## Polyomino tilings

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### Polyomino tilings

What is known about polyomino tilings in the plane? I've done a bit of preliminary work, finding tilings for every n-omino up to n=5:

B:

x = 16, y = 24, rule = //33AB3AB3AB3AB$ABABABABABABABAB$A3BA3BA3BA3B$B3AB3AB3AB3A$BABABABABABABABA$3BA3BA3BA3BA$3AB3AB3AB3AB$ABABABABABABABAB$A3BA3BA3BA3B$B3AB3AB3AB3A$BABABABABABABABA$3BA3BA3BA3BA$3AB3AB3AB3AB$ABABABABABABABAB$A3BA3BA3BA3B$B3AB3AB3AB3A$BABABABABABABABA$3BA3BA3BA3BA$3AB3AB3AB3AB$ABABABABABABABAB$A3BA3BA3BA3B$B3AB3AB3AB3A$BABABABABABABABA$3BA3BA3BA3BA! Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace! muzik Posts: 3248 Joined: January 28th, 2016, 2:47 pm Location: Scotland ### Re: Polyomino tilings muzik wrote:I don't know how this one works out: x = 38, y = 11, rule = //53.C$2.2CB3A12.B2.C2.B2.C2.B2.C$.3C2B3AD9.3B3C3B3C3B3C$2.CA3BA2DC9.3B3C3B3C3B3C$2.3ABC3D2C8.3A3D3A3D3A3D$2.2AB3CDA3C6.3A3D3A3D3A3D$2.A3B2C3AC8.A2.D2.A2.D2.A2.D$2.3BCDC2AB$.3D2C2DA3B$3D3C3D2B$.D2.C2.D.B! x = 52, y = 41, rule = //520.A$19.3AC$18.B2A3CA$17.2BAD2C3AC$16.3B2DCB2A3CA$16.AB3D2BAD2C3AC$15.3ACD3B2DCB2A3CA$14.B2A3CAB3D2BAD2C3AC$13.2BAD2C3ACD3B2DCB2A3CA$12.3B2DCB2A3CAB3D2BAD2C3AC$12.AB3D2BAD2C3ACD3B2DCB2A3CA$11.3ACD3B2DCB2A3CAB3D2BAD2C3AC$10.B2A3CAB3D2BAD2C3ACD3B2DCB2A3CA$9.2BAD2C3ACD3B2DCB2A3CAB3D2BAD2C3AC$8.3B2DCB2A3CAB3D2BAD2C3ACD3B2DCB2A3CA$8.AB3D2BAD2C3ACD3B2DCB2A3CAB3D2BAD2C3AC$7.3ACD3B2DCB2A3CAB3D2BAD2C3ACD3B2DCB2A3C$6.B2A3CAB3D2BAD2C3ACD3B2DCB2A3CAB3D2BAD2C$5.2BAD2C3ACD3B2DCB2A3CAB3D2BAD2C3ACD3B2DC$4.3B2DCB2A3CAB3D2BAD2C3ACD3B2DCB2A3CAB3D$4.AB3D2BAD2C3ACD3B2DCB2A3CAB3D2BAD2C3ACD$3.3ACD3B2DCB2A3CAB3D2BAD2C3ACD3B2DCB2A3C$2.B2A3CAB3D2BAD2C3ACD3B2DCB2A3CAB3D2BAD2C$.2BAD2C3ACD3B2DCB2A3CAB3D2BAD2C3ACD3B2DC$3B2DCB2A3CAB3D2BAD2C3ACD3B2DCB2A3CAB3D$.B3D2BAD2C3ACD3B2DCB2A3CAB3D2BAD2C3ACD$3.D3B2DCB2A3CAB3D2BAD2C3ACD3B2DCB2A3C$5.B3D2BAD2C3ACD3B2DCB2A3CAB3D2BAD2C$7.D3B2DCB2A3CAB3D2BAD2C3ACD3B2DC$9.B3D2BAD2C3ACD3B2DCB2A3CAB3D$11.D3B2DCB2A3CAB3D2BAD2C3ACD$13.B3D2BAD2C3ACD3B2DCB2A3C$15.D3B2DCB2A3CAB3D2BAD2C$17.B3D2BAD2C3ACD3B2DC$19.D3B2DCB2A3CAB3D$21.B3D2BAD2C3ACD$23.D3B2DCB2A3C$25.B3D2BAD2C$27.D3B2DC$29.B3D$31.D! EDIT: An octomino with what I believe to be not only a unique tiling, but also a unique (and also anisotropic) 3-coloring of such tiling: x = 117, y = 118, rule = //460.A2.A$59.6A$59.6C$60.CABC$56.C2.C2A2BA2.A$55.6CAB6A$55.6BAB6C$56.BCAB2A2BCABC$52.B2.B2C2ACABC2A2BA2.A$51.6BCA6CAB6A$51.6ACA6BAB6C$52.ABCA2C2ABCAB2A2BCABC$48.A2.A2B2CBCAB2C2ACABC2A2BA2.A$47.6ABC6BCA6CAB6A$47.6CBC6ACA6BAB6C$48.CABC2B2CABCA2C2ABCAB2A2BCABC$44.C2.C2A2BABCA2B2CBCAB2C2ACABC2A2BA2.A$43.6CAB6ABC6BCA6CAB6A$43.6BAB6CBC6ACA6BAB6C$44.BCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC$40.B2.B2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BA2.A$39.6BCA6CAB6ABC6BCA6CAB6A$39.6ACA6BAB6CBC6ACA6BAB6C$40.ABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC$36.A2.A2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BA2.A$35.6ABC6BCA6CAB6ABC6BCA6CAB6A$35.6CBC6ACA6BAB6CBC6ACA6BAB6C$36.CABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC$32.C2.C2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BA2.A$31.6CAB6ABC6BCA6CAB6ABC6BCA6CAB6A$31.6BAB6CBC6ACA6BAB6CBC6ACA6BAB6C$32.BCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC$28.B2.B2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BA2.A$27.6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6A$27.6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6C$28.ABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC$24.A2.A2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BA2.A$23.6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6A$23.6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6C$24.CABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC$20.C2.C2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BA2.A$19.6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6A$19.6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6C$20.BCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC$16.B2.B2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BA2.A$15.6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6A$15.6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6C$16.ABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC$12.A2.A2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BA2.A$11.6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6A$11.6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6C$12.CABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC$8.C2.C2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BA2.A$7.6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6A$7.6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6C$8.BCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC$4.B2.B2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2B$3.6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB$3.6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB$4.ABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2B$4.2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC$5.BC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6C$5.BC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA$.AB.2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2A$2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB$.AB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6B$.AB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC$2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2C$.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA$5.AB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6A$5.AB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB$4.2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2B$5.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC$9.AB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6C$9.AB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA$8.2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2A$9.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB$13.AB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6B$13.AB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB6CBC$12.2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2C$13.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA$17.AB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6CAB6A$17.AB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA6BAB$16.2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2B$17.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC$21.AB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6BCA6C$21.AB6CBC6ACA6BAB6CBC6ACA6BAB6CBC6ACA$20.2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2A$21.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB$25.AB6ABC6BCA6CAB6ABC6BCA6CAB6ABC6B$25.AB6CBC6ACA6BAB6CBC6ACA6BAB6CBC$24.2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2C$25.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC2A2BABCA$29.AB6ABC6BCA6CAB6ABC6BCA6CAB6A$29.AB6CBC6ACA6BAB6CBC6ACA6BAB$28.2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2ABCAB2A2B$29.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB2C2ACABC$33.AB6ABC6BCA6CAB6ABC6BCA6C$33.AB6CBC6ACA6BAB6CBC6ACA$32.2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2CABCA2C2A$33.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA2B2CBCAB$37.AB6ABC6BCA6CAB6ABC6B$37.AB6CBC6ACA6BAB6CBC$36.2A2BCABC2B2CABCA2C2ABCAB2A2BCABC2B2C$37.AB.2A2BABCA2B2CBCAB2C2ACABC2A2BABCA$41.AB6ABC6BCA6CAB6A$41.AB6CBC6ACA6BAB$40.2A2BCABC2B2CABCA2C2ABCAB2A2B$41.AB.2A2BABCA2B2CBCAB2C2ACABC$45.AB6ABC6BCA6C$45.AB6CBC6ACA$44.2A2BCABC2B2CABCA2C2A$45.AB.2A2BABCA2B2CBCAB$49.AB6ABC6B$49.AB6CBC$48.2A2BC2.C2B2C$49.AB5.ABCA$55.6A!

EDIT 2: A proof that the pre-pulsar heptomino is untileable:
Starting with a single heptomino:
x = 5, y = 2, rule = //5A3.A$5A! The three-cell "hollow" within the heptomino must be filled with a one-cell corner and a two-cell edge (three one-cell corners is obviously impossible): x = 5, y = 2, rule = //5AC2BA$5A!

This can take on either of the following two forms:
x = 18, y = 6, rule = //55.2B8.2B$6.B8.B$6.B8.B$5CDB3.5CB$C2.AC2BA2.C2.AC2BA$3.5A5.5A! In the first form, the marked cell cannot belong to any heptomino, as that would necessitate overlap. Therefore, the second form is the only one possible. Applying this form to the darker-orange heptomino gives this, and then this (ignoring the other heptomino as it is unnecessary for the proof): x = 18, y = 8, rule = //52.2B8.2B$2.B9.B$2.B5C4.B5C$2.BC3.C4.BC2D.C$A.2BA5.A.2BAD$5A5.5AD$15.D$14.2D!

This shows that any heptomino in the tiling must be a part of one of these larger C4_4-symmetric 28-ominoes. In addition, any heptomino must be a part of exactly one 28-omino to prevent overlap. So one can treat the heptomino tiling as simply a tiling of such 28-ominoes.

Starting from a single 28-omino, the marked cell can only belong to another 28-omino in the following way:
x = 26, y = 9, rule = //52.2A16.2B$2.A11.2A4.B$2.6A6.A5.6B$2.4ABA6.6A4B.B$A.4A8.4ABA4B$6A6.A.4A6B$5.A6.6A5.B$4.2A11.A4.2B$16.2A!

Extending this gives:
x = 15, y = 15, rule = //58.2B$2.2A4.B$2.A5.6B$2.6A4B.B$2.4ABA4B$A.4A6B$6A3.2DB$3.2CA3.D2B$3.C2A3.6D$3.6C4D.D$3.4CDC4D$.C.4C6D$.6C5.D$6.C4.2D$5.2C!

It is clearly impossible to fill the central 9-cell region using 28-ominos. Therefore, it is impossible to tile the plane using the original heptomino. QED, am I correct?

EDIT 3: Simpler Herschel tiling:
x = 32, y = 34, rule = //515.3A$13.3BA$13.B3A3C$12.3B3DC$12.3AD3C3A$10.3BA3D3BA$10.B3A3CB3A3C$9.3B3DC3B3DC$9.3AD3C3AD3C3A$7.3BA3D3BA3D3BA$7.B3A3CB3A3CB3A3C$6.3B3DC3B3DC3B3DC$6.3AD3C3AD3C3AD3C3A$4.3BA3D3BA3D3BA3D3BA$4.B3A3CB3A3CB3A3CB3A3C$3.3B3DC3B3DC3B3DC3B3DC$3.3AD3C3AD3C3AD3C3AD3C$.3BA3D3BA3D3BA3D3BA3D$.B3A3CB3A3CB3A3CB3A3C$3B3DC3B3DC3B3DC3B3DC$3.D3C3AD3C3AD3C3AD3C$2.3D3BA3D3BA3D3BA3D$5.B3A3CB3A3CB3A3C$4.3B3DC3B3DC3B3DC$7.D3C3AD3C3AD3C$6.3D3BA3D3BA3D$9.B3A3CB3A3C$8.3B3DC3B3DC$11.D3C3AD3C$10.3D3BA3D$13.B3A3C$12.3B3DC$15.D3C\$14.3D!
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

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Aidan F. Pierce

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### Re: Polyomino tilings

My friend Joseph Myers conducted arguably the most comprehensive study on tilings of polyominos, polyiamonds, and polyhexes:

https://www.polyomino.org.uk/mathematic ... rm-tiling/

This was part of an attempt to find an aperiodic monotile; whilst unsuccessful in this regard, he succeeded in breaking various records (such as finding a 16-hex where any fundamental region must have at least 10 tiles).
What do you do with ill crystallographers? Take them to the mono-clinic!

calcyman

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### Re: Polyomino tilings

calcyman wrote:My friend Joseph Myers conducted arguably the most comprehensive study on tilings of polyominos, polyiamonds, and polyhexes:

https://www.polyomino.org.uk/mathematic ... rm-tiling/

This was part of an attempt to find an aperiodic monotile; whilst unsuccessful in this regard, he succeeded in breaking various records (such as finding a 16-hex where any fundamental region must have at least 10 tiles).

Wow. Okay, forget I ever posted this topic...
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

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Aidan F. Pierce

A for awesome

Posts: 1730
Joined: September 13th, 2014, 5:36 pm
Location: 0x-1

### Re: Polyomino tilings

For every n, there is a tileable n-polyomino (certainly, since a one-cell line is one). One of these n-cell polyominoes has the greatest number m of tiling collocations.
Define M(n) as the sequence with the maximum m for every n-omino.
How does M(n) grow, or tend to grow? (linear? log? sqrt?)

PS stacking lines with offsets per line obviously counts as one (1) single disposition, because it's 1D ordering with a flat surface; think that for any 1D ordered pattern with two flat surfaces there should only be ONE valid disposition containing that given 1D ordering. Otherwise M(n)=n
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Rhombic

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### Re: Polyomino tilings

muzik wrote:Some hexes:

Any jinxes or curses?

wwei23

Posts: 935
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### Re: Polyomino tilings

wwei23 wrote:
muzik wrote:Some hexes:

Any jinxes or curses?

The latter goes against rule 2b, unfortunately.
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