My blog is a good blog. Totally untamed, but vaguely relating to my interests aka math geology music. snow is my first and only true love. I can write math now! \( e^{i\theta}=\cos\theta+i\sin\theta \)
Its the first one where I came up with the problem, and then it had a clever solution!
Say you have a number line, with a dot at every number from 1 to n. Now, draw the line connecting all of those dots. How many lines are there (easy, not answered here)? How long are all of the lines?
To explain the problem, we have:
n=1, 0 lines, 0 length
n=2, 1 line (between 1 and 2), length 1
n=3, 3 lines (between 1 and 2, 1 and 3, and 2 and 3), length 4
n=4, 6 lines (the same as above, and adding 1 and 4, 2 and 4, 3 and 4), length 10.
(caption: visualization of n=4. The 4 points are shown above and the 6 options for line length are shown below)
Answer below the cut, but I will say- the joy of this problem isn't the answer, but the interpretation(s) of it.
We can 'cheat' and just write this down in summation notation: \( \sum_{i=1}^n \sum_{j=i+1}^n j-i \). If you remember how to do summations, great! But there are more interesting ways of thinking about this problem
The written description for n=4 gave the start of a recurrence solution, where the if you add point n you add the lines of length 1,2,3,.... n-2, n-1. These are the triangular numbers. So, our sum is in some (very real) sense stacking triangles of increasing size, which imply some 3 sided triangular pyramid of the numbers. I now know this is called the tetrahedral numbers.
However, there's another way of thinking about it. The visual description of n=4 imply that instead, we should first group our segments by length, and then add them to our sum. After checking back in with the picture for n=4, we see that there are 3 of length 1, 2 of length 2, and 1 off length 3. Extrapolating back into summation notation, this would be \( \sum_{i=1}^n i*(n-i) \). But more interestingly, this 'looks' like we're stacking rectangles that slowly morph into squares, and then back into rectangles in the other direction. This counts the same thing as before, so it should be the tetrahedral number- but how?
Before, we thought of tetrahedrons as starting from the base. But if instead you think of them as starting from the edge, it turns out that these rectangles to squares to rectangles is exactly the image we need. ( This makes me think of this problem, which somehow highlights the edge arrangement of a tetrahedron for me).
Here's the nice gif to show building a tetrahedron from triangles or rectangles! We're sticking with n=4, with the one on the left being building up triangles from the base and the one on the right building up rectangles from the edge:
I later discovered it can also be done purely combinatorically, which I think provides the 'most satisfying' answer, although like most combinatorics feels a little magical. Take a line with n+1 points, and offset it by 1/2 so it fully overlaps the original line. Then, for a segment on our line with n points, its 'subsegments of length one' overlaps one of our vertices on our line with n+1 points. The left vertex of the segment on n points corresponds to the point 1/2 to the left on the line with n+1 points, and the right vertex of the segment on n points corresponds to the point 1/2 to the right on the line with n+1 point. The picture below shows one line segment with n=4, with features on the top line colored and matching and the corresponding vertices in the bottom line with the additional vertex.
You can also find this on the tetrahedron, but its harder- I tend to think of it as an extension to the 'stacking triangles' case. The last point chooses how many triangles to go 'down' the tetrahedron from the vertex, and then the first 2 points count the lengths, in this clever way. I will say making sure that 'that point in the tetrahedron really exists' is a pain, but I've convinced myself.
This last model of the problem gives the answer \( {n+1 \choose 3} \)
I love this problem, largely because it was the first 'aha!" I had in math, particularly when connecting disparate subjects.
This is my first mathblr post. I hope it shows my enthusiasm and is somewhat followable for people who've had a discrete math course. (I also hope it renders well!)
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Clarification: I don’t hate this book, I love it, it’s amazing. It’s just that taking a step back and looking it out of context is still really funny. Especially the line “We participated in a genocide, Barney.”
Bedrock is having a mayoral election. One of the candidates is a violent war mongering asshole that riles people up against the lizard people. This reminds Fred and Barney of their time in the army.
Back then the father of said violent candidate was riling people up against the “tree people”. Fred, Barney, and other soldiers fought what they believed to be a defensive measure against the tree people. Turns out, it was actually an invasion, in order to kill off the tree people and take over their forest to build Bedrock.
That’s what Fred means when he says he and Barney participated in a genocide. They literally did.
(Extra fun fact, Barney adopted a tree person baby after the war, and his son Bamm-Bamm is the last tree person.)
"Stop saying 15 year olds with weird interests are cringe, they're 15" this is true however you should also stop saying adults with weird interests are cringe because who gives a shit
I want to share some wisdom from my high school art teacher.
In my AP Art class, there was a girl who was just starting to experiment with mixed media. At this point she was still playing around, trying to decide what direction she wanted to go with her portfolio. So one critique day, she brought in an abstract canvas with some rhinestone highlights and painted and real peacock feathers. She loved sparkles and peacock feathers so she thought she’d try introducing them a *little*. And after everyone had given some input, the teacher gave her his advice, VERY roughly paraphrased here:
“So here’s the thing… I do not like this style. These are just elements that do not speak to me personally, but I see that you like them, and you’re doing interesting things with them.
“My biggest critique is, I only merely *dislike* this piece. I want you to make me HATE it. Go crazy with the things that you like. Don’t hold back trying to make it palatable to people like me. Because I am NEVER going to like it. And if the audience does not like it, it should drive them crazy seeing how much YOU love it.”
Her portfolio was chock full of neon colors and glitter and rhinestones and splashes of peacock feathers and it was a delight. Our teacher despised every piece lol, but she got great marks and I think even won some awards. And more importantly, she was happy and proud of the results. Because she didn’t limit herself by trying to appeal to people who were never going to enjoy what she enjoyed.
Takeaway here: be as cringe as you want. Don’t limit yourself based on other ppl’s tastes. They’re not you, and you are incredible 💕
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I got a 4 min long video of Kimchi dreaming today, so here's a clip
You get the whole walk cycle and the little sprint at the end.
Sometimes her sprints last for like 4 or 5 seconds and she can shoot herself off the couch or into a wall if she gets a grip with her back claws. If she does it next to a wall, her head smacking into it sounds like someone is trying to break into the house. She doesn't wake up.
Later in the dream she injured her paw and was limping, and earlier she caught something and ate it.
just got phone call from my neurologist WHO IS ON VACATION may his name never be erased, basically being like. "ok so your insurance is trying to kill you. I've referred dozens of people to Duke for this and this has never happened. they're being purposefully confusing and sending us all in circles. they keep sending me forms asking ME to agree to pay for your treatment no matter how many times I explain that I am THE REFERRING PHYSICIAN. so here's what you're going to do-" and gave us instructions on what to do when the health insurance corporation is trying to kill you
this is not his job. his job is being my neurologist.
his job is not "call patients after hours while I'm on vacation because I'm afraid if I don't give them this guidance their insurance company might murder them and so even though this isn't my job it feels like a fucking hostage situation"
anyway. he confirmed that the shit my insurance has been pulling for a month and a half is explicitly Trying To Kill The Patient behavior. told us to get everything in writing. sending us a whole packet of every single relevant form and piece of paperwork so we have it all on hand. told us to contact the patient advocate and if that doesn't work, we're going to have to go outside the company entirely and get the state insurance ombudsman involved.
so. that's fun. and a totally productive use of everyone's time and resources. I've had insurance companies be shitty to me but I've never been the Patient They're Explicitly Trying To Kill before. like I've never had them throw this much energy at blocking a procedure this lifesaving. it actually feels even worse than I thought it would!
I had a dream last night that tumblr came up with a pride flag for closeted gays and it was just a light blue flag with a shrimp on it. People would also wear shrimp pins on their lapels for some Reason???
and the vegan gays started Discourse because shrimp deserved more respect
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idk I just can't imagine caring so much about distancing myself from popular absurdist jokes among children that I avoid putting a 6 and a 7 next to each other in any context ever. you're scared of children's laughter? of joy and whimsy? do you shield your eyes from rainbows too?
One of the funniest (to me) things that I do with my Learn To Read Hebrew students is when they learn the letter Shin, and they can suddenly read the words for 6 (Sheish) and 7 (Sheva). "Sheish-Sheva! Sheish-Sheva!" I tell them, with glee in my voice. "Now you can annoy the adults in your life in Hebrew as well as in English!"
The looks on their faces, man. The glorious expression of "I don't know whether to laugh at what is suddenly an unexpectedly shared joke or to hide my face in embarrassment that a Certified Grownup™ has Cracked The 6-7 code." They get so used to adults flipping out over something minorly stupid like 6-7 that when you actively lean into it, it catches them off guard a bit.
Most of them resolve the dilemma with kind of a wry "you got me there" laugh, and then we move on to the next vocabulary slide.
i just know grace and rocky are going to be linguistically unsalvageable after so long alone together in such a high stress environment.
the mixing of eridian and human languages (and not just english! even english speakers use phrases from other languages! youre telling me grace isnt c'est la vie-ing this entire goddamn trip?) and idioms and body language and various non-word sounds is going to be a sonic nightmare for humans and eridians alike.
in the short window of the story they've already morphed several, now multiply that across the entire trip!
i keep thinking about rocky picking up gesticulation from grace, but on erid it's the equivalent of a human talking with a hand in front of their mouth.
and i shudder to imagine all the scientific short hands they come up with to speed up research! hey rocky thanks for solving the astrophage problem too bad no one knows what the fuck you're trying to explain to us!
When you get more objective about the way nature is & away from weird anthropomorphic carnivore slander it is a little funny when the baby sea turtles hatch and every other animal in like a ten mile radius goes "oh fuck yum one million mini sliders"
[ID: a screenshot of the original post's tags that read, "#real turtleheads know this is a play on sliders (turtle emoji) which are a type of terrestrial turtle. but it's also tiny hamburgers #which seems apt. #this has been another episode of Joke Explainer" /end ID]
Keeping these actually. Necessary footnote for this post
Watching a jaguar casually walk up to an adult sea turtle headed back to the water after laying her eggs and eat it like why am I in my sea turtle violence era and why are they like this. Adult sea turtles aren't especially easy to eat in the water they'll swim around with shark bites taken out of them like they're just chillin. And then on land they just kinda
Also today I learned that since the jaguar scares off other smaller predators it's actually good for the sea turtle population if there are jaguars there. Cause less babies get eaten by Everything Else. I'm gonna be honest I didn't even consider that a jaguar could be on the beach
I think it would be funny to write a murder mystery where not only did every single character involved have an obvious motive to kill this mf, they were actually all attempting to murder him first, but the murder attempts all cancelled each other out all except for one. Two people tried to poison him but the poisons just happen to work as antidotes for each other, and instead of killing him only gave him the shits, and due to having the shits he couldn't go hunting that day like he had planned, foiling the plans of the one who had conditioned his favourite hunting horse to panic and bolt at the cue of a whistle, and the other murder attempt of tampering with his gun so that it would have exploded his whole face off.
The whole mystery isn't about who could have done it or how, but who was the one who got lucky and actually succeeded.
When I was in high school a friend of mine would host murder mystery dinners once or twice a year. They were the kind you could buy as a kit -- I don't even know if they exist anymore -- and everyone was assigned (or chose) a character, then received a booklet of clues to share. The idea was to spend an evening in a one-shot LARP designed like an Agatha Christie novel.
I was a year above most of them at school so they threw a "goodbye" murder mystery for me just before graduation, and about 2/3 of the way through the game we all realized that everyone had at least attempted to kill the victim. The game then shifted from "whodunnit" to "who succeeded in dunninit" which we all felt was not only super fun but above the usual level of narrative complexity for those games.
After we solved it, we discovered that the game wasn't from a kit -- the host had written it herself and meticulously printed out the booklets in replica style of the kits. It was the best going-away party I think I could possibly have had.
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I was innocently buying a soda and a Kit Kat bar from a snack shop recently when the cashier said, "Oh, a Kit Kat! That's what I named my cat!" and then launched into An Monologue.
Nobody was behind me in line, which seemed to be a good reason for her to treat me to a five minute retelling of the identification, rescue, and argument over initial custody of Kit Kat, who was so small they thought when they first heard him crying for help that he was a bird and not a kitten in a tree, and is now fifteen pounds of "pure, sculpted lardass".
And I didn't mind, precisely, I wasn't bored or anything, but around the time she was bringing me up to speed on Kit Kat's current status it occurred to me that this woman is a cashier in a store that primarily sells candy bars and beverages. People must buy Kit Kat bars from her multiple times a day. Does she do this every time there's nobody in line behind the purchaser? Did I just have that I Own Several Cats And Will Enjoy Your Cat Stories look about me? Was it the first time it occurred to her that she sold the brand of candy bar she named her cat after? Was she new to the job of selling Kit Kat bars?
The idea that every time she sees a Kit Kat bar she is gripped by the urge, Manchurian Candidate style, to retell the story of Kit Kat the Cat, elevates her from a friendly cashier to a deep enigma. Truly there is no knowing the mind of another.
I was in the same snack shop, which I'm in, like, once a month, recently. I only recognized her because I spent five minutes listening to this monologue in sincere wonder. But I did recognize her, so as I was buying a soda and a Milky Way bar (this time) I said, without thinking about how this would come across, "Hey, how's Kit Kat?"
She looked genuinely horrified and said, "What...how?"
"Oh fuck!" I blurted. "Sorry! You told me about him last time!"
This is still quite cryptic as responses go but she gave me a frankly frantic look of sudden recognition and said, "He's fine! You bought a Kit Kat! I was unmedicated!"
I did not inform her she is small town famous on Tumblr and instead just said, "Glad you're both doing well!" and we parted as confused and mortified friends.
Gosh she's fun. I hope she's there next time. I want to reenact the Spiderman Pointing meme with her.
