# Tetromino Sudoku

An entry in Fortnightly Topic Challenge #32: Grid Deduction Hybrids

The grid below, when filled in, forms a valid Sudoku grid. It can also be filled in like a LITS (nuruomino) puzzle without the 1x4 tetrominos.

The two objectives of this puzzle are:

• Fill in the numbers in the grid.
• Identify the LITS tetrominos.

In addition to the standard rules of Sudoku and LITS, the following rules apply:

• Each 3x3 region in the Sudoku grid contains exactly one tetromino, which is either a L, T, or S tetromino.
• The tetrominos obey the rules of a LITS puzzle.
• The shaded yellow spaces are part of a tetromino.
• Each tetromino contains four numbers that add up to 20.
• No two tetrominos may contain the same set of four numbers.
• The tetrominos (each in its own 3x3 region) are arranged in a 3x3 Sudoku, using tetromino shapes L, T, and S instead of numbers. If two regions contain the same tetromino shape, then they cannot share the same horizontal alignment or vertical alignment.

Good luck!

• Question- your first and fifth clues are ambiguous, but your sixth suggests the first should be interpreted as 'each region in the grid contains exactly one tetronimo', and the fifth as 'no two tetrominoes should contain the same set of four numbers'. Is this correct? – P... Jul 18 '17 at 19:52
• @P... Yes for rule 5, no for rule 6. If you treat each 3x3 region as its own square, then you get squares in a 3x3 arrangement. These big squares form a Sudoku, using tetromino shape instead of numbers. I updated the rules to clarify this. – MikeQ Jul 18 '17 at 20:46
• I feel like I'm still missing something about the rules, as there doesn't seem to be any way to get started other than brute force. To the best of my knowledge, the rules of a LITS puzzle require that the shaded squares (forming the nine tetrominoes) form a single connected path that does not contain any 2x2 blocks of squares. There are no restrictions on unshaded squares, and the shaded path can contain loops. Is this correct? – P... Jul 21 '17 at 16:59
• With your permission, I'd like to submit a puzzle inspired by this one. I've come up with tetromino-sudoku puzzle with a unique solution using slightly more restrictive rules (Any two tetrominoes of the same type must either be rotated or flipped relative to eachother, every non-yellow space must have a direct path to the border). If you'd like I will credit you as the inspiration, and link back to this question. – P... Oct 25 '17 at 19:31
• @P... Please do! I have no ownership over this type of puzzle. – MikeQ Oct 25 '17 at 21:10

I believe I have found multiple solutions to the puzzle. It is possible that there is a nuance to the rules that I missed that disqualifies one or both of them. My understanding of the rules did not allow for a full logical deduction of the solution, so these were both found by brute force.

The sudoku is on the left, while the tetrominoes are on the right:

951 837 426 42
376 542 918 37 4 9
428 916 573 28 916 5

532 678 149 3 6 4
147 329 685 47 329 68
869 154 237 6 2

683 491 752 83 91 752
794 285 361 4 28 6
215 763 894 5

The lower-center tetromino can be flipped to:

. 1
. 85
. 6

hence the multiple solutions.

• Not the solution I had expected, however it seems correct! Good work! – MikeQ Aug 4 '17 at 23:04