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One late evening I was making my sudoku. I finally finished until water spilled all over the middle-right block.

The only thing I remember is the numbers present in the block (but not their position):

3
4
6
8

The sudoku:

![sudoku

Your task is to find the possible ways these numbers could have been laid out in this block, so that there is only one solution.

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  • $\begingroup$ Do you know that this has a "nice" solve path? This looks like an actual Sudoku puzzle with some digits blanked out [rather than a puzzle specifically designed to not have those digits]. And to me it seems very unlikely that that would lead to something approachable (except by just trying all possible cases). $\endgroup$ – Deusovi May 10 at 0:05
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There are:

four possible layouts of digits 3,4,6,8 in the middle-right block which lead to only a single solution. The layouts are:

i1 i2 i3 i4

Method of solution:

First, I solved the full grid as far as possible with the information given. That results in the following grid:

full

Then, I observed that a 3 can only be placed in one of two positions in the middle-right block: middle row-left and middle-row right. Once the 3 is placed, a 4 can only be placed in one of two remaining positions in the middle row. Thus, there are only a total of 4 unique ways to place the 3 and 4 in the middle-right block. At this point, as a computer solution is allowed, I used an online sudoku solver (sudokuwiki.org) to examine the total number of solutions for each of these four permutations. The result was 22, 34, 162, and 202 possible solutions for the four 3-4 arrangements, which means that there are a total of 420 different possible solutions of the full grid with any arrangement of 3-4-6-8 in the middle-right block. I then used a brute force approach to examine every possible layout of 3-4-6-8 in the middle right square using the same online sudoku solver, to find the four arrangements shown above which each lead to a single solution.

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