NYT hard Sudoku 21st February, 2021: Easy to understand Solution in details
In this solution to NYT hard Sudoku Feb 21, 2021, each step is explained so that even a beginner can understand. You also can solve a hard Sudoku now.
How to find the valid single digit for each empty cell in the hard Sudoku puzzle is explained using basic Sudoku rules and special patterns of digits that appear.
First to go through are meaning of a few terms used and the Sudoku techniques to solve a hard Sudoku puzzle without much effort. If you wish to skip and go to the solution direct, click here.
Terminologies used in explaining solution to the NYT hard Sudoku
Row column labels and names for 9 cell squares
The 9 by 9 Sudoku game has 81 cells in 9 rows and 9 columns.
Rows are labelled R1 to R9 and columns C1 to C9.
81 cells are further divided into 9 groups of 9 square cells. These are either referred as top - left, middle, right; bottom - left, middle, right and left middle, central middle and middle right.
These 9 square cell groups are also referred by their specific labels joining the labels for the three rows and three columns. For example the specific label for the left middle 9 cell square will be R4R5R6-C1C2C3.
Convention followed is to mention row label suffixed by the column label. For example the top leftmost cell is referred as R1C1.
9 by 9 Sudoku has three conditions for a digit to occupy a cell,
- No digit can be repeated in any row. Every row must have the 9 digits 1 to 9 without repetition.
- No digit can be repeated in any column. Every column must have the 9 digits from 1 to 9 without repetition.
- Each of the 9 numbers of 9 cell squares must be filled up by exactly 9 digits from 1 to 9 without repeating any digit.
Sudoku techniques to make a breakthrough using specific patterns in the digits
Row column scan for a digit
This is the simplest for identifying a cell where only one digit CAN be placed. No other cell in the 9 cell square, the row or the column containing this specific cell would have this VALID digit except this specific cell that you have found.
For example, if digit 5 appears in R1 (in top left 9 cell square), R2 (in top right 9 cell square), and C4, C5, the only valid cell where 5 can be placed in R3 and top middle 9 cell square will be R3C6.
Possible digit subset (DS) in a cell
For every empty cell there will be a subset of the 9 digits each of which will be a potential candidate for the cell. This is the possible digit subset or DS for a cell.
In the NYT hard Sudoku Feb 21, 2021, for R2C2, DS is [3,4,5] as these digits do not appear in the parent 9 cell square, the parent row R2 and the parent column C2.
Digit subset analysis or DSA
When no valid cell by row column scan is visible, try to find a cell for which DS has a single digit. The cell will then have a valid digit and complexity of the game will reduce a little.
For example, DSA for R2C1 in our Sudoku game is the single digit 3. Verify.
This technique is still easy and is the savior oftentimes.
Cycle of digits
If 2 digits, say [4,6], appear as a lone pair in 2 cells (without any other digits possible in the two cells) of a 9 cell square, a column or a row, a Cycle of 2 digits is formed. These two digits have to be placed in either of these two cells in the Cycle and in no other cell of the parent zones 9 cell square, the column or the row containing the lock. (a ZONE is a 9 cell square, a row or a column).
The two digits can then be eliminated from all the DSs of the parent zones of the Cycle.
A Cycle of 3 digits in 3 cells are encountered often. 4 cell Cycles can also be useful if it can be located.
Cycles are valuable patterns to identify.
Single digit lock - Conditions for single digit lock - how to identify it
Two conditions for single digit lock pattern,
- the digit can be placed in only two or three cells of a column or a row, AND,
- the locking cells must also be in SAME 9 cell square.
- the locked digit should not be present as a single cell candidate in both the adjacent two 9 cell squares through which the locked column or row passes.
In an example single digit lock for digit 1: Digit 1 not only can appear ONLY IN TWO CELLS R4C8, R5C8 of column C8, the cells are in the SAME 9 cell square R4R5R6-C7C8C9. This digit 1 lock satisfies the third condition as well because the right bottom adjacent 9 cell square doesn't have locked digit 1 as a pre-filled digit.
Even if the third condition is not satisfied, single digit lock may be present in a 9 cell square, but that cannot be used for any positive outcome.
Positive effects of single digit lock
Two positive effects of a single digit lock pattern,
- the locked digit CANNOT APPEAR IN ANY OTHER CELL OF THE 9 CELL SQUARE, and,
- also CANNOT APPEAR IN ANY OTHER CELL OF THE LOCKED COLUMN or ROW.
This reduces the number of possible digits in the corresponding cells, generally resulting in a breakthrough.
This is a powerful pattern to identify at the earliest.
Solution to the NYT hard Sudoku Feb 21, 2021 for beginners
Possible digit subset in R2 is [3,4,5,7]. With [4,5,7] in C1 → R2C1 3.
Reduced digit subset for R2 is [4,5,7]. With [4,5] in C9 and top right 9 cell square → R2C9 7.
Row column scan for 6 → 6 in R4, R6, C3 → R5C2 6.
Possible digit subset in empty cells of R6 is [2,4,7,8,9]. With [2,4,8,9] in R6 and C7 together → R6C7 7.
Possible digits in top right 9 cell square [1,3,4,6]. With [4,6] in C9, Cycle (1,3) formed in C9 → Cycle (4,6) in C8 → DS (1,3,8,9) for right middle 9 cell square → DS [8,9] in R4C9.
Row scan for 1 → 1 in R5, R6 → R4C7 1.
3 in R6, DS [3,8,9] in R6C8 → R6C8 [8,9]→ R5C8 3.
DS [7,8,9] in empty cells of R5, digits (8,9) in C4 → R5C4 7 → 8 in C3 → R5C3 9 → R5C5 8.
Row column scan for 7 → 7 in R1, C4, C5 → R3C6 7.
Column scan for 8 → 8 in C4, C5 → R1C6 8.
More simplification next stage. Results below.
With 6 in R3 → R3C8 4 → R1C8 6.
Analyze the special pattern of [1,6] in both columns C5, C6 in top middle square and central middle square. Effect of this is formation of Cycle (1,6) in R7C4 and R9C4 in bottom middle square.
This is double digit column scan for (1,6).
Combine this with single digit lock on 6 in R7C7, R7C9 by scan for 6 in C8, C9.
You get the Single digit rectangle for 6 blocking two columns C4, C6 and two rows, R7,R9. In no other cell of these two pairs of columns and rows 6 can appear.
Now do row column scan for 6 in bottom left 9 square cell → 6 effectively in R7, R9, C2, C3 → R8C1 6. This is a hard earned critical breakthrough.
We'll use yet another special pattern. Identify digit 1 in C2, C5, C7, C9.
Scan now for digit 1 in empty cells of row R8. 1 in four intersecting columns C2, C5, C7 and C9 [Cycle (1,3) blocks C9 for 1 in any other cell of the column] will leave only the single cell R8C3 for 1 → R8C3 1.
This breakthrough is by parallel scan for a digit on empty cells of a row.
Scan for 1 in top left 9 cell square → 1 in C2, C3 → R3C1 1 → R3C9 3 → R1C9 1 → R2C9 7.
More next stage.
Results till now shown below.
Row scan for 9 → 9 in R1, R2 → R3C2 9 → 9 in C2 C3 → R9C1 9.
Cycle (2,8) formed in C1 of left middle 9 cell square. This results in a second Cycle (4,7) in R4 of left middle 9 cell square.
Do possible digit analysis for R4C6 → DS [2,3,8,9] in R4, [2,8] in C6 → DS of R4C6 [3,9].
DS [3,4,5,9] in bottom middle 9 cell square, 4 in R7, 5 in C6 → DS [3,9] in R7C6.
It forms a new Cycle (3,9) in column C6 with [3,9] in R4C6 → R9C6 4.
This is a breakthrough by forced chain of Cycles.
Scan for 4 → 4 in R7, R9, C9 → R8C7 4.
Scan for 3 in top middle 9 cell square → 3 in R2, R3 → R1C5 3, Cycle (2,4) in R1, top left 9 cell square → R2C2 5 → R2C4 4 → R3C2 9.
DS [2,3,5] in C4, [3,5] in R6 → R6C4 2 → R4C4 3 → R3C4 5 → R3C5 2.
More next stage.
Results shown below.
With 2 in R6 → R6C1 8 → R4C1 2.
With 3 in R4C4 → R4C6 9 → R4C9 8 → R6C8 9 → R7C6 3.
With DS [5,9] in R9C5 and 9 in R9 → R9C5 5 → R8C5 9 → R6C5 4 → R8C2 3 → R8C9 5.
With DS [2,9] in R9C9 and 9 in R9 → R9C9 2 → R8C9 9.
DS [3,6] in R7C7 and 3 in R7 → R7C7 6 → R9C7 3 → R7C4 1 → R9C4 6 → R7C8 8 → R9C8 1.
With DS [7,8] in R9 and 8 in C3 → R9C3 7 → R4C3 4 → R4C2 7 → R1C3 2 → R1C2 4.
Being last digit in row → R9C2 8 → R7C2 2 → R7C3 5.
Comments on the NYT hard Sudoku 21 Feb, 2021
In addition to the basic Sudoku techniques and single digit lock explained at the start, three more advanced patterns and techniques have been used,
- Double digit column scan: In Stage 2, it so happened that Digits [1,6] are in both columns C5, C6 in top middle square and central middle square.
As the outcome, Cycle (1,6) in R7C4 and R9C4 In bottom middle square was formed.
This is use of single digit scan twice only. Nothing very different.
- A single digit rectangle: This pattern was identified in also Stage 2. When two single digit locks on digit 6 share two same columns and two same rows, the FOUR OCCURRENCES of the digit AT FOUR CORNERS OF THE RECTANGLE SO FORMED block appearance of the digit in any other cell of the two columns and the two rows.
The net outcome of this special pattern of digits has been the important breakthrough R8C1 6.
- Parallel scan for a single digit on the empty cells of a row or a column: in a normal column (or row) scan for a digit, the digit appears in 2 or 3 columns intersecting ONE specific 9 cell square resulting in isolating ONE single cell for the digit.
The parallel scan instead uses the fortunate pattern of a digit missing in the EMPTY cells of a row AND THE DIGIT APPEARING IN COLUMNS INTERSECTING ALL EMPTY CELLS OF THE ROW EXCEPT ONE.
In our hard Sudoku, R8 had FIVE empty cells with no digit 1 AND FOUR of the columns intersecting the empty cells C2, C5, C7, C9 had 1. Just one empty cell R8C3 was left for placing 1.
This is a variation of single or double column scan for a digit on ONE 9 cell square. When a parallel situation occurs, there would always be a Cycle in the cancelled out cells. This pattern is effective in difficult situations and on 5 empty cells spanning over more than one 9 cell squares.
The Sudoku game can be classified as extra hard.
An often asked questions is, "What makes a Sudoku puzzle hard?"
There is no metric or measurement to decide for sure that a Sudoku puzzle is hard, or not so hard.
Nevertheless, a distinction between hard, medium and easy Sudoku categories can be made.
As a ball-park figure, any Sudoku puzzle having number of filled up cells 26 and below should be taken as hard or extra hard.
In medium or easy Sudoku puzzles, number of filled up cells will be more. In easy ones it can be 36 or more.
These are rough figures drawn from experience.
For full enjoyment, avoid looking into any solution as well as the answer.
The joy of discoveries will then all be yours.
Other Sudoku puzzles you may like to go through at leisure
Hard Sudoku level 4 puzzles
New York Times Hard Sudoku puzzles
NYT hard Sudoku 21 Feb 2021: Solution for Beginners
Hard Sudoku level 3 puzzles
You may access all hard Sudoku level 3 solutions at Third level hard Sudoku.
Medium level 2 puzzles
You may read through all medium level 2 solutions at Second level medium Sudoku.
For beginners, Sudoku beginner puzzle solutions are at Beginner level Sudoku.