When the Unique Rectangle technique is unavoidable

Fri June 19 2020 9:24pmSudoku

A recent round of minor enhancements and corrections has made the CyberJerry Sudoku Analyzer capable of analyzing increasingly tough Sudokus. Two results: the "New Puzzle" control now offers a "Genius" level, and the "Hint" feature may offer extremely complex hints, containing many interdependent sub-steps of a variety of advanced strategies. The added focus on complex Sudokus has had another unforeseen result: the possibility of encountering what I call the "**Unavoidable Avoidable Rectangle**". (Not being able to find any mention of the phenomenon in any other website, I claim the right to assign this name to it.) Let me explain:

At any given point in working a valid Sudoku puzzle, there may exist only one way to solve one unsolved cell, and so proceed one step closer to a full solution. But many times there may be two, three, or more ways to solve the same cell or different cell(s). In such a situation, the Sudoku Analyzer will generally prefer a simple hint over a more complex one. But, whether simple or complex, the Analyzer always offers just one hint, randomly selecting from any number of possible hints at the given point. If you press the "Hint" button repeatedly at the same point, you may be randomly offered a wide variety of ways to proceed. To give you all possible hints at one time would be confusing, and would also be a departure from the design decision to proceed, as much as possible, as a human brain would proceed.

One of the advanced strategies is called the "Unique Rectangle" technique. A few Sudoku experts call it "Avoidable Rectangle", since it is based upon the "deadly pattern" that cannot exist in a valid single-solution Sudoku.

Ironically, in some complex Sudokus, this technique may be unavoidable. That is, at a given point in a given Sudoku, there may exist several ways to proceed. If one way involves the Unique Rectangle strategy, and you don't notice it, or choose to follow a different way, you may solve a different cell, inadvertantly cause the deadly pattern to disappear, and, with it, the Unique Rectangle step. In some cases, this means that subsequent hints may be much more difficult than if the Unique Rectangle technique had been used. In some rare cases, the Sudoku may become impossible to solve analytically without backtracking to the missed Unique Rectangle hint step.

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The Sudoku grid shown above illustrates this. Navigate to that Sudoku and click the "Hint" button. The Analyzer will offer you a hint consisting of 3 parts. The third part is a Unique Rectangle, case C, based upon avoiding the 'deadly pattern' in rectangle b2-b3-i2-i3. If you follow this hint, you will solve for cells h3 or i5, either one of which will make the rest of this Sudoku easy to solve.*

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But there is an alternative, at first sight easier. Instead of the Unique Rectangle, you could notice the Naked Pair in cells b2-b3 and use that to solve cell a2. Now you ask for another hint. This time there is only one hint available, a two-part hint involving a different Unique Rectangle (case A), at a1-a6-b1-b6, allowing you to solve cell a1. Next another hint with a case A at b2-b3-i2-i3, solving cells i2 or b3.*

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At this point, the original deadly pattern at b2-b3-i2-i3 has disappeared, and you will have to labor through some very complex steps to solve cells h3 or i5, which have turned out to be key cells in solving the whole Sudoku.
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In this respect, your human brain may have a distinct advantage over the Sudoku Analyzer, which doesn't remember past hint steps. By design, every point in the Sudoku is treated as a fresh start, calling for its own set of hints. Nor does the Analyzer bother much with hint steps that have not been randomly selected, as mentioned above. I may try to make the Sudoku Analyzer smarter in this regard. But in the meantime, be careful about overlooking a Unique Rectangle hint. It may end up being unavoidable and essential to reaching a full solution.

rev. Thu Jun 25 7:40pm |

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