Problem 96: Su Doku

(see projecteuler.net/problem=96)

Su Doku (Japanese meaning number place) is the name given to a popular puzzle concept.
Its origin is unclear, but credit must be attributed to Leonhard Euler who invented a similar, and much more difficult, puzzle idea called Latin Squares.
The objective of Su Doku puzzles, however, is to replace the blanks (or zeros) in a 9 by 9 grid in such that each row, column, and 3 by 3 box contains each of the digits 1 to 9.
Below is an example of a typical starting puzzle grid and its solution grid.

003020600
900305001
001806400
008102900
700000008
006708200
002609500
800203009
005010300

483921657
967345821
251876493
548132976
729564138
136798245
372689514
814253769
695417382

A well constructed Su Doku puzzle has a unique solution and can be solved by logic, although it may be necessary
to employ "guess and test" methods in order to eliminate options (there is much contested opinion over this).
The complexity of the search determines the difficulty of the puzzle; the example above is considered easy because it can be solved by straight forward direct deduction.

The 6K text file, sudoku.txt (right click and 'Save Link/Target As...'), contains fifty different Su Doku puzzles ranging in difficulty,
but all with unique solutions (the first puzzle in the file is the example above).

By solving all fifty puzzles find the sum of the 3-digit numbers found in the top left corner of each solution grid;
for example, 483 is the 3-digit number found in the top left corner of the solution grid above.

Algorithm

My program uses backtracking to solve a Su Doku: it looks for the first empty cell (which contains a zero).
Then it scans the row, column and the cell's 3x3 box to figure out which numbers could be placed in the cell (see available).
The algorithm tries all available numbers and recursively calls itself.
If no number can be placed in the current empty cell, then the algorithm failed and has to backtrack to the previous cell (return false).
If no empty cell can be found, then the Su Doku was solved (return true).

To speed up the program, the board is modified in-place.

Modifications by HackerRank

Just show the full solution of the current board.

My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, as well as the input data, too.

The code contains #ifdefs to switch between the original problem and the Hackerrank version.
Enable #ifdef ORIGINAL to produce the result for the original problem (default setting for most problems).

#include <string>
#include <iostream>
 
typedef unsigned int Board[9][9];
const unsigned int Empty = 0;
 
// find the first solution and store it in board
bool solve(Board& board)
{
for (unsigned int y = 0; y < 9; y++)
for (unsigned int x = 0; x < 9; x++)
{
// already solved cell ?
if (board[x][y] != Empty)
continue;
 
// track which numbers could be placed in the current cell
bool available[9+1] = { false, true, true, true, true, true, true, true, true, true };
// note: available[0] is a dummy entry, the program only uses available[1..9]
 
// same row and column
for (unsigned int i = 0; i < 9; i++)
{
if (board[i][y] != Empty)
available[board[i][y]] = false;
if (board[x][i] != Empty)
available[board[x][i]] = false;
}
// same region (3x3 area)
unsigned int rx = (x / 3) * 3;
unsigned int ry = (y / 3) * 3;
for (unsigned int i = 0; i < 3; i++)
for (unsigned int j = 0; j < 3; j++)
if (board[i + rx][j + ry] != Empty)
available[board[i + rx][j + ry]] = false;
 
// try all still available numbers
for (unsigned int i = 1; i <= 9; i++)
if (available[i])
{
board[x][y] = i;
if (solve(board))
return true;
}
// all failed, restore old board
board[x][y] = Empty;
return false;
}
 
// solved it
return true;
}
 
int main()
{
//#define ORIGINAL
#ifdef ORIGINAL
unsigned int tests = 50;
unsigned int sum = 0;
#else
unsigned int tests = 1;
#endif
 
while (tests--)
{
#ifdef ORIGINAL
// skip labels "GRID xy"
std::string dummy;
std::cin >> dummy >> dummy;
#endif
 
// read board
Board board;
for (unsigned int y = 0; y < 9; y++)
{
std::string line;
std::cin >> line;
for (unsigned int x = 0; x < 9; x++)
board[x][y] = line[x] - '0';
}
 
// and replace all zeros (=Empty) with proper digits
solve(board);
 
#ifdef ORIGINAL
// the first the cells
sum += 100 * board[0][0] + 10 * board[1][0] + board[2][0];
#else
// print full solution
for (unsigned int y = 0; y < 9; y++)
{
for (unsigned int x = 0; x < 9; x++)
std::cout << board[x][y];
std::cout << std::endl;
}
#endif
}
 
#ifdef ORIGINAL
std::cout << sum;
#endif
 
return 0;
}

This solution contains 13 empty lines, 15 comments and 12 preprocessor commands.

Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

This live test is based on the Hackerrank problem.

Input data (separated by spaces or newlines):

This is equivalent to
echo "003020600 900305001 001806400 008102900 700000008 006708200 002609500 800203009 005010300" | ./96

Output:

(please click 'Go !')

(this interactive test is still under development, computations will be aborted after one second)

Benchmark

The correct solution to the original Project Euler problem was found in 0.07 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=c++11 -DORIGINAL)

See here for a comparison of all solutions.

Note: interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without -DORIGINAL.

Changelog

March 13, 2017 submitted solution
May 6, 2017 added comments

Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler096

My code solves 21 out of 21 test cases (score: 100%)

Difficulty

Project Euler ranks this problem at 25% (out of 100%).

Hackerrank describes this problem as hard.

Note:
Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.
In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is never an option.

Links

projecteuler.net/thread=96 - the best forum on the subject (note: you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/project-euler-96-su-doku/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p096.java (written by Nayuki)
Scala: github.com/samskivert/euler-scala/blob/master/Euler096.scala (written by Michael Bayne)

Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.
yellow problems score less than 100% at Hackerrank (but still solve the original problem).
gray problems are already solved but I haven't published my solution yet.
blue problems are solved and there wasn't a Hackerrank version of it at the time I solved it or I didn't care about it because it differed too much.

Please click on a problem's number to open my solution to that problem:

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26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125
126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150
151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175
176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200
201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225
226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250
The 163 solved problems had an average difficulty of 22.2% at Project Euler and I scored 11,907 points (out of 13200) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
more about me can be found on my homepage, especially in my coding blog.
some names mentioned on this site may be trademarks of their respective owners.
thanks to the KaTeX team for their great typesetting library !