Pieces
A gallery of pieces used by puzzles. Each piece is a polycube shape.
Representing Polycubes
Compact Notation
Polycubes are represented as delimited strings of numbers.
- numbers represent stacks of cubes (in the z direction, coming out of viewing plane toward viewer)
- row of numbers are adjacent stacks (in the x direction, left to right across viewing plane)
- single slash / delimits rows (y direction, downward on viewing plane).
- double slash // delimits planes (z direction, generally only necessary to represent overhangs)
- 0s represent empty spaces. Use a 0 in a empty row or plane when there is nothing to represent (for an example, see 'hollow cube' shape). This prevents two / delimiters from being mistaken for a single // delimiter.
Naming
Every polycube gets a shorthand name.
- Name rods by length, prefixed with "I" ("I4")
- Name rectangular solids by their dimensions, delimited with "x" ("1x2x3")
- When possible, name 2D polycubes by a letter approximating their shape ("P")
- When possible, name polycubes by overhang-free faces, with bigger faces first ("PQO")
- Use + to indicate a shape where footprint has been extended, generally along the Z axis ("PLR+")
- Use +N to indicate a shape where an N-long spike has been added, generally at the top left along the Z axis ("L+3", "knot+1")
Orientation
Compact notation implies a default orientation. Best practices for choosing an orientation:
- Favor orientations that are easy to quickly assess and understand.
- Minimize the number, size, and complexity of z-axis structures extruding from the xy plane. If there is a complex 2D shape, lay it out flat on the xy plane so the viewer can see it easily.
- Don't worry about finding a canonical/minimal representation that can be algorithmically determined (eg, shortest or lexicographically earliest representation). Instead, orient in a way that is convenient/intuitive.
- If the name of a polycube uses letters to describe its shape, try to match orientation and chirality of those letters. (TODO: match chirality? of footprint or piece when viewed head-on?)
2D Polycubes (Polyominos)
I1
- Key: I1
- Compact: 1
- 1 Points: 1,1,1
- Bounds: 1x1x1 (1-1, 1-1, 1-1)
- Has overhangs: false
- Footprint area: 1
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I1
I2
- Key: I2
- Compact: 11
- 2 Points: 1,1,1|2,1,1
- Bounds: 2x1x1 (1-2, 1-1, 1-1)
- Has overhangs: false
- Footprint area: 2
- Used by Tatami Tiling - Classic (rect4x4)
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I1
I3
- Key: I3
- Compact: 111
- 3 Points: 1,1,1|2,1,1|3,1,1
- Bounds: 3x1x1 (1-3, 1-1, 1-1)
- Has overhangs: false
- Footprint area: 3
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I1
R
- Key: R
- Compact: 11/1
- 3 Points: 1,1,1|2,1,1|1,2,1
- Bounds: 2x2x1 (1-2, 1-2, 1-1)
- Has overhangs: false
- Footprint area: 3
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I1
T
- Key: T
- Compact: 111/01
- 4 Points: 1,1,1|2,1,1|3,1,1|2,2,1
- Bounds: 3x2x1 (1-3, 1-2, 1-1)
- Has overhangs: false
- Footprint area: 4
- Used by Angel Cube (cube4), Clones Tile A Rectangle - Simple Version (rect1x1_T), Clones Tile A Rectangle - Simple Version (rect2x2_T), Clones Tile A Rectangle - Simple Version (rect3x3_T), Clones Tile A Rectangle - Simple Version (rect4x4_T), Clones Tile A Rectangle - Simple Version (rect5x5_T), Clones Tile A Rectangle - Simple Version (rect6x6_T), Clones Tile A Rectangle - Simple Version (rect7x7_T), Clones Tile A Rectangle - Simple Version (rect8x8_T)
I1
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I1
O
- Key: O
- Compact: 11/11
- 4 Points: 1,1,1|2,1,1|1,2,1|2,2,1
- Bounds: 2x2x1 (1-2, 1-2, 1-1)
- Has overhangs: false
- Footprint area: 4
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I1
I4
- Key: I4
- Compact: 1111
- 4 Points: 1,1,1|2,1,1|3,1,1|4,1,1
- Bounds: 4x1x1 (1-4, 1-1, 1-1)
- Has overhangs: false
- Footprint area: 4
I1
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I1
L
- Key: L
- Compact: 1/1/11
- 4 Points: 1,1,1|1,2,1|1,3,1|2,3,1
- Bounds: 2x3x1 (1-2, 1-3, 1-1)
- Has overhangs: false
- Footprint area: 4
I1
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I1
S
- Key: S
- Compact: 011/11
- 4 Points: 2,1,1|3,1,1|1,2,1|2,2,1
- Bounds: 3x2x1 (1-3, 1-2, 1-1)
- Has overhangs: false
- Footprint area: 4
I1
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I1
P
- Key: P
- Compact: 11/11/1
- 5 Points: 1,1,1|2,1,1|1,2,1|2,2,1|1,3,1
- Bounds: 2x3x1 (1-2, 1-3, 1-1)
- Has overhangs: false
- Footprint area: 5
- Used by P-Pentominos (pent1_Px1), P-Pentominos (pent2_Px1), P-Pentominos (pent2_Px2), P-Pentominos (pent2_Px3), P-Pentominos (pent2_Px4), P-Pentominos (pent3_Px1), P-Pentominos (pent3_Px2), P-Pentominos (pent3_Px3), P-Pentominos (pent3_Px4), P-Pentominos (pent3_Px5), P-Pentominos (pent3_Px6), P-Pentominos (pent3_Px7), P-Pentominos (pent3_Px8), P-Pentominos (pent3_Px9), P-Pentominos (pent4_Px1), P-Pentominos (pent4_Px2), P-Pentominos (pent4_Px3), P-Pentominos (pent4_Px4), P-Pentominos (pent4_Px5), P-Pentominos (pent4_Px6), P-Pentominos (pent4_Px7), P-Pentominos (pent4_Px8), P-Pentominos (pent4_Px9), P-Pentominos (pent4_Px10), P-Pentominos (pent4_Px11), P-Pentominos (pent4_Px12), P-Pentominos (pent4_Px13), P-Pentominos (pent4_Px14), P-Pentominos (pent4_Px15), P-Pentominos (pent4_Px16)
I1
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I1
A
- Key: A
- Compact: 111/11/1
- 6 Points: 1,1,1|2,1,1|3,1,1|1,2,1|2,2,1|1,3,1
- Bounds: 3x3x1 (1-3, 1-3, 1-1)
- Has overhangs: false
- Footprint area: 6
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I1
3D Polycubes
S2
- Key: S2
- Compact: 022/22
- 8 Points: 2,1,1|2,1,2|3,1,1|3,1,2|1,2,1|1,2,2|2,2,1|2,2,2
- Bounds: 3x2x2 (1-3, 1-2, 1-2)
- Has overhangs: false
- Footprint area: 4
- Used by Angel Cube (cube4)
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I1
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I1
PLR
- Key: PLR
- Compact: 221/1
- 6 Points: 1,1,1|1,1,2|2,1,1|2,1,2|3,1,1|1,2,1
- Bounds: 3x2x2 (1-3, 1-2, 1-2)
- Has overhangs: false
- Footprint area: 4
- Used by 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)
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I1
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I1
PQR
- Key: PQR
- Compact: 221/11
- 7 Points: 1,1,1|1,1,2|2,1,1|2,1,2|3,1,1|1,2,1|2,2,1
- Bounds: 3x2x2 (1-3, 1-2, 1-2)
- Has overhangs: false
- Footprint area: 5
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I1
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I1
PQO
- Key: PQO
- Compact: 221/21
- 8 Points: 1,1,1|1,1,2|2,1,1|2,1,2|3,1,1|1,2,1|1,2,2|2,2,1
- Bounds: 3x2x2 (1-3, 1-2, 1-2)
- Has overhangs: false
- Footprint area: 5
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I1
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I1
VPL
- Key: VPL
- Compact: 311/11
- 7 Points: 1,1,1|1,1,2|1,1,3|2,1,1|3,1,1|1,2,1|2,2,1
- Bounds: 3x2x3 (1-3, 1-2, 1-3)
- Has overhangs: false
- Footprint area: 5
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I1
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I1
APL
- Key: APL
- Compact: 321/2
- 8 Points: 1,1,1|1,1,2|1,1,3|2,1,1|2,1,2|3,1,1|1,2,1|1,2,2
- Bounds: 3x2x3 (1-3, 1-2, 1-3)
- Has overhangs: false
- Footprint area: 4
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I1
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I1
L+3
- Key: L+3
- Compact: 411/1
- 7 Points: 1,1,1|1,1,2|1,1,3|1,1,4|2,1,1|3,1,1|1,2,1
- Bounds: 3x2x4 (1-3, 1-2, 1-4)
- Has overhangs: false
- Footprint area: 4
- Note: L plus 3-spike
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I1
PLR+
- Key: PLR+
- Compact: 43/2
- 9 Points: 1,1,1|1,1,2|1,1,3|1,1,4|2,1,1|2,1,2|2,1,3|1,2,1|1,2,2
- Bounds: 2x2x4 (1-2, 1-2, 1-4)
- Has overhangs: false
- Footprint area: 3
- Note: High footprint variation (3 to 7) may be useful for some puzzles.
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I1
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I1
knot
- Key: knot
- Compact: 322/31/11
- 13 Points: 1,1,1|1,1,2|1,1,3|2,1,1|2,1,2|3,1,1|3,1,2|1,2,1|1,2,2|1,2,3|2,2,1|1,3,1|2,3,1
- Bounds: 3x3x3 (1-3, 1-3, 1-3)
- Has overhangs: false
- Footprint area: 7
- Note: Identical footprint on 3 sides.
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I1
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I1
knot+1
- Key: knot+1
- Compact: 422/31/11
- 14 Points: 1,1,1|1,1,2|1,1,3|1,1,4|2,1,1|2,1,2|3,1,1|3,1,2|1,2,1|1,2,2|1,2,3|2,2,1|1,3,1|2,3,1
- Bounds: 3x3x4 (1-3, 1-3, 1-4)
- Has overhangs: false
- Footprint area: 7
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I1
helix
- Key: helix
- Compact: 02/011//1
- 5 Points: 2,1,1|2,1,2|2,2,1|3,2,1|1,1,2
- Bounds: 3x2x2 (1-3, 1-2, 1-2)
- Has overhangs: true
- Footprint area: 3
- Note: Overhang in every direction.
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I1
3x4x5
- Key: 3x4x5
- Compact: 5555/5555/5555
- 60 Points: 1,1,1|1,1,2|1,1,3|1,1,4|1,1,5|2,1,1|2,1,2|2,1,3|2,1,4|2,1,5|3,1,1|3,1,2|3,1,3|3,1,4|3,1,5|4,1,1|4,1,2|4,1,3|4,1,4|4,1,5|1,2,1|1,2,2|1,2,3|1,2,4|1,2,5|2,2,1|2,2,2|2,2,3|2,2,4|2,2,5|3,2,1|3,2,2|3,2,3|3,2,4|3,2,5|4,2,1|4,2,2|4,2,3|4,2,4|4,2,5|1,3,1|1,3,2|1,3,3|1,3,4|1,3,5|2,3,1|2,3,2|2,3,3|2,3,4|2,3,5|3,3,1|3,3,2|3,3,3|3,3,4|3,3,5|4,3,1|4,3,2|4,3,3|4,3,4|4,3,5
- Bounds: 4x3x5 (1-4, 1-3, 1-5)
- Has overhangs: false
- Footprint area: 12
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I1
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I1
hollowCube
- Key: hollowCube
- Compact: 333/313/333//0//0/01
- 26 Points: 1,1,1|1,1,2|1,1,3|2,1,1|2,1,2|2,1,3|3,1,1|3,1,2|3,1,3|1,2,1|1,2,2|1,2,3|2,2,1|3,2,1|3,2,2|3,2,3|1,3,1|1,3,2|1,3,3|2,3,1|2,3,2|2,3,3|3,3,1|3,3,2|3,3,3|2,2,3
- Bounds: 3x3x3 (1-3, 1-3, 1-3)
- Has overhangs: true
- Footprint area: 9
- Note: 3x3x3 cube, with middle spot empty.
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I1
kian
- Key: kian
- Compact: 403/02/151
- 16 Points: 1,1,1|1,1,2|1,1,3|1,1,4|3,1,1|3,1,2|3,1,3|2,2,1|2,2,2|1,3,1|2,3,1|2,3,2|2,3,3|2,3,4|2,3,5|3,3,1
- Bounds: 3x3x5 (1-3, 1-3, 1-5)
- Has overhangs: false
- Footprint area: 6
- Note: Kian shape
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