NYT Hard Sudoku Apr 6, 2024 needed use of the very powerful and rare quadruple digit scan Sudoku technique to break it open. Without this, it is tough.
First solve then learn from the solution. The puzzle and the solution should be enjoyable by both an amateur as well as an expert.
NYT Hard Sudoku Apr 6, 2024
This puzzle has 25 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: NYT Hard Sudoku Apr 6, 2024
Stage 1: Major breakthroughs
Hidden singles by row column scan : R7C8 3, R6C5 3, R9C1 4.
Parallel scan for 4 on C1: R9C1 4.
Quadruple digit scan for [2,3,4,5] in R3, C5 on top middle major square: this will invariably create a four digit Cycle in the major square scanned. This is and extension of double digit and triple digit scans based on same principles of all the four digits scanned in the major square must have no common digit in the major square with exactly four number of empty cells with no unique digit.
- Start of a major breakthrough: Cycle (2,3,4,5) in the major square as expected, but a second unexpected Cycle (1,6) created in C5 column along with a third Cycle (7,9) in the column and the crucial fourth Cycle (1,8) in the top middle major square itself. This creates the much awaited breakthroughs. We will show the effect of this fourth Cycle (1,8) in R3 next stage.
Stage 2: Effect of the previous stage critical breakthrough shown: No further difficult hurdles
Reductions: R1C5 6, R8C5 1, R8C3 9, R3C7 7, R3C5 9, R2C5 7.
Parallel scan for 6 on C4: breakthrough R8C4 6, reduction R8C6 3.
Hidden single scan R1C4 3.
Reduction: R8C2 2. Cycle (4,5) in R8: Reductions: R7C7 1, R7C3 8, R9C2 1, R1C2 8, R1C1 5, R2C1 9, R2C9 8, R6C1 8, R6C3 5,
R4C3 6, R2C3 1, R2C2 6.
Routine solution next stage.
Results shown.
Stage 3: One parallel scan, naked single, hidden single and rest routine DS reductions
Parallel scan for 8 on C4: R3C4 8.
Reduction from DSs: R3C6 1.
Hidden single by row column scan: R4C4 1.
Naked single by DS reduction: R6C7 4.
Hidden single by row column scan: R4C7 8.
Rest reductions from possible digit subsets (DSs).
Solution shown.
Sudoku Techniques: Based on the basic 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.
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.
Quadruple digit scan: When four digits appear in a row (or a column) and intersecting column (or row) restrict the possible cells for the four digits in the affected major square to just four, a Cycle of the four digits scanned is formed. Unaffected empty cells in the major square scanned must also be exactly four, and no digit scanned must be present in the major square scanned. This follows the same principle of double digit scan and triple digit scan.
Example: Digits [2,3,4,5] appear in R3 and intersecting column C5 both interacting with the top middle major square with exactly four unaffected empty cells and none of the scanned digits present in it. The powerful scan creates a four digit Cycle (2,3,4,5) in the four unaffected empty cells and a second Cycle in rest of the empty affected cells.
A triple digit scan or a quadruple digit scan is rare and invariably provides a major breakthrough by drastically reducing the uncertainty in the whole puzzle game. Double digit scans appear more frequently in hard Sudoku puzzles.
While solving an especially hard Sudoku puzzle, lookout should always be for multiple digit scans.
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.
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Enjoy solving Sudoku hard.