# This Sudoku Park iS LIT!

An entry in Fortnightly Topic Challenge #47: "Wacky Sudokus"

The grid below is a normal Sudoku grid with normal Sudoku clues, but the completed grid also houses a Statue Park! The completed grid admits a placement of four tetrominoes (one each of I, L, S, and T, allowing reflection and rotation) with the following properties:

• Each tetromino covers one of each even digit 2, 4, 6 and 8.
• Not only do tetrominoes not touch, but tetrominoes do not touch any other even digit, neither orthogonally nor diagonally.

I hope you enjoy!

TEXT VERSION

-------------------
|1| | | | | |8| | |
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|2| | | | | |3| | |
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|5|8|3| | | |9|4|7|
-------------------
| |4| | |7| | |2| |
-------------------
| |1| | |4| | |9| |
-------------------
| |9| | |1| | |7| |
-------------------
|4|2|1| | | |7|5|9|
-------------------
| |5| | | | | |3| |
-------------------
| |3| | | | | |8| |
-------------------


SOLVER NOTES

Without the Statue Park constraint, the Sudoku does not have a unique solution, so you will need to solve both parts of the puzzle simultaneously.

For those unfamiliar with tetromino placement puzzles, tetrominoes cover four cells in the following orientations:

I = I    L = L     S =  SS   T = TTT
I        L         SS         T
I        LL
I

For completeness, but no O-tetromino appears in this puzzle.

O = OO
OO

• Just making sure that it is possible in the final position to find other (incorrect) places where a tetronimo could cover only even digits? Jan 23, 2021 at 17:14
• @SteveV Yes, that may be possible. But there are only the designed locations that meet both conditions in the puzzle. Jan 23, 2021 at 17:17

## Looking for the I-tetromino

The top-right is the only place it can go and satisfy the constraints (every other location it could go would either contain an odd number or be adjacent to an even number). I realize I could have filled in the middle row of the top-middle box (8 9 7), but I didn't see it at this point; we get there eventually. :)

## Looking for the T-tetromino

Similarly, there is only one location for the T-tetromino. The only locations that don't enclose an odd number or put it adjacent to an even number are in the bottom-middle box. The top row of that box needs the numbers 3, 6, and 8, and the only T-shape in that box that would enclose the "6" and "8" but not the "3" are diagonal to the "4" in the middle box.
With that information and basic Sudoku rules, we can fill in the left column of boxes...

...and we can fill in a "4" and some more notes that might help us

## Looking for the L-tetromino

The L-tetromino has to go in the bottom right (again, every other location it could go would either contain an odd number or be adjacent to an even number) with the long leg at the bottom (I should have drawn this in, but didn't for whatever reason). There's no way to tell which side the short leg should be on, but regardless, we can fill in some more numbers in the bottom row like this...

...and based on that, we can fill in some more numbers and notes for the rest of the grid.

## Looking for the S-tetromino

Using the same strategy for finding tetrominoes, there's only one spot for the S-tetromino, and the "6" in it means the short leg of the L-tetromino has to be on the left. Now we can solve the right column of boxes...

...and after that the rest of the grid!

• Oof, please ignore my last. This is the correct answer. Guess what I'm asking is a little more explanation about why "there's only one place" for certain tetrominos. Not so much the I, where a quick comment would suffice, but the T is much less obvious, given the presence of an open bottom middle square. Jan 23, 2021 at 18:56
• @JeremyDover Yeah I wondered if I'd given enough explanation - I can't get to it right now, but I'll add some later today :) Jan 23, 2021 at 18:58
• @JeremyDover How's that? My strategy was just iterating through every possible location for a tetromino (the order of tetrominoes was just lucky intuition), and keeping track of what locations were valid, and there was only 1 or 2 valid spots each time I did it. Not sure how I'd run through that thought process completely without being overly verbose and redundant, but let me know if there's anything else I can elaborate on! Jan 23, 2021 at 19:51
• I think that does it...once you find the T the rest is straightforward, and your discussion on the bottom middle box is the key sticking point. A checkmark for you! Jan 23, 2021 at 19:57
• I came up with the same answer iterating s before t. Jan 23, 2021 at 23:56