# L tromino tiling

Inspired by this challenge from puzzling.SE.

Polyominoes are fun - you can fill all sorts of shapes with them. But as soon as you have limits on what polyominoes you can use, you sadly start running into shapes you can't fill.

Today we'll be looking at the L tromino - a shape made up of three conjoined squares, in such a way that all three squares are of the same size, and there is a corner which they all share. It looks like this:

Now, given a shape made out of equal-size squares as input, the challenge is to find out whether that shape can be filled using only non-overlapping L-trominoes that are placed entirely within the input figure. You may have any number of these trominoes, and you may rotate each of them by any integer multiple of 90°.

### Rules

• The challenge is to make a program or function.
• Your program will recieve as input a two-dimensional list of values.
• Each entry in the list will contain one of two consistent values of your choice - one to signify that a square should be filled, and the other to signify that it shouldn't.
• You may assume there will be at least one row and at least one column, and that every row and every column will contain at least one cell that needs to be filled. You may also either assume that all rows have been padded to be of equal length, or that they contain nothing after their last filled cell - you may also independently assume either of those for the columns. That is to say, with 1 as squares you need to fill, and 0 as squares you need to not fill, you only need to be able to handle one of these four inputs (if given one of the others, your program's behaviour is undefined):
1 1 0
1 1 1
0 1 0

1 1
1 1 1
0 1

1 1 0
1 1 1
1

1 1
1 1 1
1

• To fill some square, you must also fill two other squares. Exactly two of the three squares filled must be diagonally adjacent to (sharing a corner with) eachothers, and they must both be orthogonally adjacent to (sharing a side with) the third.
• You may assume the input shape is contiguous.
• You may assume the input's dimensions will be no greater than 5x5.
• You may not assume the number of rows and the number of columns in the input will be equal. Originally, all this question's test cases featured inputs with an equal number of rows and columns. This was a result of poorly chosen test cases and was not intended to indicate that this would be the case for all inputs.
• Your program either needs to output a single consistent value of your choice if filling exactly the squares specified by the input with L-trominoes is possible, and any other value if it is impossible, or vice versa. For example, you may decide to output false if the input shape can be filled, in which case that is the only acceptable output for shapes that can be filled, and any output except false is acceptable for input shapes that can't be filled.
• However, the output of your program must not vary between runs given identical input, if said input is valid. That means if you have output 2 for some particular valid input, any later runs with exactly that input must also output 2.
• This is - make your code as short as you can.

### Examples

The examples use 1 and 0 to represent squares that should and shouldn't be filled respectively, uses true as the only output for fillable shapes, and false as the only output for non-fillable shapes.

1  -> false

0 1
1 1 -> true

1 1 1
1 1 1
1 1 1 -> false

0 1 0
1 1 1
0 0 1  -> false

1 0 0
1 1 1
0 1 1  -> true

1 1 0
1 0 1
1 1 1
1 0 1  -> false

1 1 1 1
1 0 0 1
1 0 0 1
1 1 1 1  -> true

0 1 1 1
1 1 0 1
1 1 1 1
0 1 1 0  -> true

0 1 1 1
1 1 1 1
1 1 1 0
0 1 1 0  -> true

1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 0  -> true

1 1 1 1 1
1 0 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1  -> false

1 1 1 1 0
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1  -> true

• Can we assume that there will be a multiple of 3 squares to fill?
– Neil
Commented Oct 19, 2019 at 10:02
• @Neil, the first example has 5 1s so I'd assume it isn't always a multiple of 3. Commented Oct 19, 2019 at 11:03
• Can we assume that the input is not empty?
– G B
Commented Mar 3, 2020 at 8:06
• @GB If by "not empty" you mean either "the input is at least 1x1" or "at least one square needs filling" then as the spec states "you may assume there will be at least one row and at least one column, and that every row and every column will contain at least one cell that needs to be filled". If you mean something else when you say "not empty", I can't answer unless you clarify what you mean. Commented Mar 5, 2020 at 11:18
• @SaraJ: you already answered the question, thanks, now that I read the spec again, it makes sense.
– G B
Commented Mar 5, 2020 at 11:23

# JavaScript (ES6),  196 ... 185  182 bytes

Takes input as a binary matrix. Returns a Boolean value.

f=m=>!/1/.test(m)||m.some((r,y)=>r.some((v,x)=>v&&[0,1,2,3].some(d=>([X,V]=(g=d=>[v=x+--d%2,R=m[y+--d%2]||0,R[v]])(d))[2]&g(d+1&3)[2]&&(r[x]=V[X]=R[r[x]=V[X]=R[v]=0,o=f(m),v]=1)&o)))


Try it online!

This algorithm is too slow for the penultimate test case on TIO, but it does find the answer on my laptop in about 90 seconds.

### How?

This is a brute force search that looks for all possible trominos in the input matrix and attempts to remove them (one at a time) until the matrix is cleared or all combinations are proven to fail.

Given a reference cell at $$\(x,y)\$$ (in blue below) and a direction $$\0\le d\le 3\$$:

• We invoke the helper function $$\g\$$ with $$\d\$$ to get a first neighbor cell (in yellow) whose coordinates are x+(d-2)%2 and y+(d-1)%2.
• We invoke $$\g\$$ again with $$\(d+1)\bmod 4\$$ to get a second neighbor cell (in red).

This gives:

### Commented

f = m =>                         // f is a recursive function taking the matrix m[]
!/1/.test(m) ||                // stop if m[] does not contain any 1 anymore (success)
m.some((r, y) =>               // otherwise, for each row r[] at position y in m[]:
r.some((v, x) =>             //   for each value v at position x in r[]:
v &&                       //     abort right away if this cell is already empty
[0, 1, 2, 3].some(d =>     //     otherwise, for each direction d:
( [X, V] = (             //       get the 1st neighbor cell
g = d => [           //       g is a helper function taking a direction
//       and returning the corresponding neighbor cell:
v = x + --d % 2,   //         save the x-coordinate in v
R = m[y + --d % 2] //         extract the row R[]
|| 0,          //         (0 is used as a default object to prevent
R[v]               //         get the value of the cell
]                    //         return the tuple [v, R, value]
)(d)                   //
)[2] &                   //       get the value of the 1st neighbor cell
g(d + 1 & 3)[2]          //       get the value of the 2nd neighbor cell
&& (                     //       if both neighbor cells are set, this is a
r[x] = V[X] = R[       //       valid L-tromino:
r[x] =               //         remove it by clearing the current cell
V[X] = R[v] = 0,     //         and its 2 neighbors
o = f(m),            //         save the result of a recursive call in o
v                    //         and restore the tromino
] = 1                  //
) & o                    //         yield the result of the recursive call
)                          //     end of loop over the directions
)                            //   end of loop over the columns
)                              // end of loop over the rows


# Python 3, 136 131 bytes

lambda S:g(int((len(S)*'{:0<6}').format(*S),2))<1
g=lambda b:b*all(b>>i&t^t or g(t<<i^b)for i in range(30)for t in(67,131,193,194))


Try it online!

Input: A list of strings, each string must be a binary string. The length of the strings can be different.
Output: True if the shape can be tiled, False otherwise.

-5 bytes thanks to @Lynn!

## Big idea:

As with most code golf solutions, this is a brute-force recursive search. The program simply tries to remove an L-tromino from the shape, then recurs on the new shape.

By representing the shape using a bitboard (one integer, where each bit represents whether a cell is occupied or not), we can:

• Use bit-wise AND to check if an L-tromino fits within a shape
• Use bit-wise XOR to remove an L-shaped part from the shape.

## Program explanation

Constructing the bitboard:

int(
(len(S)*'{:0<6}').format(*S)  # left pad each row with '0' to length 6
# then concat all rows together
,2)               # interpret the result string as binary integer


Recursive function that returns 0 if the shape can be tiled, or a positive integer if the shape cannot be tiled.

g=lambda b:\  # receive a bitboard as argument
b *              # tilable if b is 0
all(             # tilable if any of the following is False, i.e. all(...)==0
b>>i&t^t or      # tromino doesn't fit shape or
g(t<<i^b)        # the remaining is not tilable, i.e. g(...)==0
for i in range(30)        # for each position in shape
for t in(67,131,193,194)  # for each tromino orientation
)


## More details

Bitboard layout

The following diagram specifies which cell in the shape correspond to which bit in the bitboard:

29 28 27 26 25 24
23 22 21 20 19 18
17 16 15 14 13 12
11 10  9  8  7  6
5  4  3  2  1  0


The size of the bitboard is 5 x 6, which is enough to store 5 x 5 shapes. If the shape has less than 6 columns, it is padded with 0 to the left. For example:

 111         ...111
1.1  ---->  ...1.1  ----> ...111 ...1.1 ...11.  ---->  0x7146
11.         ...11.

shape        padded          bitboard (binary)          bitboard (hex)


(Note: 0 is denoted by . for visual clarity)

Since each shape has at most 5 columns as specified in the problem, there will always be an empty column to the left of the bitboard. This will be important to prevent wrap-around later.

Each tromino orientation can be represented by 1 bitboard:

          ......   ......   ......   ......
......   ......   ......   ......
tromino   ......   ......   ......   ......
.....1   ....1.   ....11   ....11
....11   ....11   .....1   ....1.

bitboard     67      131      193      194
(decimal)


Check if a tromino fits within the shape

To check if an L-tromino t fits inside a certain position in shape b, we can use bit-wise AND:

......               ......                      ......
......   shift to    ....1.    AND tromino       ....1.
......   position    ...11.    with shape        ...11.
.....1   --------->  ......  -----+-------->     ......
....11               ......       |              ......
t                 t << 13      |          (t << 13) & b
|
......       |
...1..       |
..111.  -----+
...1..
......
b


We can see that the tromino fits inside the shape if and only if (t << i) & b == (t << i) (where i is the desired position). This is equivalent to:

(b >> i) & t == t


Remove an L-shaped region from the shape

We can use bit-wise XOR to remove a tromino from the shape:

 ......         ..1111           ..1111
....1.         ..1111           ..11.1
...11.   XOR   ..1111   ---->   ..1..1
......         ..1111           ..1111
......         ..1111           ..1111
t << 13            b               b2

• int((len(S)*'{:0>6}').format(*S),2) saves you some bytes.
– lynn
Commented Mar 3, 2020 at 1:17

# APL (Dyalog Unicode), 81 bytesSBCS

L←⊂⍤2⊢4 2 2⍴∘.≠⍨⍳4
{∧/~,⍵:1⋄∨/∇¨⍵∘{r c l←-⍵⋄⍺>c⌽r⊖(⍴⍺)↑L⊃⍨-l}¨⍸{∧/∘,¨L≤⊂⍵}⌺2 2⊢⍵}


Try it online!

Monadic dfn that takes a binary matrix and gives 1 if it can be tiled with L-triominoes, 0 otherwise.

A brute-force recursive search by removing one triomino at a time. Uses the fact that every orientation of an L-triomino fits within a 2x2 box, so the code tests all 2x2 submatrices of the input array to search for possible L-triomino placements. The Stencil operator ⌺ fits perfectly for this purpose.

The algorithm becomes much faster if we limit the branches to the first 5 placements (which is guaranteed to include all possible placements that includes the top-left cell) at the cost of 8 bytes (⊢↑⍨5⌊≢):

L←⊂⍤2⊢4 2 2⍴∘.≠⍨⍳4
{∧/~,⍵:1⋄∨/∇¨⍵∘{r c l←-⍵⋄⍺>c⌽r⊖(⍴⍺)↑L⊃⍨-l}¨(⊢↑⍨5⌊≢)⍸{∧/∘,¨L≤⊂⍵}⌺2 2⊢⍵}


Try it online!

⍝ 4 rotations of L-triomino; looks like
⍝ ┌───┬───┬───┬───┐
⍝ │0 1│1 0│1 1│1 1│
⍝ │1 1│1 1│0 1│1 0│
⍝ └───┴───┴───┴───┘
L←,{0@(⊂⍵)⊢2 2⍴1}¨⍳2 2
⍳2 2  ⍝ Indexes of 2-by-2 matrix
{      ⊢2 2⍴1}¨      ⍝ For each, modify [1 1][1 1] so that
0@(⊂⍵)              ⍝   that position becomes 0
,                     ⍝ Flatten top level

⍝ Main function; 1 if L-triomino tiling is possible, 0 otherwise
f←{
⍝ If no cells left, return True
∧/~,⍵: 1
~,⍵     ⍝ Boolean negation of flattened input matrix
∧/        ⍝ All; reduce by And
: 1  ⍝ If true, return 1

⍝ Get possible placements as (row_idx, col_idx, L_idx) triples
places←⍸{∧/∘,¨L≤⊂⍵}⌺2 2⊢⍵
{         }⌺2 2⊢⍵  ⍝ Map over 2x2 submatrices...
∧/∘,¨L≤⊂⍵  ⍝ Can each of L be placed on a submatrix?
L≤⊂⍵  ⍝   Compare each cell of each of L with each cell of the submatrix
⍝   so that L(1) vs. ⍵(0) gives False and anything else gives True
∧/∘,¨      ⍝   Reduce by And on all cells of each comparison matrix
⍸            ⍝ Indexes of ones
places←             ⍝ Assign to places

⍝ For each possible placement, remove the cells and
⍝ recursively try L-triomino placements
⍝ If no possible placement, ∨/ automatically gives False
∨/∇¨⍵∘{r c l←-⍵⋄⍺>c⌽r⊖(⍴⍺)↑L⊃⍨-l}¨places
⍵∘{                         }¨places  ⍝ Map each on the places...
r c l←-⍵⋄                  ⍝ Dissect the placement; negate due to built-in rotation direction
L⊃⍨-l  ⍝ Fetch the target L-triomino
(⍴⍺)↑       ⍝ Overtake to fit the size of input matrix
c⌽r⊖            ⍝ Rotate to move the L-triomino to correct position
⍺>                ⍝ Remove L-triomino cells from input matrix
∇¨  ⍝ Recursively test for L-triomino placements
∨/    ⍝ True if at least one is successful
}


# Ruby, 146 112 108 bytes

->m{g=->q{[7,11,13,14].any?{|x|t=x/4|x%4>>~m[0].size;t*=2until t>=q||q&t==t&&g[q-t];t<=q}};g[(m*?0).to_i 2]}


Try it online!