Sudoku strategies explained with graphics

Strategies


Overview Singles Naked Pairs Naked Triples Hidden Pairs Hidden Triples Naked Quads Hidden Quads Pointing Pair Pointing Triple Box Reduction X-Wing Finned X-Wing Sashimi Finned X-Wing Franken X-Wing Finned Mutant X-Wing Skyscraper Chute Remote Pairs Simple Coloring Y-Wing W-Wing Swordfish Finned Swordfish Sashimi Finned Swordfish Franken Swordfish Mutant Swordfish Finned Mutant Swordfish Sashimi Finned Mutant Swordfish Sue De Coq XYZ-Wing X-Cycle Bi-Value Universal Grave XY-Chain 3D Medusa Jellyfish Jellyfish Jellyfish Avoidable Rectangle Unique Rectangle Hidden Unique Rectangle WXYZ-Wing Firework Subset Exclusion Empty Rectangle Sue De Coq Extended SK Loop Exocet Almost Locked Sets Alternating Inference Chain Digit Forcing Chains Nishio Forcing Chains Cell Forcing Chains Unit Forcing Chains Almost Locked Sets Forcing Chain Death Blossom Pattern Overlay Bowman Bingo



Unique Rectangle


Let us define a Deadly Pattern as a configuration of candidates that leads to several solutions for a particular puzzle. The simplest example being a puzzle with only two unsolved Cells containing the same two candidates: such a puzzle has two solutions. A proper Sudoku puzzle has a unique solution, thus solving a puzzle should not lead to a "Deadly Pattern".

In this strategy, we are looking for potential Deadly Patterns involving two candidates in four Cells forming a rectangle: two Cells in two Rows and two Columns and two Squares in total.



UNIQUE RECTANGLE (TYPE 1) : If three corners of the rectangle are "Bi-Value" Cells with the same candidates and if one or both of these candidates are present in the Cell in the fourth corner, then none of these candidates can be the solution in the fourth corner; otherwise, this would lead to a Deadly Pattern.

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In this example candidates 6 and 7 may not be eliminated in H6, otherwise G1, H1, G6 and H6 would form a Deadly Pattern based on candidates 2 and 8. Now if the solution in H6 is 6 or 7, then candidates 2 and 8 can be safely eliminated in this cell.



UNIQUE RECTANGLE (TYPE 2) : If two corners of the rectangle in the same Row or Column are "Bi-Value" Cells with the same candidates and if the other two corners of the rectangle are "Three-Value" Cells with the same candidates plus one common candidate, then that common candidate must be the solution in one of the "Three-Value" Cells; otherwise, this would lead to a Deadly Pattern. Hence the common candidate can not be the solution in any Cell that "sees" (a Cell "sees" another Cell if both Cells belong to the same region) both "Three-Value" Cells.

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In this example if candidate 4 were not the solution in J4 or J6 (i.e. if it were eliminated from both Cells), then cells D4, D6, J4 and J6 would form a Deadly Pattern based on candidates 7 and 9. As candidate 4 must be the solution in either J4, or J6, then it can not be the solution in any cell that sees J4 and J6: G5 and H5 in Square 8, and J3, J7 and J9 in Row "J".



UNIQUE RECTANGLE (TYPE 3) : If two corners of the rectangle in the same Row or Column are "Bi-Value" Cells with the same candidates, if the other two corners of the rectangle are "Three-Value" Cells containing the same candidates plus one "extra" candidate, and if there is a "Bi-Value" Cell containing these "extra" candidates that can "see" the "Three-Value" Cells, then the "extra" candidates can not be the solution in any Cell that "sees" the Three-Value Cells and the Bi-Value Cell; otherwise, this would lead to a Deadly Pattern.

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In this example if candidate 2 were the solution in H8, it would eliminate candidates 2 in J8 and in G8; candidate 3 would be the solution in G8 which would eliminate candidate 3 in B8. Now B7, B8, J7 and J8 would form a Deadly Pattern based on candidates 4 and 6.
If candidate 3 were the solution in H8, it would eliminate candidates 3 in B8 and in G8; candidate 2 would be the solution in G8 which would eliminate candidate 2 in J8. Again B7, B8, J7 and J8 would form a Deadly Pattern based on candidates 4 and 6.
Hence candidates 2 and 3 can be eliminated in H8, and 3 from A8.



EXTENDED UNIQUE RECTANGLE : If a Column contains a NAKED TRIPLE and the corresponding Cells in another Column contain the same NAKED TRIPLE except for one Cell that contains one or several additional candidates, then one of these additional candidates must be the solution in that Cell; otherwise, this would lead to a puzzle with multiple solutions (another sort of Deadly Pattern, based on Naked Triples instead of Naked Pairs).
The reasoning is also applicable when you replace "Column" by "Row".

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You can practice this strategy by installing the SudokuCoach application on your Android™ device.

Get it on Google Play