Breakthroughs in NYT Hard Sudoku Apr 8, 2024 made in the first stage itself using advanced Sudoku techniques such as parallel digit scan and X wing.

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 8, 2024

This puzzle has 24 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 8, 2024

#### Stage 1: All Major breakthroughs this stage itself

**No hidden digit by single digit row column scan.**

**Double digit scan for [2,3] in C3, R8 creates Cycle (2,3)** in R7C1, R7C2 in left bottom major square. **This has been crucial.**

Hidden single R9C8 3, R3C9 3, R2C6 3, R8C2 4. *The last won't have been possible without the double digit scan first.*

**Parallel scan for 5 on C2:** R3C2 5. This won't also have been possible without the double digit scan first.

**Parallel scan for 4 on C9:** R7C9 4, R7C8 7.

**Parallel scan for 5 on R7:** R7C6 5, R7C4 8.

Cycle (2,4) in R9C5, R9C6 by double digit scan.

**Single digit lock on 6** in bottom middle major square R7C5, R8C6 and in top middle major square in R1C5, R1C6 **share columns C5, C6.** **This is asymmetric X wing**, but its effect is nonetheless significant as it removes digit 6 from DSs in shared columns C5, C6: Breakthrough R4C5 9, R6C4 6, R3C3 6 by parallel scan. Hidden single R2C3 7.

R9C3 5 by **parallel scan for 5** on R9, R6C3 8, R8C3 1, R8C1 8.

R5C7 8 by **parallel scan for 8** on C7.

Main hurdles over. Rest next stage.

#### Stage 2: Breakthroughs in the last stages solved all problems in the puzzle

Straightforward valid cells.

Rest next stage.

Results shown.

#### Stage 3: Easy and routine

Just reductions.

Solution.

### 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.

**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.

**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.

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.

### More Sudoku hard puzzles you may like to solve and learn how to solve

The updated list of **Solutions to level 3, level 4, NYT hard Sudoku** and **Expert Sudoku** puzzle games:

**How to solve Sudoku hard puzzle games full list (includes very hard Sudoku).**

*Enjoy solving Sudoku hard.*