This 23 filled cell NYTimes Sudoku hard puzzle is really hard, but use of breakthrough double digit scan Sudoku technique thrice provided a quick solution.
First solve then learn from the solution. The puzzle and the solution should be enjoyable by both an amateur as well as an expert.
The NYtimes Sudoku Hard Apr 5, 2024
This puzzle has 23 out of 81 cells filled with digits. With so low filled to empty cell ratio, you can expect the Sudoku puzzle to be hard, and so it is.
Solution: NYTimes Hard Sudoku, 5th April 2024
Stage 1: All major breakthroughs, especially by double digit scan three times
Hidden singles by row column scan: R9C4 2, R4C6 5, R3C3 5, R1C6 9, R7C4 9.
Double digit scan for [8,9] in R5, R6: Cycle (8,9) in left middle major square. First major breakthrough. Later, it proved its worth.
Double digit for [7,9] in C8, C9 and R7 on bottom right major square: Cycle (7,9) in R8C7, R9C7. Second major breakthrough.
Hidden singles: R9C9 5, R6C7 5, R5C2 5.
Single digit lock on 2 in R1C7, R2C7 in C7: breakthrough hidden single R4C8 2.
Double digit scan for [8,9] in R5, R6: Cycle (8,9) in left middle major square. Third major breakthrough resource.
Major breakthrough hidden single R5C1 7. Without double digit scan this won't have been possible. Breakthrough resource used.
This in fact, is the critical breakthrough.
Parallel digit scan for 7 on R3: R3C6 7.
Cycle (1,3,4) in R2. Hidden single R2C2 7.
Naked single R3C1 4, R3C8 1, R3C7 3. Hidden single R5C9 1.
Parallel digit scan for 3 on R5: R5C3 3.
Double digit scan for [7,8] in C6 on cells of bottom middle major square: breakthrough Cycles (7,8) in R8C5, R9C5: hidden single R4C4 7.
This is the last nail in the coffin of the puzzle. Rest straightforward.
Solution next stage.
Results shown.
Stage 2: All breakthroughs already achieved, now it is just a formality
Start from R4C9 3. Rest automatic. Only naked and hidden singles.
Solution.
Sudoku Techniques: Based on the fundamental three Sudoku rules
Hidden single: Row column digit scan: Most basic:
If a digit appears in a row and a column (or a second row) to eliminate all but one cell in the intersecting major square, the digit scanned must be placed in the single cell in the major square available for it. This is a conventional nomenclature, but basically is the simple row column scan resulting in a unique valid digit cell.
DS reductions or possible digit subset reductions:
The is used nearly at every step on the way to the solution. It specifically is useful for giving naked singles or Cycles. DS reduction for breakthrough usually occurs when DS in one zone (say row) interacts with the existing common digits of a second intersecting zone (say another intersecting column) reducing the DS in the intersected cell to just 1.
Example: DS [5,7,9] in Row R8 intersects with Column C8 containing [5,9] reducing DS of intersected cell to breakthrough R8C8 7.
Naked single by DS reductions:
When DS reduction in a specific cell by the unique digits present in the affecting row, column and the major square leaves only one possible digit for the cell, we get a unique digit valid cell. This is conventionally called a Naked Single.
Naked single may appear in many ways. When a row (or column) has eight cells filled up, the naked single automatically becomes the candidate for the ninth cells.
In a Cycle of 2 (three or higher numbers) cells, two digits are locked up. When one of the two is reduced by new appearance of the digit in an affecting zone (row, column or major square) the second cells of the Cycle gets the naked single.
In essence, naked single is the only digit eligible to be placed in a specific cell (all other eight digits eliminated by their presence in affecting zones). In abstraction, hidden single is also a type of naked single, but because of its ease of use, the basic operation of finding a hidden single by row column scan, a different name is used .
Double digit scan:
Same two digits appearing in a column and an intersecting row restrict the possible cells for the two digits in the affected major square to just two. This creates a Cycle of the two digits scanned simultaneously.
The digits scanned must not be present in the major square scanned and unaffected empty cells must be exactly two for creating the breakthrough two digit Cycle.
Parallel digit scan:
In parallel digit scan, a single digit appears in a number of rows (or columns) eliminating the cells of an intersecting column (or row) for occupancy of the digit scanned. This may leave a single cell in the affected column (or row) for the scanned digit providing a breakthrough.
Cycle:
If the same set of 2 (3, 4 or 5) digits in different combinations appear in 2 (3, 4 or 5) cells of a row (or column or a major square), no other cell of the row (or column or major square) can have these Cycled digits. Example: A Cycle of (8,9) in two cells of a row debars any other cell of the row to have the digit 8 or 9.
Single digit lock:
When a single digit appears in DSs of only two cells in a row (or column), the digit is locked in this row (or column) and its cells. No other cell in the affected row (or column) can host this locked digit. Usually, a single digit lock is sought for within a major square. This debars the cells of the major square from hosting the locked digit as well. For example: if digit 4 in R4 and R6 eliminates all cells of the central middle major square for 4 except R5C4 and R5C5, we get digit 4 lock in R5 and also in central middle major square. Digit 4 cannot appear in any other cell in R5 or the major square.
Single digit locks may occur also with same digit in three consecutive cells in a major square row (or column).
Rare is the single digit lock spread over more than one major square, but these may be of great value if a pair of such single digit locks happen to share two columns and two rows resulting in more valuable breakthrough digit pattern of X wing or still more powerful chained single digit locks.
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