Hard Sudoku New York Times 28th February, 2021: Easy to understand Solution using Sudoku patterns
How to solve hard Sudoku New York Times February 28, 2021 explains each breakthrough.
Sudoku techniques and Sudoku patterns explained separately.
Sudoku techniques are also highlighted in the solution to the NYTimes hard Sudoku 28th February, 2021.
- Sudoku terminologies used.
- Sudoku techniques for hard Sudoku puzzles.
- Solution to the hard Sudoku 28th Feb 2021 of New York Times.
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Terminologies used in explaining solution to the 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 labeled 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 right middle.
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 three parent areas: 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 digits [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, we would have a hit and the cell would be filled with the digit.
Let's 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. Git a hit.
By the term DS in the solution, possible digit subset is indicated.
Digit subset analysis or DSA and Strategy on DS evaluation for all empty cells
When no valid cell by row column scan is visible by row column scan, 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.
In the above figure, by DSA on cell R2C8 single digit candidate 5 for the cell is found. A row column scan for 5 on this cell won't have given this hit. That's the advantage of DSA over row column scan.
As a strategy,
Apply DSA on cells in an area with many filled digits in the overlapping rows, columns and 9 cell square. Focus on ONE 9 cell square.
In the example, focus has been on top right 9 cell square.
DS generally is 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.
Do not attempt to enumerate DSs for all empty cells at the start of solution unless absolutely forced.
- First reduce number of empty cells as much as possible by Sudoku techniques of Row-column scan and Digit subset analysis.
- While doing row column scan, always look for a potent single digit lock and whenever found record it.
- While doing the digit subset analysis, go on to fill the short length DSs that are easy. These short length DSs will have the advantage of discovering powerful simplifying pattern of Cycles.
These three steps are overlapped till a point is reached when no simplifying pattern can be found unless longer DSs of empty cells are evaluated. Only now start evaluating DSs of length preferably 3 (4 occasionally, never 5), but that too in promising areas.
Two points to remember,
- A hard Sudoku game from a reliable source WILL HAVE A SINGLE SOLUTION and that too a SINGLE THREADED SOLUTION.
- For such a good hard Sudoku puzzle, for hiding the SINGLE CRITICAL BREAKTHROUGH Sudoku PATTERN, CENTRAL MIDDLE SQUARE is the most promising area. In any case, there will certainly be a single critical breakthrough pattern for a good hard Sudoku game.
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.
Examples of Cycles shown.
Cycles in C9: (4,6) in R5C9, R6C9 in C9 and right middle 9 cell square, Cycle (1,2,3,5) again in C9 in cells R1C9, R4C9, R8C9 and R9C9.
Cycles in top right 9 cell square: Cycle (3,4,8) in C8 and top right 9 cell square in cells R1C8, R2C8, R3C8, Cycle (1,2) in R1 and top right 9 cell square in cells R1C7, R1C9,
Cycle in C1: Cycle (3,5) in C1 in cells R6C1, R8C1,
Cycle in R8: Cycle (3,5) in R8 in cells R8C1, R8C9.
- A Cycle may occur in a row, a column or ALSO in a 9 cell square. Occasionally a Cycle belongs to a row (or column) AND a 9 cell square.
- A Cycle of 2 digits always is on a row (or column) and a 9 cells square. This is seen often. Its effect is—elimination of all digits in the Cycle from the DSs of the mpty cells of the affected areas.
- A Cycle of 3 digits also is seen often and occasionally belongs to a row (or column) AND a 9 cell square. Its effects is same. All its digits get eliminated from the DSs of empty cells in the affected areas.
- Identify and use 2 and 3 digit Cycles as much as possible.
A 4 digit Cycle is relatively rare and its appearance in a row or column is invariably interlinked with a possible Parallel scan on the row (or column) in which the 4 digit cycle appears.
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.
For a single digit lock to be effective, 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.
Such a single digit lock satisfying the third condition also, is referred to as a Potent Single Digit Lock.
The following shows an example of a potent 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.
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.
Step by step Solution to Hard Sudoku NYTimes February 28, 2021
First breakthrough from Cycle (3,5) in R6C1, R8C1 formed from DS [1,3,5,8] for column C1, [1,8] in both R6 and R8.
Outcome → R4C1 1 → R3C1 8.
Scan 1: 1 in R4, R6, C4 → R5C6 1.
Scan 8: 8 in R7, R8, C2 → R9C3 8.
Scan 9: 9 in R1, C8 → R2C9 9 → Cycles (1,2) in R1C7, R1C9 and (3,4,8) in R1C8, R2C8, R3C8.
Double digit scan for [4,6] on left middle 9 cell square: [4,6] in R4, C7 → Cycle (4,6) in C9 and right middle 9 cell square. Additional Cyclde (1,2,3,5) in C9 is a byproduct.
Note: Double digit scan is scanning for two digits as a pair. Forming the Cycle of [4,6] is a good example of use of double digit scan.
More next stage.
Parallel scan for 5 on empty rows of R5: 5 in C3 debars R5C3, 5 in C4 debars cell R5C4, 5 in C7 debars cell R5C7 and 5 in Cycle (1,2,3,5) in C9 debars cell R5C9 → R5C2 5, the only cell left for 5 in R5.
This is an effect of a Cycle (4,6,7,8) in R5 and is not an easy breakthrough.
Note: It is easier to identify the chance of a parallel scan compared to forming a four digit long Cycle.
A few next hits.
5 in R5C2 → R6C1 3 → R8C1 5 → R8C9 3. And the game is again closed.
We'll enumerate the DSs of the empty cells that are easy to enumerate.
Attention is on the central middle square cells, the more promising area for the breakthroughs to take place.
And sure enough, the critical breakthrough Cycle (2,5,7,9) is formed in cells R4C2, R4C5, R4C8 and R4C9. Removing [2,9] from DS [2,8,9] of cell R4C4 → R4C4 8.
This is the most important breakthrough in the game.
We'll show its effects next stage.
With 8 in R4C4 → R5C4 6 → R5C9 4 → R5C3 7, R6C9 6 → R5C7 8 → R4C7 3.
Scan 7: 7 in R2, R3, C3 → R1C2 7 → R4C2 9 → R8C2 6 → R8C8 7 → R4C8 5 → R4C9 2 → R1C9 1 → R1C7 2, R9C9 5 → R7C8 6, R6C7 7 → R4C5 7, left out digit.
Rest in next stage.
No more challenges left.
DS [1,3,4] in C2 and 4 in left bottom 9 cell square creates Cycle (1,3) in R7C2, R9C2 → R3C2 4 → R3C8 3.
R6C3 4, digit left → R1C3 3 → R2C3 1, R2C5 4, R1C5 6, R2C8 8, R1C8 4 → R2C6 3 → R1C6 8.
Scan 1 in C4, C6 → R3C5 1.
DSA on R7C4 3 → R9C4 4, R7C2 1, R9C2 3 → R7C7 9 → R9C7 1 → R7C3 2 → R8C3 9.
With 9 in R8C3 → R8C6 2 → R3C6 9 → R3C4 2 → R6C4 9, R6C5 2.
With 2 in R8C6 → R6C6 5.
Last single digit candidates: R7C5 5 → R7C6 7 → R9C6 6 → R9C5 9.
This is an extra hard Sudoku puzzle that needed a number of important breakthroughs. Twice it closed down with no quick getaway visible.
If you are still reading this and you are not so 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
How to solve hard Sudoku NYTimes February 28, 2021
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.