Solve NYTimes Very Hard Sudoku 18 Apr 2024 like an expert. Five out of nine major squares are rather empty. Concentrate on the more filled major squares.
First solve then learn from the solution. The puzzle and the solution should be enjoyable to you.
This puzzle has 23 out of 81 cells filled with digits. It is a very hard Sudoku.
NYTimes Hard Sudoku 18th April 2024: Solution needed expert focus
Stage 1: All breakthroughs up to the critical
Hidden single R9C1 1, R8C1 5, R8C2 2, R8C7 6.
Double digit scan of [2,4] in R7, R8 on bottom middle major square: Cycles (2,4) and (8,9) in R9. First major breakthrough in reducing overall uncertainty in the game board.
Naked singles R9C4 2, R9C5 4.
Cycle (1,8) in left middle major square by double digit scan of [1,8] in C1, R5 on left middle major square. Second major breakthrough.
Single digit lock on 6 in R5C2, R5C3. Single digit lock on 4 on same two cells as 6 by parallel digit scan for 4 on R5 creates Cycle (4,6) in the two cells. Third major breakthrough. An interesting new way of getting an important Cycle.
Cycles (2,3) and (8,9) in R9.
Naked singles R1C1 7, R2C1 4, R1C5 8. Hidden single R8C6 8.
Double digit scan for [6,7] in R6, R1 Creates Cycle (6,7) in top middle major square. Fourth major breakthrough.
Single digit lock on 3 in R1C2, R3C2.
Naked single R9C8 9, R9C7 8. Hidden single R2C9 8.
Single digit lock on 5 in C6 will create naked single R7C6 3 and the rest will follow through as mostly naked singles and a few hidden singles. This is the tipping point.
Creating Cycles by double digit scans and single digit locks played the major roles in solving the very hard Sudoku game quickly.
Routine solution next stage.
Results shown.
Stage 2: Mostly naked and a few hidden singles
Start with R7C6 3. Rest of the cells will fall through.
Solution shown.
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.
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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