In the context of documents, the lack of font choice regarding Times New Roman could be partly attributed to the fact that it was the default font on Microsoft Word until 2007. The irony is, of course, that it was replaced by none other than Calibri.
Nice problem! I wonder if there is a generic way of testing such a problem with different board arrangements. For example, could you apply knot theory or another concept?
I haven’t seen this representation before—I suppose the vertices of the graph are the chessboard squares, the edges
are adjacency (white squares can only be adjacent to black squares and vice versa, which gives the bipartite-ness), and covering two squares corresponds to removing those two vertices from the graph?
Well, upon a closer look, one notices that the chessboard coloring is not necessary for the problem statement. It's kind of a hint actually as you could equally well just consider a blank 8x8 board and realize that this coloring arugment works. I just feel the problem is unreasonably difficult that way.
The coloring is kind of additional structure that is applied on the object you are working with. And I think this idea of "applying structure" is a very generic. You can solve similar combinatorial arrangement problems that way, but it goes beyond that.
I think that a nice, classic (and significantly more advanced) example is showing that plane and punctured plane (a plane with one missing point) are topologically different. The fundamental (homotopy) groups of these spaces are different, and hence the spaces cannot be continuously deformed to each other.
Somehow the spirit is the same, I feel. In this topology proof it's not a grid you are working with, but a topological space. And the structure you apply is not a coloring, but something quite abstract (a homotopy group). The idea in both cases is similar, though: You apply structure and this structure reveals something that's not easy to see directly.
The magic part is figuring out the structure that produces the data you need.
It's important to note that there's geographic variability in guidelines. Also, the article doesn't give enough information about the author's other risk factors. For a similar patient (based on the initial lab results), treated by a doctor adhering to the European guidelines, at least the following items would have been considered:
- Lipid lowering drugs
- ApoB testing
- Coronary CT (if the pre-test likelihood of obstructive coronary artery disease was estimated to be > 5%)
Counter-point: For someone who's used to smoking or vaping, the craving to "take a puff" can be a very strong, maybe stronger than the chemical dependence on nicotine itself.
I noticed that in myself when I was trying to quit, vaping nicotine-free liquids helped my cravings more than nicotine itself. It didn't help the physical withdrawal symptoms but it mysteriously stopped the cravings for a while.
OpenAI’s o3 was big on en dashes—one time it produced a Deep Research result containing >200 of them. I’m not aware of any other LLM using them commonly, though. I’d guess humans use them even less often; I don’t think Apple auto-inserts en dashes, and very few people (myself being one) are pedantic enough to bother.
On the other hand, I don’t think o3 was ever a common choice among people copying from LLMs, so en dashes remain infrequent regardless.
In German en dashes are more common than em dashes. I’ve been using them regularly for at least 20 years, both in German and English texts. I never liked it when people just threw in ordinary hyphen instead of an en dash, but few people note the difference.
They're very easy to type on a Mac though (opt+-). I've always used spaced en dashes without realising that that is the more common British style. Unspaced em dashes just look wrong to me.
Unspaced em dashes look wrong too me too in most web contexts, but I think it’s typography-dependency and they look good in serif text when very large and heavy compared to other elements.