Wang Tiles: Difference between revisions
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|16<ref name="YeyitoWTRocq" /> | |16<ref name="YeyitoWTRocq" /> | ||
|<code>[4,0,3,0] [3,0,4,1] [3,1,3,2] [5,2,4,0] [4,1,4,2] [4,1,3,1] [3,2,5,0]</code><ref name="YeyitoWTdcannounce">https://discord.com/channels/960643023006490684/1529586687926210691/1567754354671493152</ref> | |<code>[4,0,3,0] [3,0,4,1] [3,1,3,2] [5,2,4,0] [4,1,4,2] [4,1,3,1] [3,2,5,0]</code><ref name="YeyitoWTdcannounce">https://discord.com/channels/960643023006490684/1529586687926210691/1567754354671493152</ref> | ||
|[[File:BusyBeaverWangTiles CurrentChampion 7.png|alt=The best known champion for the largest valid square patch ( | |[[File:BusyBeaverWangTiles CurrentChampion 7.png|alt=The best known champion for the largest valid square patch (16 by 16) made by a set of 7 Wang tiles.|border|center|frameless]] | ||
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|8 | |8 | ||
|≥ 17<ref name="YeyitoWTRocq" /> | |≥ 17<ref name="YeyitoWTRocq" /> | ||
|<code>[4,0,3,0] [3,0,4,1] [3,1,3,2] [5,2,4,0] [4,1,4,2] [4,1,3,1] [3,2,5,0] [0,0,3,1]</code><ref name="YeyitoWTdcannounce" /> | |<code>[4,0,3,0] [3,0,4,1] [3,1,3,2] [5,2,4,0] [4,1,4,2] [4,1,3,1] [3,2,5,0] [0,0,3,1]</code><ref name="YeyitoWTdcannounce" /> | ||
|[[File:BusyBeaverWangTiles CurrentChampion 8.png|alt=The best known champion for the largest valid square patch ( | |[[File:BusyBeaverWangTiles CurrentChampion 8.png|alt=The best known champion for the largest valid square patch (17 by 17) made by a set of 8 Wang tiles.|border|center|frameless]] | ||
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|9 | |9 | ||
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|10 | |10 | ||
|≥ 212 | |≥ 212<ref name="JeandelRao2021">Emmanuel Jeandel, Michaël Rao, "[https://arxiv.org/pdf/1506.06492 An aperiodic set of 11 Wang tiles]". January 2021. Accessed 7 November 2026.</ref> | ||
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Latest revision as of 15:55, 7 November 2026
Wang Tiles (or Wang dominoes) are a class of formal mathematical systems first proposed by Hao Wang in 1961. They consist of equal-sized square tiles with colored edges that are arranged side-by-side on a regular grid.
When placing tiles to cover an infinite plane, you must strictly follow three structural constraints:
- Orientation: Tiles must be placed with a fixed orientation and cannot be rotated or reflected.
- Edge Matching: Adjoining edges of touching tiles must share the exact same color.
- Grid Alignment: Every tile must occupy exactly one square in a standard square grid.
Busy Beaver function
Define the score of a set of Wang tiles as the largest d such that it can tile a dxd square, or 0 if there is no largest d.
Then BBWT(n) = the maximum score over all n-tile sets.
The current format to describe a set of tiles is a list of tuples that have 4 nonegative numbers, each representing a side color. Example: [TOP,RIGHT,BOTTOM,LEFT]
| n | Value | Champion | Image |
|---|---|---|---|
| 1 | 1 | [1,0,0,0]
|
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| 2 | 2 | [1,0,2,0] [2,0,3,0]
|
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| 3 | 3 | [1,0,2,0] [2,0,3,0] [3,0,4,0]
|
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| 4 | 4 | [1,0,2,0] [2,0,3,0] [3,0,4,0] [4,0,5,0]
|
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| 5 | 6[1] | [1,0,2,0] [2,0,3,0] [3,0,4,0] [4,6,1,5] [4,5,2,6]
|
 |
| 6 | 8[1] | [1,0,2,0] [2,0,3,0] [3,0,4,0] [4,0,5,0] [5,7,1,6] [5,6,2,7]
|
 |
| 7 | 16[1] | [4,0,3,0] [3,0,4,1] [3,1,3,2] [5,2,4,0] [4,1,4,2] [4,1,3,1] [3,2,5,0][2]
|
 |
| 8 | ≥ 17[1] | [4,0,3,0] [3,0,4,1] [3,1,3,2] [5,2,4,0] [4,1,4,2] [4,1,3,1] [3,2,5,0] [0,0,3,1][2]
|
 |
| 9 | ≥ 17 | [1,0,2,0] [2,0,3,0] [3,0,4,0] [4,0,5,0] [5,0,6,0] [6,0,7,0] [7,9,1,8] [7,8,2,9] [6,8,3,9]
|
 |
| 10 | ≥ 212[3] |
Analysis
TODO
References
- ↑ 1.0 1.1 1.2 1.3 Yeyito, "BB_WT in Rocq: BB_WT(5) = 6, BB_WT(6) = 8, BB_WT(7) = 16, BB_WT(8) ≥ 17". Github document, November 2026. Accessed 7 November 2026.
- ↑ 2.0 2.1 https://discord.com/channels/960643023006490684/1529586687926210691/1567754354671493152
- ↑ Emmanuel Jeandel, Michaël Rao, "An aperiodic set of 11 Wang tiles". January 2021. Accessed 7 November 2026.