First, rough solution -- works up to a point

Makefile

@@ -0,0 +1,11 @@
+CXX = g++
+CXXFLAGS = -Wall
+
+q: main.o
+	$(CXX) $(CXXFLAGS) -o q main.o
+
+main.o: main.cpp
+	$(CXX) $(CXXFLAGS) -c main.cpp
+
+clean :
+	-rm *.o q

main.cpp

@@ -0,0 +1,104 @@
+#include <iostream>
+#include <list>
+#include <chrono>
+#include <algorithm>
+
+#define BOARD_SIZE 81
+#define BOARD_DIMENSION 9
+
+class Sudoku {
+private:
+    int board[BOARD_SIZE];
+ 
+public:
+ 
+    Sudoku(std::string s) {
+        for (unsigned int i = 0; i < s.length(); i++) {
+            board[i] = (int) (s[i] - '0');
+        }
+    }
+ 
+    void solve() {
+        // Very naive first attempt: Fails because it sets stuff wrongly and doesn't backtrack
+        for (int y = 0; y < BOARD_DIMENSION; y++) {
+            for (int x = 0; x < BOARD_DIMENSION; x++) {
+                if (get(x, y) == 0) {
+                    auto possibilities = get_possibilities(x, y);
+
+                    set(x, y, possibilities.front());
+                }
+            }
+        }
+
+        // Real search:
+        // 1. Sort possibilities by how many new constraints they create
+        // 2. Start with the one creating the smallest amount of new constraints
+        // 3. Continue until dead end
+        // 4. If dead end: Try the next one
+    }
+
+    int coordinates_to_id(int x, int y) {
+        return y * BOARD_DIMENSION + x;
+    }
+
+    int get(int x, int y) {
+        return board[coordinates_to_id(x, y)];
+    }
+
+    void set(int x, int y, int value) {
+        board[coordinates_to_id(x, y)] = value;
+    }
+
+    std::list<int> get_possibilities(int x, int y) {
+        std::list<int> possibilities{1, 2, 3, 4, 5, 6, 7, 8 ,9};
+
+        // Remove all from this row
+        for (int dx = 0; dx < BOARD_DIMENSION; dx++) {
+            possibilities.remove(get(dx, y));
+        }
+
+        // Remove all from this column
+        for (int dy = 0; dy < BOARD_DIMENSION; dy++) {
+            possibilities.remove(get(x, dy));
+        }
+
+        // Remove all from this quadrant
+        for (int dy = 0; dy < 3; dy++) {
+            for (int dx = 0; dx < 3; dx++) {
+                possibilities.remove(get((x - x % 3) + dx, (y - y % 3) + dy));
+            }
+        }
+
+        return possibilities;
+    }
+
+    void print_board() {
+        std::cout << board[0] << " ";
+		for (unsigned int i = 1; i < BOARD_SIZE; i++) {
+			if (i % 3 == 0 && i % 9 != 0) {
+				std::cout << "| ";
+			}
+			if (i % 9 == 0) {
+				std::cout << std::endl;
+			}
+			if (i % 27 == 0) {
+				std::cout << "------+-------+------" << std::endl;
+			}
+			std::cout << board[i] << " ";
+		}
+
+        std::cout << std::endl;
+    }
+};
+
+int main() {
+	//char str[82];
+	//cin >> str;
+	//Sudoku solver(str);
+	Sudoku solver("006001849030000000000020006000400320400003000600008000010060003000005004029074005");
+	//Sudoku solver("850002400720000009004000000000107002305000900040000000000080070017000000000036040");
+	solver.solve();
+	solver.print_board();
+
+    return 0;
+}