Pentomino Tiling

This is a P-pentomino:

This shape is chiral: flipping it creates a new shape that can't be restored by shifting or rotating. That means there is a 'regular' and 'flipped' version. We'll call the version above P because it looks like an upper case letter P.

The flipped version looks like a lower case Q:

Multiple Ps and Qs can be arranged to form a larger version of the original P shape.

Here two Ps and two Qs are used:

This tiling of the space is satisfying: everything fits perfectly, with no gaps or overlaps. Lovely!

We can instead fill this space with one P and three Qs:

But some combinations aren't possible. For example, there's no way to arrange four P shapes to make a larger P.

Scaling Up

Let's say our original P-pentomino shape is made of five unit squares. It has area 5, width 2, height 3.

The larger board we tiled above has four shapes (area 20, width 4, height 6).

There are many possible larger boards: for any N={1,2,3,...}, we can draw a board of area 5N², width 2N, and height 3N.

Here's a board of size 3:

For each larger board, we can ask:

• Can we tile the board with tile set of N² P tiles?
• What about a tile set of N²-1 P tiles and 1 Q tile?
• What about a tile set of N²-2 P tiles and 2 Q tiles?
• What about a tile set of N²-3 P tiles and 3 Q tiles?
• ...
• What about a tile set of 0 P tiles and N² Q tiles?

In effect, we have a boundless number of tiling puzzles, where each puzzle is a fixed tile set and board size. Some will have no solutions; some will have many. It's not immediately obvious if there will be any pattern, but this seems like an interesting space to explore.

Predictions

Before I programmatically explore the space, I'll make some predictions:

1. Will there be a pattern to which boards/tile sets have solutions?

   Prediction: No, solutions will be more common as area grows, without a clear pattern.

2. Will large boards be completely possible/impossible regardless of tile set?

   Prediction: No, large boards will have a mix of possible and impossible tile sets.

3. Will large boards have unique solutions?

   Prediction: Yes, we'll find at least one puzzle with a unique solution on every board we look at.

4. Will there be a repeating interior pattern in large board solutions?

   Prediction: Yes, there will be repeating interior patterns when there are many solutions. Unique solutions will have chaotic interiors.

Quality Rubric

Ultimately, I'm hoping to find a puzzle that is fun to solve.

In descending priority order, I'd like to find a board/tile set that is:

• Possible: There is at least one solution.
• Unique: There is exactly one solution.
• Chiral: The tile set can only make one board shape (it can't assemble a mirror board).
• Tricky: There are many 'near misses': solutions for the same size board that use one more or fewer P tiles.

We can quantify these and automatically rank solutions by quality as we find them.

There are other factors to consider:

• Avoid repetitive patterns (tedious)
• Avoid very large/small boards (too hard/easy)

These are harder to quantify up front. We'll just have to play it by ear after we find some solutions.

Colors

As we solve individual board/tileset combinations we'll color-code them:

• Deep purple for unknown states
• Red for puzzles with 0 solutions (worst)
• Orange for puzzles with multiple solutions
• Yellow for unique puzzles: exactly 1 solution but not chiral-unique; tile set can assemble a flipped board.
• Green for chiral-unique puzzles: exactly 1 solution; can't assemble a flipped board. (best)

Visualizing Results

Each board size N has N²+1 possible tile sets corresponding to (0,1,...N²) P tiles.

To visualize the solution space, we'll group solutions by board size. In each group, the single top square represents a tile set with zero P tiles (all Q tiles). The N² squares below will represent tile sets with (1 ... N²) P tiles.

This is what the map of solutions for N=[1,2,3,4] will look like:

Don't read too much into the spatial arrangement here: like the periodic table of elements, it's an arbitrary/aesthetic layout decision to make the data easy to scan. Hopefully this layout will help us spot interesting solutions.

The fact that N=2 forms a P-Pentomino shape is a coincidence (and hopefully not too confusing)

Results

Let's start with the smallest board size (N=1) and work our way up.

We can do the first few by hand.

Size 1 Boards

It's easy to reason about N=1. These are tile sets with 1 tile total and the board is the size of a single tile.

• 0 P tiles means we only have 1 Q tile. There's no solution, so this square is red.
• 1 P tile is trivally solvable. Since we can't use the tile to make a flipped board, the solution is chiral-unique. Green.

Size 2 Boards

You can work out N=2 in a few minutes with a pen and paper. These are tile sets with 4 tiles total. The two solutions described are the ones illustrated earlier in this post.

• 0 P tiles is impossible (red)
• 1 P tile is chiral-unique (green)
• 2 P tiles is unique (yellow).
• 3 and 4 P tiles are impossible (red)

As the board grows, it gets harder to work out solutions. I wrote a program that solves tiling puzzles and will use this going forward.

Size 3 Boards

This content was generated 100% automatically. It's reassuring to see that the solutions for size 1 and 2 match the manual results above.

Size 4 Boards

No unique solutions! A few extreme cases have no solutions, but most have multiple solutions.

Size 5, 6, 7 Boards

More of the same: no unique solutions, and all but a few cases have multiple solutions.

Analysis

• were my predictions right?
• examine the best solutions
• anything else interesting here?
• do the solutions have repeating structure?
• is recursion apparent? (ie: are N=2 solutions embedded in N=4?)

Further Reading

• Rep-tile (wikipedia): a shape that can be dissected into smaller copies of the same shape.

Appendix: Constraints and Solutions