NYT hard Sudoku 25th February, 2021: Easy to understand Solution in details
In this solution to NYT hard Sudoku Feb 25, 2021, each step is explained for even a beginner to understand. Hard Sudoku not any more beyond reach.
How to find the valid single digit for empty cells and achieving breakthroughs by Sudoku techniques shown.
- Sudoku terminologies used.
- Sudoku techniques to solve hard Sudoku puzzles.
- Solution to the hard Sudoku 25th Feb 2021 of New York Times.
To skip the terminologies and move on to Sudoku techniques, click here.
And to skip both and move on 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.
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.
In the Sudoku puzzle figure below, by row column scan for digit 1 on top right 9 cell square R1R2R3-C7C8C9, with 1 in row R3, and 1 in column C8, 4 out of 5 empty cells in the 9 cell square are invalid for placing 1. As a result, the digit 1 can be placed in the single remaining cell → R1C9 1. It is the single digit candidate for the single cell R1C9.
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 possible DS for an empty cell.
In the hard Sudoku puzzle figure below, possible DS for R3C7 is [3,4,6]. These three digits do not appear in the parents: top right 9 cell square, the parent row R3 and the parent column C7.
In the 9 cell square, filled DS is [2,7,8,9]. Cell R3C7 is in two more affecting areas, the row R3 that has filled DS [1,7,8,9] and the column C7 with filled DS [2,5,8,9]. These two together has [1,5] extra to what the 9 cell square has.
The filled DS for R2C7 becomes [1,2,5,7,8,9]. Possible digit subset [3,4,6] are missing in this set of filled digits and these are the only digit candidates or possible DS for R2C7 that can occupy the empty cell.
Instead of three possible digits, if it were a single possible digit, the cell would be filled with the digit and we would have a hit.
Enumerate the possible digit for cell R2C8. Filled DS for this cell is [1,2,3,4,6,7,8,9] with only 5 missing. Possible DS is single digit 5 → R2C8 5. A hit obtained.
By the term DS in the solution, possible digit subset is indicated.
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.
As a strategy, DS is evaluated when promise of a simplifying pattern is expected. DS is generally evaluated in heavily filled up zones to keep its length short. This speeds up solution and reduces labor.
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.
The following shows and example of single digit lock of 5 in cells R7C1 and R9C1.
How a single digit lock is formed
Look at columns C1, C2 and C3 in the bottom left 9 cell square R7R8R9-C1C2C3. Out of 3 empty cells, the cell R7C3 is debarred for placing digit 5 as column C3 has a 5 and it lights up the cell for digit 5.
5 can appear only in two cells in column C1, R7C1 and R9C1 and in no other cell in the 9 cell square or the column C1.
It is locked inside these two cells in C1 and 9 cell parent square.
How a Sudoku single digit lock is used - What it does
The locked digit 5 eliminates itself from the DSs of the other two empty cells R5C1 and R6C1 and a new Cycle (2,3) is created in C1.
Focus again on the bottom left 9 cell square. With Cycle (2,3) in C1, another Cycle (5,9) is formed in the two cells of the 9 cell square. As a result, digit 1 becomes the only digit left and cell R7C3 only cell left for it in the 9 cell square.
Still more happens. With 1 in C3 now, digit 9 now must occupy the cell R6C3.
These two single digit candidates obtained by the single digit lock of 5 affects other cells and breaks the bottleneck.
As a strategy, always form a single digit lock as soon as it is discovered.
It is a powerful digit pattern. Even if its effect is not immediate, it should have positive results later.
Solution to the NYT hard Sudoku Feb 25, 2021 for beginners
Row column scan for 8 on central middle 9 cell square: 8 in R5, R6, C6 → R4C5 8.
Column scan for 2 on Right middle 9 cell square: 2 in C7, C9 → R6C8 2.
Row scan for 3 on right middle 9 cell square → 3 in R4, R5 → R6C7 3.
6 in R2, C4 → single digit lock in R3C5, R3C6 equivalent to 6 in R3.
Row column scan for 6 on top right 9 cell square: 6 in locked row R3, in C7, C8 → R1C9 6.
Column scan for 8 on top right 9 cell square: 8 in R5, R6, C6 → 8 in C7, C8 → R3C9 8.
Cycle (1,9) in right middle 9 cell square in C9 → A second Cycle (3,4,5) in left bottom 9 cell square in C9 → DS in three empty cells of the right bottom square [1,7,9], [7,9] in R9 → R9C7 1 → R7C8 9, R8C8 7.
Cycle (1,3,4) in C8 in top right 9 cell square → DS in three empty cells of the square [5,7,9] and [5,9] in R1 → R1C7 7 → R1C4 1.
Row column scan for 8 on top left 9 cell square: 8 in R2, R3, C1 → R1C3 8.
DSA on R7C4 → R7C4 8, being the only cell left to occupy the cell.
R9C2 8 Row column scan for 8 in left bottom 9 cell square: 8 in R7, R8, C1, C3 → R9C2 8.
8 in R7C4 → R9C4 2. Row column scan for 2 on central middle 9 cell square: 2 in R6, C4, C6 → R5C5 2.
More next stage.
With no easy single digit hits available, DSs of most of the cells enumerated.
With this data on board, the Cycle (1,3,5,9) is identified in R2, creating a single cell candidate → R2C3 7.
Double digit scan on C1 cells with (5,6) in top left 9 cell square and R4 eliminates (5,6) from the DS [1,3,4,9,5,6] in first four cells R1C1, R2C1, R3C1, R4C1.
Another 4 cell 4 digit Cycle (1,3,4,9) is created in these 4 cells. And as a result, a third Cycle at this stage (5,6) is created in C1 in cells R5C1 and R7C1.
These are not for immediate use.
Searching for single digit lock now (with Cycles exhausted, breakthrough must now be in a well-hidden single digit lock, that's why),
Identify the key breakthrough digit pattern in Single digit lock on 1 in central middle 9 cells square in cells R4C6, R5C6.
This lock forms because of scan on 1 in R6 and C4.
Potent single digit locks are formed usually by a cross-scan on a digit in a row and a column.
What is its effect?
It removes the 1 from the DS [1,3,4] in R7C6. The cell with reduced DS [3,4] in turn creates the destroyer Cycle (3,4) with cell R7C2 in R7.
A deluge of single digit candidates will now simplify the game.
We'll show the simplifications next stage.
With Cycle (3,4) in R7 → R7C9 5 → R7C1 6 → R5C1 5 → R7C3 1 as the only remaining digit in the row R7.
With 5 in R7C5 → Cycle (7,9) is formed in R5C2 and R5C4 → R5C9 1, R4C9 9 → R4C1 4 → R1C1 3 → R1C8 4.
More simplification next stage.
With 1 in R5C9 And Cycle (7,9) in R5 → R5C6 6 → R6C6 4, R4C6 1 being the only 1 in central middle 9 cell square → R7C6 3 → R7C2 4 → R9C3 5 → R8C5 1 being the only cell for digit 1 left in the bottom middle 9 cell square → R9C5 4 → R9C9 3 → R8C9 4.
With Cycle (1,9) in C1 and top left 9 cell square, DS in R3C2, R3C3 is [2,4] and 4 in C2 → R3C2 2 → R3C3 4.
DSA on cell R4C2 → R4C2 7 only digit left for the cell → R5C2 9 → R5C4 7, R6C3 6 → R6C5 5 → R6C4 9 → R4C3 2 → R8C3 9 → R8C2 3.
With 7 in R5C4 → R3C4 5 → R3C7 9 → R2C7 5, R3C1 1 → R2C1 9, R2C5 3 → R3C5 6, R3C6 7 → R3C8 3 → R2C8 1.
The hard Sudoku gave away easy single digit candidates at the start but then it raised a block that forced use of possible digit enumeration for most of the cells.
And breakthrough provided by the single digit lock on 1 in the central middle square during second stage analysis. Single digit lock identification and use play such an important role so often!
It is a good hard Sudoku puzzle.
If you are still reading this and you are not very experienced in solving hard Sudoku puzzles, just leave aside the solution and solve the puzzle yourself.
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 25 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.