## Quick solution to NYT Sudoku Hard February 24, 2021 by advanced Sudoku techniques

Easy solution to NYT Sudoku Hard February 24 2021 in three stages. Though hard, this game solved quickly by early breakthroughs using advanced techniques.

### The NYT Sudoku Hard, 24th February, 2021

Before going through the solution solve the puzzle first.

### Step by step solution to the NYT Sudoku Hard 24th February, 2021: Stage 1: Breakthroughs by Parallel digit scan and Single digit lock

First few valid cells are by row column scans.

**R1C9 1** by row column scan for 1 in R3, C8 -- **R8C7 1** by scan for 1 in R9, C8, C9.

Next **R9C2 5** by scan for 5 in R7, R8.

Next **R2C8 5** by DSA reduction of [3,4,6], [3,6] in R2 and 4 in C8, from possible digit subset DS [3,4,5,6] in top right major square -- **R3C3 5** by scan for 5 in R1, R2.

Now is the time for a **breakthrough by a PARALLEL DIGIT SCAN FOR 6 ON EMPTY CELLS OF R3:**

6 in C9 eliminates R3C9 for 6 and 6 in top middle major square eliminates the other two cells R3C4, R3C5 for 6 leaving the single cell R3C7 for 6 --

R3C7 6.

The debarring of the three out of four empty cells of R3 by parallel digit scan for 6 is shown by colored arrows. By this debarring 6, only one cell in R3, namely R3C7 is left for placement for 6.

An interesting **property of parallel digit scan** is the formation of a Cycle as a byproduct after you place the scanned digit in its single valid cell. In this game, the Cycle will naturally be formed by the three leftout digits (2,3,4).

You could have first formed this Cycle instead by individual cell DSA, and obtained the breakthrough also. But that would have been slower.

That's why we always prefer to use a parallel digit scan if we can.

The breakthrough is followed by **R1C8 3** by reduction of 4 in C8 -- **R3C9 4** by exception in top right major square.

**R6C8 6** by scan for 6 in C7, C9.

Now we'll use the second advanced digit pattern of **single digit lock to get the second important breakthrough.**

Formation: Single digit lock on 6 formedin R4C2, R4C3 by cross scan for 6 in R6, C1.

Property: By this lock, digit 6 cannot appear in any empty cell of row R4.

Use for breakthrough:By this property, the single digit lock can be used for a scan for 6 in R4--The single digit lock is now used for the breakthroughR5C6 6by scan for 6 by 6 in R6, C4 and lock on 6 in R4.

This breakthrough helps to get **R5C7 7** by DSA reduction of [1,2,5] in C7 from possible digit subset [1,2,5,7] in R5 --** R5C9 9** by reduction of [1,2] in C9 from DS [1,2,5] in R5 -- **R4C7 3** by DSA reduction of [8,9] from DS [3,8,9] in right middle major square -- **R9C7 4** by exception in C7.

**R2C6 8** by scan for 8 in R1, R3, C5.

**R4C1 5** by scan for 5 in R5, C2, C3 -- **R6C4 5** by scan for 5 in R4, R5, C5, C6. Digit 5 over.

We'll again apply parallel digit scan but this time for digit 8 on empty cells of C1 for the **third important breakthrough.**

R7C1 8by parallel scan for 8 on empty cells of C1: 8 in top left major square eliminates two cells R1C1, R2C1 for 8 and 8 in R5 eliminates R5C1 for 8 leaving the single cell R7C1 for 8 -- R7C1 8. Cells disallowed for 8 shown by colored arrows.

With this breakthrough, **R7C8 2** by DSA reduction of [8,9] in R7 from DS [2,8,9] in C8.

Three important breakthroughs are marked by blue background in the cells and white colored digits in contrast.

The exercise has the focus to get quick breakthroughs by advanced Sudoku digit patterns rather than by time-consuming enumeration of possible digit subsets in empty cells and then use the DSs for identifying breakthroughs.

Results of the actions taken shown.

### Solution to the NYT Sudoku Hard, 24th February, 2021: Stage 2: Breakthrough by effective Single digit lock and Parallel digit scan

First valid cell by row column scan: **R9C5 6** by scan for 6 in R7, R8, C4, C6.

Next we'll **apply parallel scan again** to *get an important breakthrough quickly.*

As a preparatory step first form the **single digit lock on 2** in R8C2, R8C3 **by row scan for 2 in R7. **

Now do a **parallel scan for 2 on empty cells of C5**: **Cells eliminated are: **

R2C5 by 2 in R2, R6C5 by 2 in R6, R7C5 by 2 in R7, R8C5 by single digit lock on 2 in R8.

Resulting breakthrough:R4C5 2.

With this breakthrough of R4C5 2 we get a series of valid cells:

R5C4 1by reduction of 2 --R5C1 2by exception in R5 Possible digit subset in C1 reduced to [1,4] --R1C14 by reduction of 1 in R1 --R2C1 1by exception in C1 --R6C2 1by scan for 1 in R4, C3 --R2C2 7by reduction of 7 from DS [4,7] in R2 --R2C5 4(color still red to preserve the earlier breakthrough status).

R1C4 7by scan for 7 in R3, C6 --R1C6 9by scan for 9 in R3.

Formation and effect of single digit lock on 2 as well as parallel scan for 2 on empty cells of C5 are shown by red arrows in the game status figure shown below.

### Solution to the NYT Sudoku Hard 24th February: Final Stage 3: Last breakthrough by DSA technique

Ultimate **breakthrough** in **R9C6 2 by DSA reduction** of [1,3,4] from possible digit subset DS in C6 [1,2,3,4].

**Let's explain the DSA breakthrough R9C6 2** on the basisĀ of the above game status picture.

The possible digit subset DS in four empty cells of C6 and hence in R9C6 is [1,2,3,4]. Observe that out of these four digits missing in C6, digits [1,3,4] already exist in R9 and this reduces the DS of [1,2,3,4] to single digit [2] for R6C2.

No further breakthrough needed and rest are easy picks.

**With R9C6 2,**

R3C6 3 by reduction of 2 -- R3C4 2 by exception -- R6C6 4 by reduction of 3 -- R6C3 8 by reduction of 4 -- R6C9 9 9 by reduction of 8 -- R6C5 3 by exception in R6 and by reduction -- R7C5 7 by reduction of 3 -- R8C5 9 by exception in C5 and reduction.

**With 7 in R7C5**, R7C3 4 by reduction -- R8C2 2 by reduction -- R8C3 7 by exception in bottom left major square and by reduction.

**With 4 in R7C3,** R4C3 6 by reduction -- R4C2 4 by reduction of 6 -- R4C4 9 by reduction and by exception in central middle major square -- R4C9 8 by reduction and exception in R4. With 2 in R8C2, R1C2 6 by reduction of 2 -- R1C3 6 by reduction and exception in R1.

R7C6 1 by scan for 1 in C4 -- R7C4 3 by exception in R7.

R9C8 9 by scan for 9 in R8 -- R8C8 8 by exception in C8 -- R8C4 4 by exception in R8 -- R9C4 8 by exception in whole game.

Though the Sudoku puzzle is fairly hard it could be solved it in three stages quickly without disturbing breakthrough digit patterns. It must surely be a quick solution.

This Sudoku game is a hard Sudoku puzzle that needed multiple breakthroughs by advanced Sudoku techniques - more than once Single digit lock and Parallel digit scan provided the main breakthroughs.

Focus on early breakthroughs and avoidance of unnecessary enumeration of possible digit subsets in empty cells speeds up the solution.

Though the number of filled cells 25 in this Sudoku hard is a little high for a hard Sudoku game, the digit arrangements in this Sudoku puzzle made sure that enough challenges are posed to the solver.

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