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NYTimes Very Hard Sudoku 16 Apr 2024: Solve Sudoku Like an Expert

NYT Hard Sudoku 16 Apr 2024: Solve Sudoku Like an Expert

Solve Sudoku like an expert. After a long series of breakthroughs by double digit scans and parallel scans, at a late stage, the critical X wing is formed.

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 hard Sudoku.

NYT Hard Sudoku April 16, 2024

NYT Hard Sudoku 16th April 2024: Solution needed special breakthroughs

Stage 1: All breakthroughs ending at the critical one

Hidden singles R5C5 4, R7C4 6, R6C2 8. Naked single R4C5 8.

Double digit scan for [1,7] in R4 on central middle major square creates Cycle (1,7) in C4. Cycles (3,5) and (2,6) in R4. This is the first major breakthrough.

Naked single R7C6 1.

Cycles (1,6) and (2,7) in C5 by double digit scan for (1,6) in C4, C6 on top middle major square.

Hidden single R2C4 9, R1C4 2.

Single digit lock on 3 in R3C4, R3C6 in R3: hidden single R1C1 3.

Naked singles R8C2 7, R3C2 1, R2C2 5, R9C2 3.

Double digit scan for [1,2] in C3 on left bottom major square creates Cycles (1,2) in C1 and (4,5,9) in C3. This is the second major breakthrough.

Parallel scan for double digit [6,7] in C4, C6 and C7 reduces [6,7] from R3 DS [3,4,8,6,7] to create breakthrough Cycle (3,4,8) in R3: breakthrough hidden single R2C1 4. This is a little complex major third breakthrough by improvised double digit parallel scan on a row with a goal to create a breakthrough Cycle.

Naked singles R2C6 8, R3C4 3, R3C6 4, R8C6 5, R9C4 8, R4C6 3, R4C4 5, R3C7 8, R8C3 9, R7C3 4, R9C3 5.

At this late stage of this very hard Sudoku, the breakthrough X wing on 2 is formed by single digit locks on 2 in R3C7, R3C9 and R7C7, R7C9 sharing columns C7 and C9: Critical breakthrough: R4C9 6.

With this breakthrough the game is thrown wide open. Rest of the cells are easy to fill.

Solution next stage.

Results shown.

NYT Hard Sudoku April 16, 2024 Solution Stage 1

Stage 2: No sweat solution

Start with R4C8 2. Rest of the cells will be easy.

Solution shown.

NYT Hard Sudoku April 16, 2024 final Solution Stage 2


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.

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.

An asymmetric X wing of two single digit locks sharing just two columns, but not the rows (or rows) will reduce the locked digit from the DSs of the column (or row).

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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