## Base Equation Resolution Theory

Advanced methods and approaches for solving Sudoku puzzles

### Re: Base Equation Resolution Theory

Noumenon wrote:Also, to me, some people make great mental gymnastics to hide the fact they are using the vilified "brute force" methods to solve. Every time a conditional "what if" proposition is used, and the outcome eliminates a candidate, brute force is being used. One might even say that even a path length of 1 is doing this with each clear elimination of a numeral in a family which already contains it: "what if I place a numeral '2' here?" "I can't because this row already has a 2."

You seem to have in mind that "brute force" == "bad+ugly", should not being used. A brute-force elimination starts with a target candidate set to be true and is looking for a contradiction when applying single-eliminations. If there is no immediate contradiction the resolution state branches into several sub-states, sub-sub-states and so on. If all branches of the tree end in a contradiction, the initial candidate is confirmed as elimination. This works for ANY candidate that is not part of the final solution. If we happen to pick a solution candidate as target, a solution grid will be found. The branches require to make copies of the grid. This is fine for computers - they are good at it - but cumbersome for manual solvers. This is maybe a reason for the bad reputation.
You observed correctly that the principal idea of "brute-force" remains the same if the contradiction path is short and comes without branches. It is always "assumption -> contradiction". Is this a bad approach hidden by mental gymnastics? NO.
If we call the logical network made of candidates and links a "sub-puzzle", we can always transform and rewrite the sub-puzzle in a way that the very same elimination is justified straightforward without assumptions and branches. That means that "brute-force" or "assumption->contradiction" or "straightforward logic" are not inherent properties of the sub-puzzle, but only different procedures to verify the elimination target.
This means there is absolutely nothing to vilify. Contradiction logic and straightforward logic are complementary and two sides of the same coin. It's like reading from left to right or from right to left.

About uniqueness: This forum community strongly insists that a Sudoku grid has a unique solution. For me it's a non-issue. The main point is: once you decided to solve grids with eliminations only, uniqueness is required, otherwise you are unable to resolve the grid completely.

logel

Posts: 57
Joined: 29 April 2012
Location: Germany

Previous