪═══╣\n",
|
|
"║ │ │ ║ │ │ ║ │ 6 │ ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ │ 1 │ ║ │ 3 │ ║ 4 │ │ 9 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ │ │ 2 ║ │ │ 7 ║ │ │ ║\n",
|
|
"╠═══╪═══╪═══╬═══╪═══╪═══╬═══╪═══╪═══╣\n",
|
|
"║ │ 9 │ ║ │ 4 │ ║ 5 │ │ 3 ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ 1 │ │ ║ 9 │ │ ║ │ │ ║\n",
|
|
"╟───┼───┼───╫───┼───┼───╫───┼───┼───╢\n",
|
|
"║ │ │ ║ │ │ ║ │ │ 8 ║\n",
|
|
"╚═══╧═══╧═══╩═══╧═══╧═══╩═══╧═══╧═══╝\n",
|
|
"unsat\n",
|
|
"No solution\n"
|
|
]
|
|
}
|
|
]
|
|
}
|
|
]
|
|
} |