I just managed to finish another installment of our Sudoku Compilation. To be honest, I am not as happy with today’s six puzzles as I was with those from the first two groups in the series. But then, it so happens that I find Clone Sudokus generally less exciting than other variants. They belong to a category one might call “Equality Sudokus”, meaning that cells are denoted which must contain the same entry (Palindrome Sudokus are another somewhat popular representative of the same category).
As I see it, such variants do not have much to offer when it comes to additional solving techniques. Basically, they are using the same stuff that is already known from Standard Sudokus (Naked and Hidden Singles, followed by Pointing Pairs, X-Wings, etc.), just in more confusing ways. You are welcome to disagree with my assessment, of course. Anyway, here is the new set: Clone Sudoku
Rules: Enter numbers from 1 to 6 (1 to 4 in the example) so that each number appears exactly once in every row, every column and every outlined region. Shaded regions with the same shape and orientation must contain identical entries in corresponding positions.
Example and solution:
When it comes to the rules, it should be clarified what counts as a “region”. As far as I know, it is considered standard to interpret only that which is orthogonally connected as a single region; marked areas which touch each other diagonally are treated as separate regions. The latter situation is present several times in the PDF file, and in particular in our example as well.
Next, let me stress that numbers may repeat within a Clone region. This is something which cannot be seen in the example and should therefore be emphasized. (For example, the Clone constellation in today’s fifth puzzle is such that the puzzle would – even without the givens! – have no solution at all unless the shaded 3×2 block has one pair of repeating entries.)
It should also be noted that the Clone regions may appear more than twice within a puzzle. Once again, this is a situation I have omitted in the example due to the very limited space on a 4×4 grid, but the first 6×6 puzzle features such occurrences. Observe, however, that the different orientations of the domino-shaped regions in the same puzzle give rise to two separate groups of Clones.
There is, as I mentioned above, little to say about solving logic. It is usually a good idea to study the candidates for shaded cells, using the Clone rule. It may well turn out that certain positions in the Clone regions admit only one entry. The first Sudoku from the PDF sheet can be solved using hardly anything else. What we have here is the Clone equivalent of Naked Singles. Such puzzles can be quite boring, but in fairness, the same argument can be brought up for (easy) Standard Sudokus, too.
I have attempted to design the remaining puzzles slightly harder, so that at least the Clone equivalent of a Pointing Pair is occasionally required. In the end you should not expect too much, though; this variant is not what I would describe as “rich”.
Regarding the theory of Clone Sudokus, I would like to mention one more thing. Unlike the first two variants we have presented, Thermo Sudokus and Killer Sudokus, this variant has the property that all the number values are interchangeable. Neither the natural order of integers nor any arithmetic operations are used in the instructions. As a consequence, at least N-1 givens are required in an NxN Clone Sudoku to ensure a unique solution.
Several of today’s puzzle will do with five givens. In general, this has to do with the size and number of the Clone Regions. You see, more and larger regions provide more constraints, so that in extreme cases the entire grid can be broken down into N sets of N cells each (which must contain the same number). This is a little similar to Irregular Sudokus: The more geometry one adds, the less entries are needed.
Such extremes can quickly get boring once more. I think, as with other variants, Clone Sudoku should ideally have a moderate amount of shaded regions, so that the Clone rule is substantially required during the solve, but not so much that the Clone dynamics take over.
Difficulty-wise, I think none of the six puzzles in the PDF file is entirely trivial, but none of them is very hard either. As usual, you may find that a bifurcation is the quickest way to success on grids of this size. And as usual, I am urging you to try using only logic instead. Here is a small hint for the last puzzle:
Hint: You can at once determine the entry in the cell pair R2C4/R5C3. After that, you may want to look for Hidden Singles regarding the entry value of 1; there are several Sudoku regions with only one possible position.
Enjoy the puzzles!