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Imagine the following Sudoku:

  • Imagine the Sudoku has a 6x6 grid with standard boxes, 2 rows and 3 columns each; and standard digits, 1 to 6.
  • Imagine it has an antiknight rule: identical digits cannot be chess knight move away from each other.
  • Imagine it has an antidouble rule: no digit is orthogonally adjacent to its double.
  • Imagine the top-leftmost and bottom-rightmost digits are equal, so do the top-rightmost and bottom-leftmost.
  • Imagine in Row 5, there exist 3 adjacent digits that are numerically adjacent too in increasing order from left to right.
  • Imagine in Column 1, there exist 3 adjacent digits that are numerically adjacent too in increasing order from top to bottom.

Can you imagine that?


(P.S. This can be fully done logically but might need some "special" knowledge.)

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First step (some simple logical deduction):

enter image description here

Now, if we focus on this highlighted 2x3 grid:

enter image description here we see two B's. If we choose the red B we would go against the antiknight-rule in the 2x3 grid below the yellow one, so we now know we have to choose the blue B.

That leads to:

enter image description here This red B. And the rest of the boxes highlighted in yellow will follow.

Let's clean this up:

enter image description here

Now, using some more logic, we can place:

enter image description here This red D. And the rest of the highlighted yellow boxes will follow.

Let's clean up again:

enter image description here We now have the whole grid in terms of letters. Time to look how we can translate these to numbers, 1-6, using the rules.

From the information given by the rules we can conclude that:

Column1 and row5 must be of the form, 2-3-4, 3-4-5 or 4-5-6 (note that the possibility 1-2-3 is excluded since 2 is 1 doubled) where its possbible along the column/row. We can see that the string of numbers must include a 4. Let's compare column1 possibilities with the corresponding letter-position in row5 and see what happens:

A-D-C: enter image description here This string won't work since it's impossible to form a 3 digit consecutive string in row 5.

D-C-B: enter image description here Like-wise this string won't work since it's impossible to form a 3 digit consecutive string in row 5.

C-B-E: Now this one is possible if we choose them to be 3-4-5. The other possibilities (2-3-4 and 4-5-6) don't work. enter image description here

Lastly, B-E-F: enter image description here Same as the two first possibilities we can see that this is impossible.

This means that the third possibility, C-B-E, is the only one that works.

We now replace the letters with the corresponding numbers and get the full grid:

enter image description here

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  • $\begingroup$ In the first step, why isn't D an option for the first row last column? $\endgroup$
    – Pedro A
    Oct 16 at 1:20
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    $\begingroup$ Nevermind, figured it out: top-rightmost and bottom-leftmost are equal. $\endgroup$
    – Pedro A
    Oct 16 at 1:52
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    $\begingroup$ Great job and well-explained! I'm giving this a tickmark as it's easier to follow. This is just for an additional info: I mentioned about a "special" knowledge. This knowledge is actually regarding an antiknight 6x6 Sudoku: you can paint them in checkerboard then each color will only have 3 distinct numbers each. Using this knowledge, it will speed up a bit to fill the variables and to know which sequence is okay for Row 5 and Column 1. But yours work pretty well too! :D $\endgroup$
    – athin
    Oct 16 at 9:57
  • $\begingroup$ @athin Thanks! And nice puzzle :) $\endgroup$ Oct 16 at 14:28
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The answer is

Yes, I can imagine it.

enter image description here

The first step was

to use the fact that opposite corners were identical digits and that there was an antiknight restriction. This gave me the following:

enter image description here

The opposite corner digits can only propogate in the grid in two ways. In the above, only the yellow has been shown in both configurations.

The next step was

to realize that the most restricted digit is the 4. There are two digits it cannot be adjacent to and it must be part of any 3 digit sequence. I therefore decided to find where the 4 could be in the first column, starting at the bottom:

enter image description here In the first configuration, the placement of the 3 and 4 in row 5 means no 3-digit sequence is possible. In the other configuration, the 1 and the 2 are forced to be adjacent. Conclusion: the 4 cannot be at the bottom.

Moving along,

I tried the 4 in the 5th row, first using a 2-3-4 sequence:

enter image description here

Again, this was not possible as shown. I then tried a 3-4-5 sequence:

enter image description here
This too is impossible (no matter the yellow configuration) as the 4 can only be part of a 4-5-6 sequence and the 5 is too far away.

Moving up

I tried the 4 in the 4th row, first using a 2-3-4 sequence:
enter image description here
Didn't work in either yellow configuration as no 3-digit sequence could be formed in the 5th row.

And finally

I tried the sequence 3-4-5 which gave
enter image description here
I.e. the second yellow configuration worked!

Nice puzzle! If there is an easier method, I'd love to hear it.

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    $\begingroup$ I know you answered before me, but wanted to show my solving method since I worked on it quite a lot today! $\endgroup$ Oct 15 at 17:33
  • $\begingroup$ Well done! An "easier" method to avoid some case bashing has been done by @Prim3numbah: you can actually put all the variables first on the grid and see the only substring that can be for Row 5 and Column 1. :D $\endgroup$
    – athin
    Oct 16 at 9:51

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