All breakthroughs, though not easy, still could be achieved in the first stage itself. Advanced Sudoku techniques of double digit scan and X wing are used.
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 7, 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, 7th April 2024
Stage 1: All Major breakthroughs this stage itself
Hidden singles by row column scan: R5C3 5, R7C6 8.
Double digit scan for [4,6]: Cycle (4,6) in left middle major square: Naked single R4C3 3, R7C3 7.
Hidden single R4C5 7, R5C5 8, R6C6 4, R5C6 1.
Naked single R5C8 9, R5C9 2. Hidden single R6C8 6. Naked single R7C8 3.
Parallel scan for 5 on C6: R1C6 5. Hidden single R8C5 3.
Cycles (2,9) in R4, R7 share C1 to form third Cycle (2,9) and Cycles (3,4,6) and (7,8) in C1.
Cycle (1,3,4,6) in bottom left major square restricts DS in R9C2 to [2,9] and creates unexpected breakthrough Cycle (1,2,7,9) in R9 cells R9C2, R9C6, R9C8 and R9C9 in R9:
Breakthrough: valid digit R9C4 6 by reduction of [2,7,9], R9C3 4, R9C1 3, R8C1 6, R8C3 1, R5C1 4, R5C2 6.
Single digit lock on 1 in R9 and R4 share columns C8, C9. This is the much sought after X wing digit pattern. It reduces all 1s in the shared rows and columns.
Reduction of 1 from DS in R1C8 forms [7,8]. It joins with [7,8] in R1C1 to form Cycle (7,8) in R1.
The X wing on 1 provides the critical breakthrough. Rest are routine reductions.
Stage 2: Last breakthrough left the puzzle wide open
Routine effortless valid cells fill up most of the empty cells.
Rest also routine. Shown next stage.
Results shown.
Stage 3: Formality
Just reductions.
Solution.
Sudoku Techniques: All based on the three fundamental 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.
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.
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.
X wing digit pattern: When two single digit independent locks share both rows and column, an X wing (like a large X) is formed by the four locked digits. Its power is: the pattern reduces all occurrences of the locked digit from the shared pairs of rows and columns.
This is a truly advanced digit pattern primarily based on the good old single digit locks and almost always provides the critical breakthrough in the puzzle. An effective X wing is less frequently appearing than double digit scan or parallel digit scan both of which are highly useful.
Example: A single digit lock is formed in row R4 in the pair of cells R4C8, R4C9. Another second single digit lock on 1 is formed in row R9 in cells R9C8, R9C9. It is apparent that the two independent single digit locks in two rows also share the two columns R8 and R9. Already digit 1 was barred from the two rows. Now because of column sharing, digit 1 is barred (eliminated from DSs) in other cells of the two columns C8 and C9 as well. Breakthrough occurs by reduction of 1 from DS of R1C8 giving DS [7,8]. This joins with a second DS [7,8] in R1C1 and eliminates [7,8] from all cells of R1. A cascade of valid digits is the result.
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