The rare gem of NYTimes 20th April 2024 Very Hard Sudoku has only one entry point
Solve NYT Very Hard Sudoku 20 Apr 2024 with no hidden single or any kind of naked single. The entry point of this Sudoku is just one. Expertise needed.
First solve the very hard Sudoku. Then learn from the solution. The puzzle and the solution should be very interesting to you.
NYT Hard Sudoku 20 Apr 2024
This puzzle has 24 out of 81 cells filled with digits. It is an extremely rare kind of very hard Sudoku puzzle with only one entry point at the start.
NYT Very Hard Sudoku 20 Apr 2024: Finding the entry point with no naked or hidden singles was tough
Stage 1: All breakthroughs ending at the critical one
Not a single naked or hidden single or a naked single by single digit lock or parallel digit scan. This one is truly a very hard Sudoku, at least in the beginning. The entry point of this rare Sudoku is only one.
Double digit scan of [4,8] in R7, R9 on bottom middle major square: Cycles (4,8) in the major square.
A difficult to visualize R2C6 4 by parallel scan for 4 on C6.
Triple digit scan of [1,3,7] in C6 on the bottom middle major square again creates a new Cycle (2,6) in C6 in the major square and another Cycle (8,9) in C6 with residual digits.
Single digit lock on 8 in C8 pairs up with the lock on 8 in Cycle (8,9) sharing rows R5, R6: forms X wing on 8: Critical early breakthrough naked single R5C1 5 followed by a second naked single R5C2 2 and hidden single R4C1 8. Cycle (4,9) in C3.
Hidden single R7C3 2, naked singles R7C6 6, R9C6 2.
DS [1,4,6,7] in C1 reduced by [4,6] in bottom left major square to create Cycle (1,7) in R8C1, R9C1: and Cycle (3,5) in R8C2, R9C2 in C2: naked singles R2C1 6, R1C1 4, R1C2 8, R3C2 7 and Cycle (1,3,5) in C3.
Hidden single R2C8 7. Naked single R2C4 2.
DS [2,4,3,6,7] in central middle major square is reduced by [2,4,7] in C5 to form major potential breakthrough Cycle (3,6) in R5C4, R5C4: major breakthrough naked single R4C5 4.
Rest easy. Shown next stage.
Results shown.
Stage 2: Only naked and hidden singles, no problems
Start with R8C5 8. Rest tedious but certain with no difficulty at all. Only naked singles and hidden singles.
Solution shown.
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.
X wing digit pattern:
When two single digit independent locks share both rows and column, an X wing (like a large X) is formed by the four locked digits. Its power is: the pattern reduces all occurrences of the locked digit from the shared pairs of rows and columns.
This is a truly advanced digit pattern primarily based on the good old single digit locks and almost always provides the critical breakthrough in the puzzle. An effective X wing is less frequently appearing than double digit scan or parallel digit scan both of which are highly useful.
An asymmetric X wing of two single digit locks sharing just two columns, but not the rows (or rows) will reduce the locked digit from the DSs of the column (or row).
Example: A single digit lock is formed in row R4 in the pair of cells R4C8, R4C9. Another second single digit lock on 1 is formed in row R9 in cells R9C8, R9C9. It is apparent that the two independent single digit locks in two rows also share the two columns R8 and R9. Already digit 1 was barred from the two rows. Now because of column sharing, digit 1 is barred (eliminated from DSs) in other cells of the two columns C8 and C9 as well. Breakthrough occurs by reduction of 1 from DS of R1C8 giving DS [7,8]. This joins with a second DS [7,8] in R1C1 and eliminates [7,8] from all cells of R1. A cascade of valid digits is the result.
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