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+# PDP-10 simple solver for Sudokus
+
+## What is a sudoku?
+
+[Sudoku](https://en.wikipedia.org/wiki/Sudoku) is a logic based number-placement
+puzzle game originally from Japan. The word *sudoku* literally means
+*digit-single*. The objecttive is to fill a 9x9 grid with numbers between 1 and
+9, without having a number repeat in a row, column or a 3x3 subgrid. Normally,
+there are a few numbers preset. In a regular sudoku, the pre-filled numbers make
+sure, that exactly one solution for the puzzle exists.
+
+The number of possible valid solutions to a sudoku is finite. In a classical
+Sudoku, there are 6,670,903,752,021,072,936,960 possible valid grids. A number
+that reduces to 5,472,730,538 essentially different solutions. The Sudokus you
+will find in a modern day puzzle book, are always one of those roughly 5
+billion, in order to guarantee a solution.
+
+### Rules
+
+A Sudoku is presented in a 9x9-grid:
+
+```
+||===|===|===||===|===|===||===|===|===||
+||   |   | 2 || 9 |   | 3 || 8 |   |   ||
+||---|---|---||---|---|---||---|---|---||
+||   |   |   || 4 | 6 | 8 ||   |   |   ||
+||---|---|---||---|---|---||---|---|---||
+|| 4 |   |   ||   |   |   ||   |   | 1 ||
+||===|===|===||===|===|===||===|===|===||
+|| 5 | 6 |   ||   |   |   ||   | 4 | 9 ||
+||---|---|---||---|---|---||---|---|---||
+||   | 2 |   ||   | 5 |   ||   | 7 |   ||
+||---|---|---||---|---|---||---|---|---||
+|| 7 | 8 |   ||   |   |   ||   | 1 | 3 ||
+||===|===|===||===|===|===||===|===|===||
+|| 2 |   |   ||   |   |   ||   |   | 7 ||
+||---|---|---||---|---|---||---|---|---||
+||   |   |   || 3 | 8 | 4 ||   |   |   ||
+||---|---|---||---|---|---||---|---|---||
+||   |   | 6 || 2 |   | 1 || 3 |   |   ||
+||===|===|===||===|===|===||===|===|===||
+```
+
+The rules are the following:
+
+- Every row must contain every number between 1 and 9 exactly once.
+- Every vertical column must contain all the numbers between 1 and 9 exactly
+  once.
+- Every 3x3 sub-grid must contain every number between 1 and 9 exactly once.
+
+## Solving Sudokus programatically
+
+### Backtracking (naïve approach)
+
+Solving a Sudoku can be achieved in serveral ways. The most simple algorithm, is
+a backtracking algorithm. With this a sudoku can be resolved basically by
+brute-forcing it. In this case, we walk through each cell of the grid
+recurseivly, and add a number. Then we check, if the grid is still valid. If it
+is, we recursivly move on to the next column. If we find a number that doesn't
+match, we return to the previous incarnation, and try another number. In order
+to check, if our grid is still valid, we can use simple function:
+
+```C
+// Function to check, if it is save to place num at mat[row][col]
+int isSafe(int[][] *mat, int row, int col, int num) {
+  // Check if num exists in the row
+  for (int x = 0; x <= 8; x++) {
+    if (*mat[row][x] == num) {
+      return 0;
+    }
+  }
+
+  // Check, if num exists in column
+  for (int x = 0; x < 8; x++) {
+    if (*mat[x][col]) {
+      return 0;
+    }
+  }
+
+  // Check if num exists in the 3x3 sub matrix
+  int startRow = row - (row % 3);
+  int startCol = col - (col % 3);
+
+  for (int i = 0; i < 3; i++) {
+    for (int j = 0; j < 3; j++ {
+      if (*mat[i + startRow][j + startCol] == num) {
+        return 0;
+      }
+    }
+  }
+
+  return 1;
+}
+```
+
+The only bit in this code, not straight forward, is the last part. By taking the
+modulus of the current row and column, we can find the top left element in each
+sub-grid, and adding the according number to it, we can then walk through.
+
+The solving algorithm itself, is surprisingly simple:
+
+```C
+// Function to solve the sudoku problem by backtracking.
+int solveSudokuRec(int[][] *mat, int row, int col) {
+  int n = sizeof(*mat);
+
+  // base case: Reached the nth column of the last row. We're done.
+  if (row == n - 1 && col == n) {
+    return 1;
+  }
+
+  // if last column of a row, go to next row
+  if (col == n) {
+    row++;
+    col=0;
+  }
+
+  // if cell is already occupied, move on
+  if (*mat[row][col] != 0) {
+    return solveSudokuRec(mat, row, col + 1);
+  }
+
+  for (int num = 1; num <= n; num++) {
+
+    // if it is safe to place num at current position, do so.
+    if (isSafe(mat, row, col, num)) {
+      *mat[row][col] = num;
+      if (solveSudokuRec(mat, row, col + 1)) {
+        return 1;
+      }
+
+      *mat[row][col] = 0;
+    }
+  }
+
+  return 0;
+}
+```
+
+And that's it.
+
+## Using the PDP-10 to solve the Sudoku
+
+The PDP-10 should be able to solve the sudoku using backtracking. The above
+algorithm will be implemented directly in PDP-10 assembly, using the MIDAS
+assembler, of the ITS operating system.