Clones Tile A Square
Premise
Puzzle: you are given N identical polycubes and asked to tile a square board of size B.
- Each piece must touch the board.
- The board must be completely covered (no gaps, no spillover outside the boundaries)
- Pieces in the same orientation can't touch (at face, edge or corner)
For what pieces and values B, N is this a good puzzle?
Configuration
Clones Tile A Rectangle - Restricted Neighbors
Copies of a single polycube tile a rectangle boardKey. Matching orientations can't touch.
Key: Clones_Tile_A_Square (Template)
Children: Clones Tile A Rectangle - Restricted Neighbors (rect1x1_PLR), Clones Tile A Rectangle - Restricted Neighbors (rect2x2_PLR), Clones Tile A Rectangle - Restricted Neighbors (rect3x3_PLR), Clones Tile A Rectangle - Restricted Neighbors (rect4x4_PLR), Clones Tile A Rectangle - Restricted Neighbors (rect5x5_PLR), Clones Tile A Rectangle - Restricted Neighbors (rect6x6_PLR), Clones Tile A Rectangle - Restricted Neighbors (rect7x7_PLR), Clones Tile A Rectangle - Restricted Neighbors (rect8x8_PLR), Clones Tile A Rectangle - Restricted Neighbors (rect9x9_PLR), Clones Tile A Rectangle - Restricted Neighbors (rect10x10_PLR)
...
Clones Tile A Rectangle - Simple Version (rect4x4_T)
Copies of a single polycube tile a rectangle boardKey. Simplest config; no constraint flags.
Key: Clones_Tile_A_Square_Simple_rect4x4_T (Generated)
Parent: Clones_Tile_A_Square_Simple
Board: rect4x4
Pieces: T x ∞
Constraints: defaults.
Result
Found 1 solution.
In 0 seconds and 9 steps, solver v0.1.1 ran on 8/14/2024 and found 2 total solutions, which reduced to 1 unique solutions. Completed without bailing early.
Pieces
Here's the piece you start with. We'll examine 1-25 of these.
Here are the three valid faces:
Possible Boards
For each N, which board sizes are possible?
- N=1: (3..5) --> board size: 2
- N=2: (6..10) --> 3
- N=3: (9..15) --> 3
- N=4: (12..20) --> 4
- N=5: (15..25) --> 4 or 5 (area 16 or 25)
- ...
- N=25 (75..125) --> 9, 10, or 11
TODO: make this a table. Only show allowed board sizes.
Color coding:
- Red = 0 solutions
- Yellow = 2+ unique solutions
- Green = 1 solution
Headers:
- Columns: no header
- Rows: number of pieces (N)
Cell Text (without highlight)
- Board Size (B) (or maybe exclude for cleaner view?)
When highlighting each cell we see:
- number of pieces (N)
- board size (B)
- preview of 1 solution (if applicable)