<< problem 95 - Amicable chains | Large non-Mersenne prime - problem 97 >> |

# 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 `#ifdef`

s 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.

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

Output:

*(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:*

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |

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 |

My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

<< problem 95 - Amicable chains | Large non-Mersenne prime - problem 97 >> |