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In an earlier question (Can a standard Sudoku puzzle contain a miniature Sudoku puzzle?), I asked whether a miniature 4x4 Sudoku could be embedded in a standard 9x9 Sudoku.

Inspired by the following comment on that question by quarague:

Looking at this I would guess that it is possible to put a 4x4-sudoku inside a giant 16x16 sudoku. The check would work the same way but it seems like you have enough degrees of freedom to fill everything.

I ask a new question:

In the following grid, can you fill each empty square with a number (1,2,3,...,16) so that there are no duplicated numbers in any row, column or 4x4 block.

enter image description here


I realize that if filling the grid is possible, it can be filled in many ways.

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    $\begingroup$ Seems kinda trivial? $\endgroup$
    – Bass
    Jul 21, 2023 at 7:41
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    $\begingroup$ @Bass Easy or not, it is wonderful to see beautiful answers like that given by Lezzup. Also I occasionally like to post simple questions; I imagine that people who are new to puzzle solving would appreciate such puzzles. $\endgroup$ Jul 21, 2023 at 7:54
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    $\begingroup$ This is similar to Killer Sudoku. sudoku.com/killer There's even an Amazon listing for a Monster Killer Sudoku puzzle book that has similar looking 16x16 puzzles. $\endgroup$ Jul 21, 2023 at 16:49

2 Answers 2

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The answer is:

Yes!:

solution

As you can see here, with this pattern you can easily fill the 16x16 grid. In the top left corner, I have filled in the mini sudoku in pink. In the other 3 corners, the 4 pink squares are places in such a way, that no pink square aligns with a pink square in another corner. That way we can fill in the pink squares with exactly the same numbers as the mini sudoku in the top left corner. Basically still mini sudoku's, but taken apart.
The other 3 colors work exactly the same, and could be filled in with the same numbers. For example, you can put the 5,6,7,8 in the yellow squares, the 9,10,11,12 in the green squares and the 13,14,15,16 in the purple squares.

@John Bollinger shows in a really nice way how many different solutions there are, just with this setup: We are effectively dividing the 16x16 grid into 16 independent 4x4 puzzles, split evenly among 4 color-coded categories. With 288 distinct 4x4 sudoku (including label permutations) that makes for 288^16 total ways to fill out such a grid. Even if you divide by the 16! permutations of the labels, that still gives you about 10^26 solutions of this form.

Bonus: you now have 4 mini sudoku's in 1 big sudoku.

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  • $\begingroup$ could you have a mini sudoko in the middle of the bottom right quarter? $\endgroup$
    – Grump.
    Jul 23, 2023 at 16:29
  • $\begingroup$ @Grump. Each color can be a separate mini-sudoko $\endgroup$ Jul 23, 2023 at 18:46
  • $\begingroup$ That doesn't answer my question @MooingDuck $\endgroup$
    – Grump.
    Jul 23, 2023 at 22:21
  • $\begingroup$ @Grump. which bottom right corner did you mean? Regardless: the answer is yes. $\endgroup$ Jul 24, 2023 at 13:24
  • $\begingroup$ I didn't say corner @MooingDuck - I $\endgroup$
    – Grump.
    Jul 25, 2023 at 18:12
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Adding to Lezzup's answer:

When you fill in the numbers, you get the following solution:

filled in sudoku

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    $\begingroup$ I believe the point of Lezzups answer is that there's 24 ways to fill out the numbers in the yellow, green, and red squares, and pretty much any of them is valid. So there's ~13824 solutions, not just this one. $\endgroup$ Jul 21, 2023 at 20:42
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    $\begingroup$ I think there are a lot more than that, @MooingDuck. Lezzup effectively divides the 16x16 grid into 16 independent 4x4 puzzles, split evenly among 4 color-coded categories. With 288 distinct 4x4 sudoku (including label permutations) that makes for 288^16 total ways to fill out a Lezzup-style grid. Even if you divide by the 16! permutations of the labels, that still gives you about 10^26 solutions of this form. $\endgroup$ Jul 23, 2023 at 15:10
  • $\begingroup$ Or perhaps that's a generalization of what Lezzup had in mind, but it's true in any case. $\endgroup$ Jul 23, 2023 at 15:22
  • $\begingroup$ Ah. There are 13824 solutions where all the pink squares are identical, all the green, all the yellow, and all the grey too. But you're right that there's many more where they aren't identical. I'd overlooked those. $\endgroup$ Jul 23, 2023 at 18:48
  • $\begingroup$ @JohnBollinger Initially, I wanted to create not just one example, but a nice pattern to simplify the proof how it could be possible. Turns out, that the pattern was very elegant and symetrical. However, I didn't thought about how many solutions it had, but 10^26 is mindboggeling! Thanks for calculating! $\endgroup$
    – Lezzup
    Jul 24, 2023 at 13:32

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