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63S-55, when N

Posted By: Nack Ballard
Date: Monday, 8 July 2024, at 10:04 p.m.

In Response To: 63S-55, when N (Timothy Chow)

I first learned about 63S-55N at -2-4 around 2010 and was surprised at the size of the margin. Perhaps what struck me most is that the rollout I had at the time didn't (quite) support N at gammon save. Later, I saw Neil's confirming -2-4 rollout in 2016. Then you apparently discovered/noticed it around 2018(?), then I independently ran across it again a couple of years ago, then you posted it in this thread.

The day before I read your post, an aging math professor friend happened to e-mail me:

Tiling the bathroom floor is easy, using square tiles, or using hexagonal tiles. It can extend as far as you want, but it will be *periodic, the pattern will *repeat, and *repeat.

You probably have seen some aperiodic tiling like this, created by the famous Roger Penrose:

[Picture omitted]

But, it's built from *two types of tiles. Could a single tile create an aperiodic pattern?

Mathematicians and non-mathematicians have spent years trying to find *one single tile shape that could tile your bathroom floor but give an *aperiodic pattern.

This year a non-mathematician found such a tile:

[Picture omitted]

Surprised everyone! Translating to German: one = ein, and stone = stein, it is now named the Einstein tile.

I wrote back: "But I thought you also told me *last year that that happened. Is it a periodic occurrence?"

He is ten years older and has even more senior moments than I do, but I knew he would take it in good humor. Actually, I think he's told me twice before. I enjoyed his reply: Congratulations. You passed my (annual) dementia test. When you consider it a new result there might be some cause for concern.

With that exchange fresh in my mind, I tossed out the term "Penrose Tile" kind of hoping you'd take the bait and tell me something about it that I don't already know. I don't have anything particular in mind. :)

Nack

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