[ I started thinking about this about twenty years ago, and then writing it down in 2019, but it seems to be obsolete. I am publishing it anyway. ] The canonical division of the year into seasons in the northern temperate zone goes something like this: Spring: March 21 – June 21 Summer: June 21 – September 21 Autumn: September 21 – December 21 Winter: December 21 – March 21 Living in the mid-Atlantic region of the northeast U.S., I have never been happy with this. It is just not a good description of the climate. I begin by observing that the year is not equally partitioned between the four seasons. The summer and winter are longer, and spring and autumn are brief and happy interludes in between. I have no problem with spring beginning in the middle of March. I think that is just right. March famously comes in like a lion and goes out like a lamb. The beginning of March is crappy, like February, and frequently has snowstorms and freezes. By the end of March, spring is usually skipping along, with singing birds and not just the early flowers (snowdrops, crocuses, daffodil) but many of the later ones also. By the middle of May the spring flowers are over and the weather is getting warm, often uncomfortably so. Summer continues through the beginning of September, which is still good for swimming and lightweight clothes. In late September it finally gives way to autumn. Autumn is jacket weather but not overcoat weather. Its last gasp is in the middle of November. By this time all the leaves have changed, and the ones that are going to fall off the trees have done so. The cool autumn mist has become a chilly winter mist. The cold winter rains begin at the end of November. So my first cut would look something like this: Months Seasons January February March April May June July August September October November December Winter Spring Summer Autumn Winter Note that this puts Thanksgiving where it belongs at the boundary between autumn (harvest season) and winter (did we harvest enough to survive?). Also, it puts the winter solstice (December 21) about one quarter of the way through the winter. This is correct. By the solstice the days have gotten short, and after that the cold starts to kick in. (“As the days begin to lengthen, the cold begins to strengthen”.) The conventional division takes the solstice as the beginning of winter, which I just find perplexing. December 1 is not the very coldest part of winter, but it certainly isn't autumn. There is something to be said for it though. I think I can distinguish several subseasons — ten in fact: Dominus Seasonal Calendar Months Seasons Sub-seasons January February March April May June July August September October November December Winter Spring Summer Autumn Winter Midwinter Late Winter Early spring Late spring Early Summer Midsummer Late Summer Early autumn Late autumn Early winter Midwinter Midwinter, beginning around the solstice, is when the really crappy weather arrives, day after day of bitter cold. In contrast, early and late winter are typically much milder. By late February the snow is usually starting to melt. (March, of course, is always unpredictable, and usually has one nasty practical joke hiding up its sleeve. Often, March is pleasant and springy in the second week, and then mocks you by turning back into January for the third week. This takes people by surprise almost every year and I wonder why they never seem to catch on.) Similarly, the really hot weather is mostly confined to midsummer. Early and late summer may be warm but you do not get blazing sun and you have to fry your eggs indoors, not on the pavement. Why the seasons seem to turn in the middle of each month, and not at the beginning, I can't say. Someone messed up, but who? Probably the Romans. I hear that the Persians and the Baha’i start their year on the vernal equinox. Smart! Weather in other places is very different, even in the temperate zones. For example, in southern California they don't have any of the traditional seasons. They have a period of cooler damp weather in the winter months, and then instead of summer they have a period of gloomy haze from June through August. However I may have waited too long to publish this article, as climate change seems to have rendered it obsolete. In recent years, we have barely had midwinter, and instead of the usual two to three annual snows we have zero. Midsummer has grown from two to four months, and summer now lasts into October.

Suppose a centrifuge has slots, arranged in a circle around the center, and we have test tubes we wish to place into the slots. If the tubes are not arranged symmetrically around the center, the centrifuge will explode. (By "arranged symmetrically around the center, I mean that if the center is at , then the sum of the positions of the tubes must also be at .) Let's consider the example of . Clearly we can arrange , , , or tubes symmetrically: Equally clearly we can't arrange only . Also it's easy to see we can do tubes if and only if we can also do tubes, which rules out . From now on I will write to mean the problem of balancing tubes in a centrifuge with slots. So and are possible, and and are not. And is solvable if and only if is. It's perhaps a little surprising that is possible. If you just ask this to someone out of nowhere they might have a happy inspiration: “Oh, I'll just combine the solutions for and , easy.” But that doesn't work because two groups of the form and always overlap. For example, if your group of is the slots then you can't also have your group of be , because slot already has a tube in it. The other balanced groups of are blocked in the same way. You cannot solve the puzzle with ; you have to do as below left. The best way to approach this is to do , as below right. This is easy, since the triangle only blocks three of the six symmetric pairs. Then you replace the holes with tubes and the tubes with holes to turn into . Given and , how can we decide whether the centrifuge can be safely packed? Clearly you can solve when is a multiple of , but the example of (or ) shows this isn't a necessary condition. A generalization of this is that is always solvable if since you can easily balance tubes at positions , then do another tubes one position over, and so on. For example, to do you just put first four tubes in slots and the next four one position over, in slots . An interesting counterexample is that the strategy for , where we did , cannot be extended to . One would want to do , but there is no way to arrange the tubes so that the group of doesn't conflict with the group of , which blocks one slot from every pair. But we can see that this must be true without even considering the geometry. is the reverse of , which impossible: the only nontrivial divisors of are and , so must be a sum of s and s, and is not. You can't fit tubes when , but again the reason is a bit tricky. When I looked at directly, I did a case analysis to make sure that the -group and the -group would always conflict. But again there was an easier was to see this: and clearly won't work, as is not a sum of s and s. I wonder if there's an example where both and are not obvious? For , every works except and the always-impossible . What's the answer in general? I don't know. Addenda 20250502 Now I am amusing myself thinking about the perversity of a centrifuge with a prime number of slots, say . If you use it at all, you must fill every slot. I hope you like explosions! While I did not explode any centrifuges in university chemistry, I did once explode an expensive Liebig condenser. Condenser setup by Mario Link from an original image by Arlen on Flickr. Licensed cc-by-2.0, provided via Wikimedia Commons. 20250503 Michael Lugo informs me that a complete solution may be found on Matt Baker's math blog. I have not yet looked at this myself. Omar Antolín points out an important consideration I missed: it may be necessary to subtract polygons. Consider . This is obviously possible since . But there is a more interesting solution. We can add the pentagon to the digons and to obtain the solution $${0,5,6,10,12,18, 20, 24, 25}.$$ Then from this we can subtract the triangle to obtain $${5, 6, 12, 18, 24, 25},$$ a solution to which is not a sum of regular polygons: Thanks to Dave Long for pointing out a small but significant error, which I have corrected. 20250505 Robin Houston points out this video, The centrifuge Problem with Holly Krieger, on the Numberphile channel.

Content warning: rambly Given the coordinates of the three vertices of a triangle, can we find the area? Yes. If by no other method, we can use the Pythagorean theorem to find the lengths of the edges, and then Heron's formula to compute the area from that. Now, given the coordinates of the four vertices of a quadrilateral, can we find the area? And the answer is, no, there is no method to do that, because there is not enough information: These three quadrilaterals have the same vertices, but different areas. Just knowing the vertices is not enough; you also need their order. I suppose one could abstract this: Let be the function that maps the set of vertices to the area of the quadrilateral. Can we calculate values of ? No, because there is no such , it is not well-defined. Put that way it seems less interesting. It's just another example of the principle that, just because you put together a plausible sounding description of some object, you cannot infer that such an object must exist. One of the all-time pop hits here is: Let be the smallest [real / rational] number strictly greater than … which appears on Math SE quite frequently. Another one I remember is someone who asked about the volume of a polyhedron with exactly five faces, all triangles. This is a fallacy at the ontological level, not the mathematical level, so when it comes up I try to demonstrate it with a nonmathematical counterexample, usually something like “the largest purple hat in my closet” or perhaps “the current Crown Prince of the Ottoman Empire”. The latter is less good because it relies on the other person to know obscure stuff about the Ottoman Empire, whatever that is. This is also unfortunately also the error in Anselm's so-called “ontological proof of God”. A philosophically-minded friend of mine once remarked that being known for the discovery of the ontological proof of God is like being known for the discovery that you can wipe your ass with your hand. Anyway, I'm digressing. The interesting part of the quadrilateral thing, to me, is not so much that doesn't exist, but the specific reasoning that demonstrates that it can't exist. I think there are more examples of this proof strategy, where we prove nonexistence by showing there is not enough information for the thing to exist, but I haven't thought about it enough to come up with one. There is a proof, the so-called “information-theoretic proof”, that a comparison sorting algorithm takes at least time, based on comparing the amount of information gathered from the comparisons (one bit each) with that required to distinguish all possible permutations ( bits total). I'm not sure that's what I'm looking for here. But I'm also not sure it isn't, or why I feel it might be different. Addenda 20250430 Carl Muckenhoupt suggests that logical independence proofs are of the same sort. He says, for example: Is there a way to prove the parallel postulate from Euclid's other axioms? No, there is not enough information. Here are two geometric models that produce different results. This is just the sort of thing I was looking for. 20250503 Rik Signes has allowed me to reveal that he was the source of the memorable disparagement of Anselm's dumbass argument.

(Previously: [1] [2]) Welcome to Philadelphia! We have a lot of political corruption here. I recently wrote about the unusually corrupt Philadelphia Traffic Court, where four of the judges went to the federal pokey, and the state decided there was no way to clean it up, they had to step on it like a cockroach. I ended by saying: One of those traffic court judges was Willie Singletary, who I've been planning to write about since 2019. But he is a hard worker who deserves better than to be stuck in an epilogue, so I'll try to get to him later this month. This is that article from 2019, come to fruit at last. It was originally inspired by this notice that appeared at my polling place on election day that year: (Click for uncropped version) VOTES FOR THIS CANDIDATE WILL NOT BE COUNTED DEAR VOTERS: Willie Singletary, candidate for Democratic Council At-Large, has been removed from the Primary Ballot by Court Order. Although his name appears on the ballot, votes for this candidate will not be counted because he was convicted of two Class E felonies by the United States District Court for the Eastern District of Pennsylvania, which bars his candidacy under Article 2, Section 7 of the Pennsylvania Constitution. That's because Singletary had been one of those traffic court judges. In 2014 he had been convicted of lying to the FBI in connection with that case, and was sentenced to 20 months in federal prison; I think he actually served 12. That didn't stop Willie from trying to run for City Council, though, and the challenge to his candidacy didn't wrap up before the ballots were printed, so they had to post these notices. Even before the bribery scandal and the federal conviction, Singletary had already lost his Traffic Court job when it transpired that he had showed dick pics to a Traffic Court cashier. Before that, when he was campaigning for the Traffic Court job, he was caught on video promising to give favorable treatment to campaign donors. But Willie's enterprise and go-get-it attitude means he can't be kept down for long. Willie rises to all challenges! He is now enjoying a $90,000 annual salary as a Deputy Director of Community Partnerships in the administration of Philadelphia Mayor Cherelle Parker. Parker's spokesperson says "The Parker administration supports every person’s right to a second chance in society.” I think he might be on his fourth or fifth chance by now, but who's counting? Let it never be said that Willie Singletary was a quitter. Lorrie once made a remark that will live in my memory forever, about the "West Philadelphia local politics-to-prison pipeline”. Mayor Parker is such a visionary that she has been able to establish a second pipeline in the opposite direction! Addendum 20250501 I don't know how this happened, but when I committed the final version of this article a few days ago, the commit message that my fingers typed was: Date: Sat Apr 26 14:24:19 2025 -0400 Willie Wingletsray finally ready to go And now, because Git, it's written in stone.