Burt’s Bees

The inside joke manifested fairly early on in Darrel and Heather’s relationship.

It was a warm Sunday morning amidst a brutal winter, and the happy couple decided to spend it on the patio. During a lull in the conversation, Heather pulled out her Burt’s Bees lip balm and smeared some over her dried lips.

“Would you like some cha –”

A songbird, who up til then, was silently sheltering underneath the awning, decided it was its turn to hop into the conversation.

“CHAA peee CHAA peee CHAAA peee,” it trilled, before promptly flying into the woods.

“Was it just me, or did that bird really just say ‘chapy’ like in chapstick?” asked Darrel.

“I definitely heard it too,” Heather quickly replied.

And so it was born. For the remainder of the cold winter, the two never called it a “lip balm” or the corresponding proprietary eponym “ChapStick” anymore. Rather, whenever one of their lips were dry, they would yodel “chapy chapy” at each other followed by an inevitable giggling.

But Darrel and Heather didn’t last. She wanted someone who was more assertive, and he wanted someone who would open up more. Or so it went. When they finally split up two winters later, the silly little inside joke had nowhere to go, and so disappeared alongside the conjunction between their two names. Still, even now, whenever Darrel or Heather pull out their individual lip balm, their internal dialogue still yodels that silly little sound bite; their minds wander to dry winters long ago.

“Why do you always smile at your chapstick after putting it on?” their current partners would ask.

“Oh, it’s nothing.”

A Fibonacci Identity

From Putnam 1996:

Define a selfish set to be a set which has its own cardinality as an element. Find, with proof, the number of subsets of $\{1,2,\ldots,n\}$ which are minimal selfish sets, that is, selfish sets none of whose proper subsets is selfish.

First solution:

We can easily show that a selfish set of length $k$ cannot be minimally selfish if it contains any numbers smaller than $k$. We can conversely show that a selfish set of length $k$ containing only elements greater than or equal to the number $k$ is minimal. Thus we can iterate over the length of the subsets, and simply choose from the remaining numbers to fill the set

\sum_{k=0}^n \binom{n-k}{k-1}

Lastly, notice that $n-k \le k – 1 \implies n+1 \ge 2k$ which means the upper bound can be changed to the floor of $(n+1)/2$.

Second solution:

Let’s consider a inductive solution. Let $s_n$ be the number of minimal selfish set for the set $A_n = \{1,2,\ldots,n\}$. We first note that all minimal selfish set of $A_n$ are also minimal selfish sets in $A_{n+1}$; note that all minimal selfish sets of $A_n$ do not contain the element $n+1$. Thus, the question now is counting how many minimal selfish sets also contains $n+1$.

Actually computing the minimal selfish sets for a few small $n$s suggests that we should be looking for the Fibonacci numbers. Indeed, consider a minimal selfish set $k \subsetneq A_{n-1}$; if we add 1 to each number in $k$, and then append $n+1$, we see that it is also a selfish set, but is it minimal?

Assume that it is not minimal, that is, there exists a strict subset $f$ of $k +1 \cup \{n+1\}$ which is also selfish. The first observation is to note that if $f$ contains $n+1$, then taking $n+1$ out of $f$ then subtracting 1 contradicts the assumption that $k$ is a minimal selfish set. But if $f$ doesn’t contain $n+1$, then removing an arbitrary item and subtracting 1 will again contradict the assumption that $k$ is minimally selfish. We remark that $f$ can be of length of 1 since in the construction, we added 1.

Thus, $f_n = f_{n-1} + f_{n-2}$ and we have that the sum in the first solution is equal to the Fibonacci sequence.

A Reminder

After I got my Echo some four and a half years ago, I developed a bedtime routine of playing the “Flash Briefing.” What that entailed was Alexa would play the latest NPR hourly news update followed by the latest news update from WGBH (the Public’s Radio didn’t interface well with the device at the time; can we also discuss how Rhode Island’s public radio brand doesn’t start with the letter “W”? What is up with that?).

At the time, this ritual served two purposes: allowed me to keep up with local news (too bad I missed the news of my murdered landlord) and also the NPR newscaster all have amazingly soothing voices. Oh, how I missed those days when I would lightly chuckle at the events before drifting off to bed. When the news is light, there would be a shade of snark in the scripts. Now the news update is a constant, desolate wasteland.

In times like this, I have to remind myself a little quote from Mr. Rogers:

“My mother would say to me, ‘Look for the helpers. You will always find people who are helping.’ To this day, especially in times of disaster, I remember my mother’s words, and I am always comforted by realizing that there are still so many helpers — so many caring people in this world.”

It’s so easy, in the ugliness of this world, to not look for the beauty in things. It’s so easy to see the violence and havoc, rather than the unity and hope. It’s so easy to find the disrupters and not the helpers of the world. I want to choose to see more of the beauty in this world. After all, it’s the only one we have.


There are a few names repeated a lot these days; when they were alive, they probably did not expect to hear their names uttered by the masses. They were regular citizens of these United States with dreams, fear and ambitions. They probably wanted what we all desire: George Floyd moved to Minneapolis to work, Ahmaud Arbery just wanted to run outside, Breonna Taylor just wanted to live in her own home. They all wanted to live.

I hope this time it’s different.

Top Secret


  1. “My greatest strength is probably the ability to deal with worst case scenarios”
  2. “Don’t worry Sid, the portfolio is structured for long term growth. It’ll probably be only a short term correction.”
  3. “Let’s rebrand Pangaea… something like Pan-gone?”


Crossword Letter Frequencies

I scrapped NYT crossword puzzle answers and plotted the frequencies of the letters. It turns out that the actual data is a bit taboo, with many past projects having to remove them from the internet (see here or here). I used a site like this one, which have been aggregating answers for a few years.

The data consisted of a total of 85687 answers; some answers were pruned due to numerical answers and other oddities from the data source. The tools used were Python for wrangling and Seaborn for plotting.

It’s clear that there is an abundance of A, E and S in crosswords, while T and H in the English language is comparatively higher in frequency (use of the word “the” probably). I would love to have cleaner data and see the frequency changes along Monday through Sunday similar to this and this.



A List

One of those stream of consciousness lists.

– Intellectual curiosity: I like to think that I am reasonably intelligent with a natural inclination to learn. It is a great trait to have (mister self-aggrandizing here) , and I honestly find that it is far easier to hold and maintain a fun conversation with an inquisitive person. The banter can flow from one topic to another with little turbulence, and what is a relationship without nice back-and-froths?

The $64000 question is how does one find this quality in another person? There are no marked physical manifestation to my knowledge (I presume a tattoo of “Reading Rainbow” might count, though it has to be done tastefully). I reckon a  shallow answer is to look for degrees at elite institutions like Brown. From my time there, people either fall into one of two categories: the ones I seek, and the ones with multiple Canada Goose jackets.

– Reasonably active: one of my clarinet teacher from high school had a great remark on her Facebook back in the day: “doing my best to avoid morbid obesity.” This describes me to a fairly large degree as well. For awhile, I was walking a fine line between trying to bulk up and not looking like Thor in the first part of Endgame. Even my parents mentioned how I looked like I was getting a bit fat.

I guess the message here is that I stay active, and active people actively seek out other active people to do fun activities with.  For now, since I’ve been doing yoga basically everyday ever since this whole quarantine business started, I really want to try some of those “couple poses.” Maybe I’m just seeking a fellow yogi? And a good one at that; my down dog can use a lot of help.

– Can sport a pixie cut: it’s probably the case that pandemics will become more common in the future. It’s an unfortunate fact due to the density of humans paired with our propensity to breed lots of animals in grotesque and cramped spaces, and then subsequently eating them in various delicious ways. I still haven’t gave up on meat completely if you can’t tell. This will last until my BYND stocks have doubled or tripled.

Nevertheless, the pandemics will continue. Hence it’s probable that I would need to give a haircut to my partner in the future. While the pixie cut is not the easiest to do, someone self-confident enough sporting one would allow me to cut her hair. Maybe I just think of haircuts as a proxy for self-confidence?

– Friends with Daniel Radcliffe: I just really want to be friends with Daniel. There are so many questions I want to ask him. Like why Swiss Army Man? Did you  method act and visit a Victorinox factory? What is why and how this scene:

– Attended Hogwarts School of Wizardry: well not in the literal sense. I find that ever since undergrad, there’s been a lack of “magic” in my life. A sense of wonder that I … oh wait, I was just describing Emma Watson wasn’t I?

American Gods

Finishing a good work of fiction is always sad for me. It doesn’t matter if it’s an anime, a short story or, pertinent to this case, a novel. It was the case with Where the Crawdads Sing and certainly the case with the Dutch House. Unfortunately, American Gods didn’t elicit that sense of wonder or yearning when I read the last word. Don’t get me wrong, the book was fascinating, but there was no pang of longing as I left the world of the walking gods.

It was certainly an epic in every sense of the word: gods and men fighting, fantastical beings, themes of sacrifice and honor, and, since I was reading the special edition, some 600 pages of writing. The prose was solid, and at points, remarkably beautiful, but the most poignant passages were not about our protagonist Shadow, but rather describing tales of how gods wondered onto America. There was a fascinating short story about about how the first god arrived onto the new world through the Bering strait. Another interlude described the atrocities of the slave trade, and the subsequent transplanting of African traditions to Haiti and New Orleans.

I reckon it’s the measly amount of world building that Gaiman did that caused the disinterest. As someone who enjoys the superhero franchises, the premise is absolutely enticing for me. What powers would they have? Can they influence major world events? Is there going to be a tier list of gods somewhere on YouTube? But I didn’t as much as bat an eye at the feats of gods from the book. Most of the gods were more a plot vehicle, rather than a focus of development, and i would love to have read the opposite of that. Maybe I’ve just been spoiled by Stan Lee and the Russo brothers.

The book did make me want to start up coin magic, so there’s that.



There is a section in American Gods which detailed the early slave trade in a gruesome fashion. It follows a pair of twins as they were transported from their home, onto the slave ship and finally as they were separated in the New World. They never saw each other again, with the brother dying in the slave revolt of Haiti and the sister eventually finding a semi-fulfilling life (maybe?) in New Orleans.

I’m not sure why this particular passage hit me so deeply. Part of the reason might be my personal interactions with twins, and seeing how they can be so intertwined. Forcefully separating any family is a travesty, but destroying the bonds of fraternal or identical twins seems somehow worse. The chapter reminded me of Three Identical Strangers, and how even though the triplets were separated, they presented similar personality traits when the reunited years later.

Maybe it’s how Gaiman described the entire affair. While he did write about the physical tortures that characters had to suffer through, he also detailed all the new names that the character had assigned to them. This little detail really made me pause. I guess I’ve never really thought about how humanizing a name is, how an identity is so important to self worth.

If I write enough, hopefully I can better express how poignant passages touches me.

Silent Synergy Graph

Sort of inspired by the video by GMTK,  I spent a few hours last week making the following graph, showing synergy between the Silent cards. The arrow between card A and card B means that A makes B better.

Some of them are pretty obvious, but others, less so (example: Piercing Wail and Well-Laid Plans is quite good). There are also other cases where a card is “good” with anything: Burst is obviously “good” with any card, but particularly good with Catalyst and After Image can be procced by any card for example.

This definitely isn’t complete, and the Silent is my worst character in terms of heart kills so I would appreciate feedback.