Rules of Sudoku

By Dr Arf

Introduction

Sudoku puzzles have become ubiquitous and many have found enjoyment in the mental exercise of solving them. Although puzzle grids appear everywhere it seems, surprisingly the rules of Sudoku are not commonly known. Unlike crossword puzzles, where almost everyone knows how since childhood, the Sudoku puzzle still has an aspect of mystery. Perhaps some are embarrassed to admit not knowing how, about something so popular. Or, perhaps Sudoku has little visual appeal, without knowledge of the rules (just some numbers in a grid). For whatever reason, the knowledge of Sudoku is not as ubiquitous quite yet.

Many sources provide the rules for Sudoku puzzles. The author’s knowledge of Sudoku rules came from an oral explanation provided by a complete stranger in a public setting. In that tradition, the rules are offered without reference.

The Grid

A Sudoku puzzle is a two dimensional grid of boxes or squares. Customarily, it is a 9×9 grid. Each position of the grid is a placeholder for a single digit or number. The solver enters digits in the open squares until the grid is filled. It’s like a crossword puzzle, but instead of letters entered at each box, numbers are entered.

The 9×9 grid is graphically divided into nine sectors of nine squares.  In a properly constructed Sudoku grid, these sectors are visually apparent and are easily identified. To make it a puzzle, some of the squares have already been given digits as a starting point. The grid below shows a fresh Sudoku puzzle, ready to be solved, with only the given entries (see Figure 1).

Fresh Sudoku Puzzle

Figure 1 – An example of a Sudoku puzzle grid.

Rules

The solver endeavors to complete the grid by entering digits that obey these four rules:

  1. Each square of the grid must contain a single digit or number (a numeral 1 through 9).
  2. A numeral, 1 through 9, can occur only once in each row of the grid.
  3. A numeral, 1 through 9, can occur only once in each column of the grid.
  4. A numeral, 1 through 9, can occur only once in each sector of the grid.

The easiest way to understand these rules is to view a solved puzzle. An example of a completed grid or solved Sudoku puzzle is below. The given digits are black. The solvers entries are blue. (see Figure 2).

Solved Puzzle

Figure 2 – An example of a solved Sudoku puzzle.

In the example solution, the row that is shaded green has each numeral only once (see Figure 3).

Only Once in Row

Figure 3 – Row has each numeral only once.

The next illustrations show that each numeral uniquely occurs in the shaded column and sector (see Figure 4 and Figure 5).

Only Once in Column

Figure 4 -  Column has each numeral only once.

Only Once in Sector

Figure 5 -  Sector has each numeral only once.

Single Solution

One more thing that is worth mentioning, a properly constructed Sudoku puzzle must have one, and only one, solution. That is to say, that for every proper Sudoku puzzle, there is a single solution.

The solver is assured that a Sudoku puzzle can be solved. Further, the solver can develop strategies that rely on knowing a proper puzzle will have only one solution.

Conclusion

The rules of Sudoku are simple. Like many things in life, simple is the best.

July 15, 2012Permalink 4 Comments

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