NYT Very Hard Sudoku 22 Apr 2024 could be broken open quite late in the solution process with identification of the critical breakthrough.
First solve the very hard Sudoku. Then verify from from the solution if you need.
NYT Hard Sudoku 22nd April 2024
This puzzle has 23 out of 81 cells filled with digits. This is a very hard Sudoku with its resistance to be opened up early.
NYT Very Hard Sudoku 22 Apr 2024: Solution needs lots of patience
Stage 1: All breakthroughs made out
Hidden singles R7C5 1, R6C5 2.
Triple digit scan for [2,5,8] on left bottom major square creates Cycle (2,5,8) in the major square: breakthrough hidden singles R7C2 9, R7C8 6, R2C4 3. First hammer blow on the game.
Double digit scan for [6,8] in R1, C6 on top middle major square creates Cycle (6,8) in the major square. Second blow on the game difficulty.
Double digit scan for [1,6] in C2, C3 leaves DS [1,6] in R4C1, R6C1: Naked singles: R6C1 6, R4C1 1, R6C7 7, R9C7 9, Cycles (7,8) in R8, (4,8) in R6. Now we get the results.
Hidden singles R5C9 6, R4C8 5.
Cycle (1,3,9) in R5. Single digit lock on 9 in C6.
Major breakthrough naked single R8C6 2.
Naked singles: R8C3 5, R9C1 8, R9C2 2.
Hidden singles: R9C4 5, R1C4 2, R3C1 5, R1C9 5.
Parallel scan for 1 on R1: R1C6 1. Major breakthrough.
Hidden single R2C3 2.
This is the tipping point or critical breakthrough. It opens up the game completely with only naked singles and hidden singles left.
Solution next stage.
Results shown.
Stage 2: Only naked and hidden singles, no problems
Start with hidden single R3C8 2.
All the rest are naked singles or hidden singles with no obstacles any more.
The tipping point of the critical breakthrough (after which all are naked and hidden singles) came quite late in this very hard Sudoku. In hindsight, the first triple digit scan followed by two double digit scans broke the back of the very hard Sudoku. Even then we had to wait for the critical breakthrough.
Solution.
Sudoku Techniques: Based on the fundamental 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.
Naked single may appear in many ways. When a row (or column) has eight cells filled up, the naked single automatically becomes the candidate for the ninth cells.
In a Cycle of 2 (three or higher numbers) cells, two digits are locked up. When one of the two is reduced by new appearance of the digit in an affecting zone (row, column or major square) the second cells of the Cycle gets the naked single.
In essence, naked single is the only digit eligible to be placed in a specific cell (all other eight digits eliminated by their presence in affecting zones). In abstraction, hidden single is also a type of naked single, but because of its ease of use, the basic operation of finding a hidden single by row column scan, a different name is used .
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.
Triple digit scan:
When three digits appear in a row (or column) intersecting a major square where none of the three digits scanned exist as filled digit cell AND only three cells unaffected by the intersection are empty, only these three cells can take the scanned three digits. Result is a breakthrough Cycle of the three scanned digits in the major square scanned. This is an extension of the double digit scan.
This is a very rare and extra-powerful digit pattern with the potential to break open a very hard Sudoku game completely. Same way, you can look for a four digit or even five digit scans when you spot more number of digits bunched together in one row or column.
Example: Triple digit scan of [2,6,7] in C1 on bottom left major square (the major square has none of the three scanned digits in any of its nine cells, three of the six unaffected cells have [3,4,9] and rest three unaffected cells by the scan of [2,6,7] in C1 are empty): creates breakthrough Cycle (2,6,7) in R9C1, R8C3, R7C3. Breakthroughs are many starting with R9C2 6 because of [2,7] in C2.
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.
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.
Single digit lock:
When a single digit appears in DSs of only two cells in a row (or column), the digit is locked in this row (or column) and its cells. No other cell in the affected row (or column) can host this locked digit. Usually, a single digit lock is sought for within a major square. This debars the cells of the major square from hosting the locked digit as well. For example: if digit 4 in R4 and R6 eliminates all cells of the central middle major square for 4 except R5C4 and R5C5, we get digit 4 lock in R5 and also in central middle major square. Digit 4 cannot appear in any other cell in R5 or the major square.
Single digit locks may occur also with same digit in three consecutive cells in a major square row (or column).
Rare is the single digit lock spread over more than one major square, but these may be of great value if a pair of such single digit locks happen to share two columns and two rows resulting in more valuable breakthrough digit pattern of X wing or still more powerful chained single digit locks.
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