# Monument Valley Sudoku

Another Sudoku monstrosity from the brains of Yours Truly. No rules are given, part of the challenge is trying to work out what the rules are :)

• I think I know what to do with this but the one issue I have is I can’t tell what’s a 6 and what’s a 9. Is it possible to perhaps draw a line under each to show orientation? Commented Jul 14, 2020 at 9:43
• @BeastlyGerbil I think that's the point. (Specifically, (rot13)gur qvtvg frg vf mreb, bar, qnfu, rvtug, vasvavgl, fvk, avar, gbc unys bs pnapre flzoby, obggbz unys bs pnapre flzoby, naq qvtvgf ner vagrecergrq qvssreragyl ng gur tevq vagrefrpgvbaf.)
– Deusovi
Commented Jul 14, 2020 at 11:54

### Now Complete Answer

The first thing to notice is:

The symbols used cannot all represent digits directly, since several rows have multiple 1's or 8's. However, leveraging the Monument Valley theme of the puzzle, we realize that orientation matters, and that in fact the symbols forming the Sudoku are
0 | — 8 ∞ 9 6 ᓂ ᓄ
Moreover, where the yellow-pink and yellow-green junctions are, the orientation is the same for both connecting sudokus. However in the yellow-green junction, the characters will be rotated...but our character set is invariant under 90 degree rotations, so we'll be OK.

To represent the solution:

I've first turned the green grid blue because I like primary colors. Then I physically split the yellow and blue grids to make the diagram lie flat. The upper right square of the yellow grid and the upper left square of the blue grid are both shown...we just need to make sure that the blue square is the yellow square rotated clockwise by 90 degrees, both the cells in the square and the symbols in the cells.

The first steps:

There are a number of easy fill-ins one can get by identifying rows, squares or columns where there is only one possible place for a symbol to go. Utilizing the initial conditions and adding these fill-ins gives the grid, with just a couple requiring logic beyond simple reduction of possibilities based on observation. The first comes in filling pink grid, bottom-center square, upper-left cell, where we note that in the middle pink square, the 0 must be in the right-most column, forcing the 0 in the bottom-middle square to be in the left-most column. The second comes in filling yellow grid, bottom-left square, upper-right cell. Here we note first that the 0 in the upper-right square of the pink grid must be in the bottom row, which forces the 0 in the upper-left pink square, which is also the lower-right yellow square, to be in the middle row. At this point, the grid looks as follows:

Moving on:

After a couple more easy fill-ins, we hit a couple of important logic jumps. First, note that the ᓄ in the blue grid, lower-left square must be in the left-hand column. Since this is identical to the upper-right square of the pink grid, we find that the ᓄ must appear in the right column of the right-middle square of the pink grid. We continue looking in the upper-right square of the pink-grid, specifically the upper-right cell. By direct observation, we see the only possible values for this cell are 6 and 9. Looking in the yellow grid, we see that the 6 in the lower-right square must be in its top row, and since this is identical to the upper-right square of the pink grid, we eliminate 6 as an option, forcing this cell to be a 9. By listing possible candidates to fill each cell of this square, we can fill most of it, proceeding from right-middle to lower-left to lower-right to lower-middle to upper-middle. At this point the grid looks like:

At this point:

About half the pink grid fills in by inspection. After completing the right-middle and lower-right squares, observe that the 9 and the | in the lower-middle square must occupy the remaining two cells in its middle row. Thus the right-middle cell of the pink lower-left square must be —. Numerous other simple fill-ins ensue in the pink grid, and also in the blue. This leaves us in the following state:

Using the funky square:

The top-most row of the yellow grid has a ∞, so the top-right square of the yellow grid cannot have this symbol in its top row. Translating this across the funky square, this implies that the upper-left square of the blue grid cannot have an 8 in its right-most column. This forces the upper-right cell of the left-middle square of the blue grid to be an 8. We can perform several fill-ins, including several across the funky square, and particularly can place several — symbols in the yellow grid. We are aided in this by noting that the — symbol in the lower-right square of the yellow grid must be in the middle column, as forced by the squares in the pink grid. This leaves us with the grid:

Some more funky square logic:

Look at the middle column of the upper-left square of the blue grid, where there are two open cells. These cells cannot be 0 or a 9, since there is already a 0 and a ᓂ in the middle row of the top rank of squares in the yellow grid. Moreover these cells cannot be ᓂ as this symbol already appears in this column. Hence these two cells must contain 8 and 6, in some order. Moving down to the left-middle square of the blue grid, this forces the two empty cells in its middle column to be 0 and 9, which we can place unambiguously, and lets us finish off the left-middle and center squares of the blue grid, and some other easy fill-ins, leaving the grid:

Some perspective after sleep:

Let's work in row 2 of the yellow grid. Looking in the fourth column of the yellow grid, we see that the 6 in this column must be in the left column of the upper-middle square, which means the middle cell of this square, row 2, column 5 cannot be a 6. We've already determined that the cells in this row in the upper-right square must be ∞ and ᓄ, in some order, so this forces the 6 in this row to be in the second column, and also forces a | in the fifth column in this row. With some additional fill-ins, we get:

One final logical step:

Work in the first column of the pink grid, which joins with the seventh column of the yellow grid. In the middle cell of the shared pink-yellow square in this column, the only possible values are 0 and 9. Suppose that this is a 0. Then the bottom left cell of the bottom left square of the pink grid has to be ∞. This finally forces the upper-left cell of the shared pink-yellow square to be 6. But recall our analysis of the funky square shared between yellow and blue: the bottom row has to contain 0, 6 and ᓂ in some combination, but all three of these values are now blocked for the lower-left cell. This is a contradiction, forcing the left-middle cell of the shared pink-yellow square to be 9. With this deduction, the rest of the grid is fill-ins.

• Can you please explain what you mean by “leveraging the Monument Valley theme of the puzzle”? I don’t see any connection to desert sandstone buttes. Commented Jul 16, 2020 at 1:48
• This refers to the game Monument Valley, which has a number of spatial illusions where you need to suppress what your mind wants to see, and treat the game literally. See en.m.wikipedia.org/wiki/Monument_Valley_(video_game) Commented Jul 16, 2020 at 11:07
• To further illustrate the MV theme, imagine you start with an 8 in the pink region, lower right corner. You move it up the green region then down the orange region and back to the pink region, lower left corner. Lo and behold, the 8 has turned into infinity! In MV, the protaganist Ida must make similar moves, often ending up in the same position but with gravity rotated 90 degrees :) If you're into topology, you could think of this Sudoku as an example of a non-orientable surface (maybe not an exact analogy, but it's near enough). en.wikipedia.org/wiki/Orientability Commented Jul 16, 2020 at 12:33
• @JeremyDover Thanks for the link. That makes much more sense now! Commented Jul 16, 2020 at 13:13
• Lovely puzzle reminds me of a favourite theme of M.C.Escher. Commented Jul 16, 2020 at 18:11