[HN Gopher] Polyominoes (2019)
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Polyominoes (2019)
Author : fatneckbeardz
Score : 22 points
Date : 2022-10-15 15:18 UTC (7 hours ago)
(HTM) web link (www.researchgate.net)
(TXT) w3m dump (www.researchgate.net)
| dynamic_sausage wrote:
| This text seems to quickly get rather technical, with theorems
| and stuff. I would suggest looking at Golomb's classical
| _Polyominoes_ first
| https://archive.org/details/polyominoes00golo/page/n7/mode/2...
|
| If you like playing with polyominoes, there is a really fun
| chesslike table game inspired by them called Blokus, and someone
| wrote an open-source engine for it:
| https://pentobi.sourceforge.io/
|
| Of course, as already noted in another comment, there's the
| ultimate tetromino game: Tetris (literally defined as tetromino
| tennis)...
| cableshaft wrote:
| There's tons of polyomino games that have come out in the past
| 8 or so years. I wouldn't necessarily say they do a better job
| than Blokus did (imo, I really like Blokus), but it's
| interesting to see what game designers have been doing with the
| concept lately.
|
| Uwe Rosenberg in particular has designed a bunch of games
| playing around with the polyomino concept, starting with
| Patchwork, then a trilogy of games Indian Summer, Cottage
| Garden, and Spring Meadow, and then culminating in a rules-
| heavy game A Feast for Odin.
|
| Phil Walker-Harding designed a very popular one called
| Barenpark, where you cover spots on a board which unlock
| different types of tiles and new boards.
|
| A recently released game that's been well received that
| utilizes a lazy susan for tile selection is called Planet
| Unknown.
|
| There's also a tile drafting game called Isle of Cats that's
| quite popular as well.
|
| A game that combines polyominoes with roll and write games
| that's quite popular is Cartographers.
|
| Patchwork kicked off this new trend in board games, and was
| released in 2014.
| tyleo wrote:
| Upon reading the first few pages and seeing the pictures I
| realized that all Tetris pieces are made of shapes with 4
| adjacent cells. Additionally, all permutations of 4-cell shapes
| are represented in Tetris. It also clicked that the prefix is
| similar to tetra.
|
| I suspect many other folks realized this already but it was a
| charming lightbulb moment for myself.
| fatneckbeardz wrote:
| A 404 page book on polyominoes
| [deleted]
| tartakovsky wrote:
| No wonder it doesn't load.
| alcover wrote:
| Polyominoes have been obsessing me. Not tiling but counting them.
| No formula has been yet found. Their _free_ numbers (i.e up to
| rotations and flipping) go 1, 1, 2, 5, 12, 35,
| 108, 369, 1285, ...
|
| Meaning for ex. there are 5 tetrominoes : #
| # # # # #
| # # # # # # #
| # # # # # # #
|
| The myterious sequence follows a ~4 growth rate.
|
| I've been (nowhere as a mathematician) exploring this problem for
| years, generating them and trying in vain to find patterns in
| their properties.
|
| I'd love readers with a high view in combinatorics telling what
| they think of this problem. Do you think a formula will one day
| be found or rather that there can't be a closed one for some
| necessary reason ?
| lkozma wrote:
| This is a very nice and deep question. Are you aware of the
| published research on the problem?
|
| The growth rate is indeed around 4 and now known to be strictly
| above 4: https://page.mi.fu-
| berlin.de/rote/Papers/pdf/Lambda-4.pdf
|
| If I remember correctly, there was a non-rigorous argument for
| a concrete conjectured value not far above 4.
| alcover wrote:
| Thanks for the link. we obtain 4.00253176 as
| a certified lower bound
|
| How intringing it's strictly just above 4 !
|
| Maybe it has to do with symmetries. The proportion of
| symmetric polys in successive generations goes towards zero.
| I suspect that this 'pollutes' the asymptot.
|
| About published research, I just glanced at the little I
| found, (as I often can't do more than glance, being limited
| in Maths.)
|
| Also I prefer to go at it naively first.
| NKosmatos wrote:
| A000105: Number of free polyominoes (or square animals) with n
| cells
|
| https://oeis.org/A000105
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