Theorems

This page is for the discussion of Sudoku theorems. They start with fairly straightforward observations, and we hope that they will develop in time.

Definitions:

1. A 'well-formed' Sudoku puzzle is one that you never have to guess the next step in the solution, and will always have a unique answer.

2. A 'badly formed' Sudoku puzzle is one where you have to make a guess at some point in the solution, and may (or may not) have more than one solution.

3. An 'impossible' Sudoku puzzle is one that has some internal contradiction, so it cannot be solved logically unless you change at least one of the initial numbers.

Badly formed puzzle with fewest spaces

Question: what is the smallest number of spaces that can be left and the puzzle is still 'badly formed'?

Answer: 4. See example:

Theorem 1 example

You still have to guess where to place a 1 or a 2 - once you place one number, the others become fixed. You can't get lower than 4 blanks, as you need by symmetry two unknowns across a row, and across the matching columns (and thus the symmetrical final row). This leads to a minimum of 4 squares that must be blank.

Well-formed puzzle that is missing a number

Question: is it possible to set a well-formed puzzle which has no reference in its initial conditions to one of the nine numbers?

Answer: yes. Consider:

Theorem 2 graphic

This puzzle is also symmetrical, as required by the best Sudoku puzzles.