Mark Dominus

How many ways are there to tile a 2×2n rectangle with n Tetris tiles?

The answer turns out to be the square of the n'th #Fibonacci number: 1, 1, 4, 9, 25, 64, 169, …

https://shreevatsa.net/post/tetris-tilings-fibonacci/ by @svat

Mar 9 at 10:21pmWeb
Show threadShreevatsa RMar 10
@mjd Thank you, I'm not sure about the etiquette of boosting a mention of my blog post, but I was considering reposting this myself (such a cool fact!) as a top-level post (toot?) instead of as a reply, so I'll just reuse yours. :)
Show threadarclightMar 10
@svat @mjd In this age of laundering attribution, it good to see credit where it's due. Nice work!
Show threadn-gonsMar 10
@svat @mjd it's cool and you should boost your things!
Show threadjcreedMar 10
@mjd @svat nice blog post! I tentatively suspect (but haven't proved) that the following picture visually summarizes the same bjection as the one in the post: https://mastodon.social/@jcreed/116155046190192425
Show threadjcreedMar 10
@mjd @svat (it does so by describing a bijection
indecomposable tilings of a 2x2N by 1x1 and 1x2 blocks <->
indecomposable tilings of a 2x2N by tetris pieces; a general tiling is the horizontal composition of some sequence of indecomposable tilings)
Show threadShreevatsa RMar 10
@jcreed @mjd Thank you, I had missed this thread from the original discussion; looks intriguing and I will think about it!
Show threadjcreedMar 10
@svat @mjd it's a fun puzzle! :)
Show threadjcreedMar 11
@svat @mjd Here's a js implementation: https://jsbin.com/yilucacavu/1/edit?html,output
JS Bin

A live pastebin for HTML, CSS & JavaScript and a range of processors, including SCSS, CoffeeScript, Jade and more...

Show threadShreevatsa R6d ago

@jcreed @mjd Thank you. I thought about it some more... Being totally lacking in any visual imagination, I had to translate your picture to symbols and automata, but now I finally understand it and can see that it's the “same” bijection, though stated differently. (And it's cool that you could just think of it directly.)

I've added a new section at the bottom of the post mentioning your bijection (let me know if I should call you something other than "@jcreed"), search for the part that says "Added later 2026-03-12": https://shreevatsa.net/post/tetris-tilings-fibonacci/#:~:text=Added%20later%202026%2D03%2D12

(cc @robinhouston )

Tilings with Tetris pieces, and squared Fibonacci numbers | Blog

Show threadjcreed6d ago
@svat @mjd @robinhouston neat! very cool that there are so many angles to look at the same problem from. thanks for collecting them together!