From @mmalex :
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i cant believe i never realised before that to make a bitmask with set bits in [x0, x1), you can just do `mask = (1u<<x1)-(1u<<x0);` (obviously be sure that 0<=x0<=x1<32). ive used the degenerate case with x0==0 a billion times, but never thought to extend it. doh.
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Here's a chart of how close we can get to a frequency ratio of 3/2 using N-tone equal temperament.
See how great 12-tone equal temperament is?
There are also some neat patterns. See the stripes of even numbers and stripes of odd numbers? That's not a coincidence. For more charts like this, and much more cool stuff along these lines, go here:
https://johncarlosbaez.wordpress.com/2023/10/19/perfect-fifths-in-equal-tempered-scales-part-2/
(13/n)
I don't know who first came up with this, but it's really hard to un-see it.
Attends les Epstein files ep. 1 sont décaviardisables trivialement, avec une technique vieille comme Internet, et on a mis une semaine à réaliser qu'on peut avoir accès à une bonne partie du texte censuré ??
New result: you can build a universal computer using a single billiard ball on a carefully crafted table!
More precisely: you can create a computer that can run any program, using just a single point moving frictionlessly in a region of the plane and bouncing off the walls elastically.
Since the halting problem is undecidable, this means there are some yes-or-no questions about the eventual future behavior of this point that cannot be settled in a finite time by any computer program.
This is true even though the point's motion is computable to arbitrary accuracy for any given finite time. In fact, since the methodology here does *not* exploit the chaos that can occur for billiards on certain shaped tables, it's not even one of those cases where the point's motion is computable in principle but your knowledge of the initial conditions needs to be absurdly precise.
This result is not surprising to me - it would be much more surprising if you *couldn't* make a universal computer this way. Universal computation seems to be a very prevalent feature of sufficiently complex systems. But still it's very nice.
• Eva Miranda and Isaac Ramos, Classical billiards can compute, https://arxiv.org/abs/2512.19156.
From my happy folder.