Susam's Miscellaneous Pages

Five Minutes of Prime Time

2026-05-16T00:00:00Z

Let me share a very silly story from roughly 18 years ago! In 2008, I joined RSA, the network security company named after the initials of the inventors of the RSA algorithm, Rivest, Shamir and Adleman, who were also the founders of RSA, the company.

There was a bit of a nerd culture in the workplace where topics like prime numbers, combinatorics, probability theory, etc. were discussed fervently. A prime-number employee number was considered a lucky charm. I had a rather nice large five-digit prime number as my employee number, which I remember being quite pleased about. There were internal forums for almost all kinds of topics. A few I remember fondly were a mathematics forum where colleagues would challenge each other with mathematical puzzles and a similar physics forum where, for some reason, crafting contrived paradoxes using special and general relativity and putting them up for debate was a common activity. The participants would analyse each paradox to determine if it truly was one or if it could be resolved into something that was no longer a paradox. In fact, my Langford Pairing (2011) post was the result of a question I had stumbled upon in the mathematics forum.

The human resources (HR) department used to organise afternoon games once every month where people would self-organise into teams and solve a small challenge. There was a cash prize for the winning team each time. The HR folks were very well aware of the nerd culture and the fascination with prime numbers, but they probably did not know enough about exactly what type of problems we were fascinated with. So in one of the monthly game events, the HR team gave us this challenge.

Write as many prime numbers between 1 and 1000 as you can in 5 minutes.

The 5-minute timer started immediately. Really, that was the challenge. We all looked at each other in surprise, wondering if that really was the problem. The game was already on. No constraints were set. Were we allowed to look up a list on the web? Probably not, otherwise what would be the point of the game?

There were five or six teams with a total of about 30 or so participants. We began working on the challenge. The first thing I did was load up the list of all 168 prime numbers from 1 to 1000 on my mobile phone and begin copying them meticulously to our sheet of paper.

I would learn later that another team had followed the same approach too. The person writing their list decided to omit a few numbers so that it did not look like they had copied the numbers verbatim from an external source. I think they were being overcautious because there were no rules, and all is fair in love and listing prime numbers.

I completed writing our list with about half a minute to spare. We turned in our sheets after the timer ran out. The HR folks then evaluated the sheets. The winner was announced. Our team won with all 168 numbers written accurately. We won the cash prize. One of the HR folks asked me later what formula we had used to generate all the prime numbers so accurately. I told them that we had used the ancient formula of looking up a list of prime numbers compiled by someone else and writing it down. We spent the cash prize immediately and ordered pizza and soft drinks for everyone.

The monthly afternoon games were not usually this silly. The other games were more interesting, more challenging and better designed. But I don't remember what those games were at all now. It is only this silly game that remains etched in my memory.

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The Problem of Pedagogy in Advanced Mathematics

2026-05-11T00:00:00Z

It is a commonly held opinion that educational institutions could do more to improve the pedagogy of mathematics. This is especially applicable to primary and secondary schools, where students are first exposed to mathematics as a formal subject, along with other new subjects. Poor exposition can turn students away from mathematics for a lifetime. Only the highly motivated ones continue to engage with the subject. This is very unfortunate because mathematics is a beautiful subject and it is filled with wonder. It also teaches rigour in reasoning, clarity of thought and the discipline of constructing arguments from first principles to obtain intricate and often beautiful results.

What is perhaps less known is that pedagogy is a problem even for graduate-level mathematics students and professional mathematicians. The proofs in many graduate-level mathematics textbooks are, in my humble opinion, not really proofs at all. They are closer to high-level outlines of proofs. The authors simply do not show their work. The student then has to put in an extraordinary amount of effort to understand and justify each line. Sometimes a 10-line argument in a textbook might expand into a 10-page proof if the student really wants to convince themselves that the argument works.

I am not a mathematician, but out of personal interest, I have worked with professional mathematicians in the past to help refine notes that explain certain intermediate steps in textbooks (for example, Galois Theory by Stewart, in a specific case). I was surprised to find that it was not just me who found the intermediate steps of certain proofs obscure. Even professional mathematicians who had studied the subject for much of their lives found them obscure. It took us two days of working together to untangle a complicated argument and present it in a way that satisfied three properties: correctness, completeness and accessibility to a reasonably motivated student. There is a reason why jokes like 'proof by obviousness' and 'proof by intimidation' are so funny. They are funny because they are true, more true than one would like.

I don't mean that the books merely omit basic results from elementary topics like group theory or field theory, which students typically learn in their undergraduate courses. Even if we take all the elementary results from undergraduate courses for granted, the proofs presented in graduate-level textbooks are often nowhere near a complete explanation of why the arguments work. They are high-level outlines at best. I find this hugely problematic, especially because students often have to learn a topic under difficult deadlines. If the exposition does not include sufficient detail, some students might never learn exactly why the proof works, because not everyone has the time to work out a 10-page proof for every 10 lines in the book.

The situation is even more dire when it comes to research papers but that would be a topic for another discussion. I'll focus on books for now. I completely understand why an advanced textbook cannot provide a justification for every argument, because if a textbook tries to do so, then a 200 page book would turn into a 2000 page book. No student or teacher has the time or patience to read through thousands of pages of uninteresting 'technical' arguments. So the authors choose to focus on the interesting parts and expect the student to work out all the elisions. Even so, I find it painful to see just how many such elisions exist in a typical textbook and how big they can sometimes be.

Many good universities provide accompanying notes that expand the difficult arguments by giving rigorous proofs and adding commentary to aid intuition. I think that is a great practice. I have studied several graduate-level textbooks in the last few years. While these textbooks are a boon to the world because textbooks that expose the subject are better than no textbooks at all, I am also disappointed by how inaccessible such material often is. If I had unlimited time, I would write accompaniments to those textbooks that provide a detailed exposition of all the arguments. But of course, I don't have unlimited time. Even so, I am thinking of at least making a start by writing accompaniment notes for some topics whose exposition quality I feel strongly about, such as s-arc transitivity of graphs, field extensions and related topics.

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Touch Typing Number Keys

2026-05-01T00:00:00Z

I learnt touch typing about two decades ago when I was still at university. Although I took some typewriter lessons as a child, those lessons did not stick with me. It was at university, when I found a Java applet-based touch typing tutor on the web, that I really learnt to touch type. Since then, touch typing has been an important part of my computing life. I've sometimes read arguments on the web downplaying touch typing as a skill, with claims like 'typing isn't the bottleneck, thinking is'. While that may be true, I still consider touch typing a useful skill, since it makes writing documents, code and email feel much more fluid and pleasant. It's like playing a musical instrument with the correct technique, rather than simply getting by without it. One feels smooth and expressive and the other feels raw and laboured.

Later in life, I also wrote a tool named QuickQWERTY so that I could share the joy of touch typing with my friends. The tool teaches typing only with the QWERTY layout. I wrote it at a time when I had a rather limited view of the computing world, so I was not even aware that other keyboard layouts existed. As a result, only QWERTY is supported. The tool is free and open source, so motivated individuals can modify the lessons to support other keyboard layouts. Some people have indeed done so over the years. Several of my friends used this tool. I know at least a few who benefitted from it and shared similar sentiments about how touch typing made their computing experience smoother.

Back in my university days, I had learnt a method in which the digits 1 and 2 are typed with the left little finger, 3 with the left ring finger and so on. In this approach, the digits 1 to 6 are typed with the left hand and 7 to 0 with the right.

There is an alternative method in which only 1 is typed with the left little finger, 2 with the left ring finger and so on. In this approach, the digits 1 to 5 are typed with the left hand and 6 to 0 with the right.

Both methods require typing 1 with the left little finger. I have often felt that this may not be the most efficient way to type 1. The little finger is shorter than the others and reaching 1 often requires shifting the whole hand slightly diagonally upwards. I have therefore felt that using the left ring finger for 1 might be more comfortable.

Last month, I trained myself to use the left ring finger to type both 1 and 2. This goes against almost every typing guide out there, but I decided to forgo established practices and explore on my own to find what feels right. At first, I was sceptical about whether I would be able to learn this method, since it meant overcoming 20 years of muscle memory that I have relied on almost every day. However, developing the new muscle memory has been surprisingly easy.

In fact, both the old and the new muscle memories now coexist and I can switch between them at will without much trouble. It is remarkable how the brain can store conflicting muscle memories so effortlessly. So far, I am finding this new way of typing 1 and 2 more comfortable than either of the two popular methods I described above. I will continue typing this way for the rest of this month and see how it feels.

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Nerd Quiz #4

2026-02-22T00:00:00Z

Nerd Quiz #4 is the fourth instalment of Nerd Quiz, a single page HTML application that challenges you to measure your inner geek with a brief quiz. Each question in the quiz comes from everyday moments of reading, writing, thinking, learning and exploring.

This release introduces five new questions drawn from a range of topics, including computing history, graph theory and Unix. Visit Nerd Quiz to try the quiz.

A community discussion page is available here. You are very welcome to share your score or discuss the questions there.

Read on website | #web | #miscellaneous | #game

Deep Blue: Chess vs Programming

2026-02-15T00:00:00Z

I remember how dismayed Kasparov was after losing the 1997 match to IBM's Deep Blue, although his views on Deep Blue became more balanced with time and he accepted that we had entered a new era in which computers would outperform grandmasters at chess.

Still, chess players can take comfort in the fact that chess is still played between humans. Players make their name and fame by beating other humans because playing against computers is no longer interesting as a competition.

Many software developers would like to have similar comfort. But that comfort is harder to find, because unlike chess, building prototypes or PoCs is not seen as a sport or art form. It is mostly seen as a utility. So while brain-coding a PoC may still be intellectually satisfying for the programmer, to most other people it only matters that the thing works. That means that programmers do not automatically get the same protected space that chess players have, where the human activity itself remains valued even after machines become stronger. The activity programmers enjoy may continue but the recognition and economic value attached to it may shrink.

So I think the big adjustment software developers have to make is this: The craft will still exist and we will still enjoy doing it but the credit and value will increasingly go to those who define problems well, connect systems, make good product decisions and make technology useful in messy real-world situations. It has already been this way for a while and will only become more so as time goes by.

This note reproduces a recent comment I posted in a Lobsters forum thread about LLM-assisted software development at at lobste.rs/s/qmjejh.

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Three Inverse Laws of AI

2026-01-12T00:00:00Z

Introduction

Since the launch of ChatGPT in November 2022, generative artificial intelligence (AI) chatbot services have become increasingly sophisticated and popular. These systems are now embedded in search engines, software development tools as well as office software. For many people, they have quickly become part of everyday computing.

These services have turned out to be quite useful, especially for exploring unfamiliar topics and as a general productivity aid. However, I also think that the way these services are advertised and consumed can pose a danger to society, especially if we get into the habit of trusting their output without further scrutiny.

Introduction
Pitfalls
Inverse Laws of Robotics
Conclusion

Pitfalls

Certain design choices in modern AI systems can encourage uncritical acceptance of their output. For example, many popular search engines are already highlighting answers generated by AI at the very top of the page. When this happens, it is easy to stop scrolling, accept the generated answer and move on. Over time, this could inadvertently train users to treat AI as the default authority rather than as a starting point for further investigation. I wish that each such generative AI service came with a brief but conspicuous warning explaining that these systems can sometimes produce output that is factually incorrect, misleading or incomplete. Such warnings should highlight that habitually trusting AI output can be dangerous. In my experience, even when such warnings exist, they tend to be minimal and visually deemphasised.

In the world of science fiction, there are the Three Laws of Robotics devised by Isaac Asimov, which recur throughout his work. These laws were designed to constrain the behaviour of robots in order to keep humans safe. As far as I know, Asimov never formulated any equivalent laws governing how humans should interact with robots. I think we now need something to that effect to keep ourselves safe. I will call them the Inverse Laws of Robotics. These apply to any situation that requires us humans to interact with a robot, where the term 'robot' refers to any machine, computer program, software service or AI system that is capable of performing complex tasks automatically. I use the term 'inverse' here not in the sense of logical negation but to indicate that these laws apply to humans rather than to robots.

It is well known that Asimov's laws were flawed. Indeed, Asimov used those flaws to great effect as a source of tension. But the particular ways in which they fail for fictional robots do not necessarily carry over to these inverse laws for humans. Asimov's laws try to constrain the behaviour of autonomous robots. However, these inverse laws are meant to guide the judgement and conduct of humans. Still, one thing we can learn from Asimov's stories is that no finite set of laws can ever be foolproof for the complex issues we face with AI and robotics. But that does not mean we should not even try. There will always be edge cases where judgement is required. A non-exhaustive set of principles can still be useful if it helps us think more clearly about the risks involved.

Inverse Laws of Robotics

Here are the three inverse laws of robotics:

Humans must not anthropomorphise AI systems.
Humans must not blindly trust the output of AI systems.
Humans must remain fully responsible and accountable for consequences arising from the use of AI systems.

Non-Anthropomorphism

Humans must not anthropomorphise AI systems. That is, humans must not attribute emotions, intentions or moral agency to them. Anthropomorphism distorts judgement. In extreme cases, anthropomorphising can lead to emotional dependence.

Modern chatbot systems often sound conversational and empathetic. They use polite phrasing and conversational patterns that closely resemble human interaction. While this makes them easier and more pleasant to use, it also makes it easier to forget what they actually are: large statistical models producing plausible text based on patterns in data.

I think vendors of AI based chatbot services could do a better job here. In many cases, the systems are deliberately tuned to feel more human rather than more mechanical. I would argue that the opposite approach would be healthier in the long term. A slightly more robotic tone would reduce the likelihood that users mistake fluent language for understanding, judgement or intent.

Whether or not vendors make such changes, it still serves us well, I think, to avoid this pitfall ourselves. We should actively resist the habit of treating AI systems as social actors or moral agents. Doing so preserves clear thinking about their capabilities and limitations.

Non-Deference

Humans must not blindly trust the output of AI systems. AI-generated content must not be treated as authoritative without independent verification appropriate to its context.

This principle is not unique to AI. In most areas of life, we should not accept information uncritically. In practice, of course, this is not always feasible. Not everyone is an expert in medicine or law, so we often rely on trusted institutions and public health authorities for guidance. However, the guidance published by such institutions is in most cases peer reviewed by experts in their fields. On the other hand, when we receive an answer to a question from an AI chatbot in a private chat session, there has been no peer review of the particular stochastically generated response presented to us. Therefore, the onus of critically examining the response falls on us.

Although AI systems today have become quite impressive at certain tasks, they are still known to produce output that would be a mistake to rely on. Even if AI systems improve to the point of producing reliable output with a high degree of likelihood, due to their inherent stochastic nature, there would still be a small likelihood of producing output that contains errors. This makes them particularly dangerous when used in contexts where errors are subtle but costly. The more serious the potential consequences, the higher the burden of verification should be.

In some applications such as formulating mathematical proofs or developing software, we can add an automated verification layer in the form of proof checker or unit tests to verify the output of AI. In other cases, we must independently verify the output ourselves.

Non-Abdication of Responsibility

Humans must remain fully responsible for decisions involving AI and accountable for the consequences arising from its use. If a negative outcome occurs as a result of following AI-generated advice or decisions, it is not sufficient to say, 'the AI told us to do it'. AI systems do not choose goals, deploy themselves or bear the costs of failure. Humans and organisations do. An AI system is a tool and like any other tool, responsibility for its use rests with the people who decide to rely on it.

This is easier said than done, though. It gets especially tricky in real-time applications like self-driving cars, where a human does not have the opportunity to sufficiently review the decisions taken by the AI system before it acts. Requiring a human driver to remain constantly vigilant does not solve the problem that the AI system often acts in less time than it takes a human to intervene. Despite this rather serious limitation, we must acknowledge that if an AI system fails in such applications, the responsibility for investigating the failure and adding additional guardrails should still fall on the humans responsible for the design of the system.

In all other cases, where there is no physical constraint that prevents a human from reviewing the AI output before it is acted upon, any negative consequence arising from the use of AI must fall entirely on the human decision-maker. As a general principle, we should never accept 'the AI told us so' as an acceptable excuse for harmful outcomes. Yes, the AI may have produced the recommendation but a human decided to follow it, so that human must be held accountable. This is absolutely critical to preventing the indiscriminate use of AI in situations where irresponsible use can cause significant harm.

Conclusion

The three laws outlined above are based on usage patterns I have seen that I feel are detrimental to society. I am hoping that with these three simple laws, we can encourage our fellow humans to pause and reflect on how they interact with modern AI systems, to resist habits that weaken judgement or blur responsibility and to remain mindful that AI is a tool we choose to use, not an authority we defer to.

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Circular Recursive Negating Acronyms

2026-01-05T00:00:00Z

One of my favourite acronyms from the world of computing and technology is XINU. It stands for 'XINU Is Not Unix'. The delightful thing about this acronym is that XINU is also UNIX spelled backwards.

For a given word W, a recursive acronym that both negates W and reverses it is possible when W has the form W = '?NI?' where each '?' denotes a letter. Some fictitious examples make this clearer:

LINA Is Not ANIL.
TINK Is Not KNIT.
OINO Is Not ONIO.

Words of the form '?N?' also work if we are happy to contract the word 'is' in the acronym. In fact, in this case we can even obtain circular recursive acronyms:

ANI's Not INA. INA's Not ANI.
ONE's Not ENO. ENO's Not ONE.

In each pair, the two acronyms negate each other, making them circular while also being reverses of one another. Such acronyms could serve as amusing names for expressing friendly banter between rival projects.

Further, if we make the acronyms refer to themselves, we get paradoxes too:

ANA's Not ANA
XNX's Not XNX

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Nerd Quiz #3

2025-12-23T00:00:00Z

Nerd Quiz #3 is the third release of Nerd Quiz, a single-page HTML application that invites you to test your nerd level through a short quiz. Each question is inspired by everyday moments of reading, writing, thinking, learning and exploring.

This release introduces five new questions drawn from a range of topics, including computing history, graph theory and Unix. Visit Nerd Quiz to try the quiz.

A community discussion page is available at Discuss Nerd Quiz. You are very welcome to share your score or discuss the questions there.

Read on website | #web | #miscellaneous | #game

Nerd Quiz #2

2025-11-18T00:00:00Z

Nerd Quiz #2 is the second release of Nerd Quiz, a single-page HTML application that invites you to gauge your nerd level through short, focused quiz sets. Each question is carefully crafted from everyday moments of reading, writing, thinking, learning and exploring.

This release introduces five new questions drawn from a range of topics such as chess, graph theory and linguistics. To try the quiz, visit nq.html.

Read on website | #web | #miscellaneous | #game

Nerd Quiz #1

2025-10-16T00:00:00Z

Nerd Quiz #1 is the first release of Nerd Quiz, a single-page HTML application that lets you test your nerd level through short, focused quiz sets. Each question in the quiz is handcrafted from everyday moments of reading, writing, thinking, learning and exploring.

To try the quiz now, visit nq.html.

The source code of Nerd Quiz is available at github.com/susam/nq under the terms of the MIT licence.

Read on website | #web | #miscellaneous | #game

Million Times Million

2025-07-03T00:00:00Z

Is a million times a million a billion or is it a trillion? For my entire childhood, it was a billion, only for me to grow up and realise, as an adult, that it had better be a trillion!

Growing Up With the Long Scale

As a child, I stumbled upon an old dictionary lying around our house and that was where I discovered the names of large numbers. The dictionary used the long scale system, which is based on powers of a million. According to the long scale system,

a million times a million is a billion,
a million times a million times a million is a trillion,
a million multiplied by itself four times is a quadrillion,
multiply it five times and we get a quintillion
and so on.

These long scale names made a lot of sense to me. Each name told us exactly how many times we multiplied a million by itself. The little prefix at the start, like bi-, tri-, quadri-, quinti-, etc. matched the number of times a million appeared in the multiplication: twice for a billion, three times for a trillion and so on. If I came across a number like quintillion, I immediately knew it meant a million multiplied by itself five times, so a quintillion had to be \( (10^6)^5 = 10^{30}. \) It was simple and intuitive.

Now imagine my disappointment when I left home for university, got access to computers and the World Wide Web and discovered that the names I had learnt were off by several orders of magnitude compared with what everyone else was using. The long scale names I had grown up with were irrelevant in this new world I was stepping into. It was the short scale that had taken over most of the technology world, especially in computing. In the short scale, a million times a million is no longer a billion. Instead, somehow, it is a trillion! Similarly, a million multiplied three times is no longer a trillion. It is a quintillion! What on earth was going on?

Naturally, I had to go through a period of adjustment, letting go of my old intuition for the long scale names and getting used to the short scale ones. At the same time, I found a way to trick my mind into making sense of the short scale. Before I get into the mental gymnastics that made the short scale click for me, let us first take a quick look at the long scale and short scale names side by side.

Same Names, Different Numbers

The long scale and short scale are simply two different ways of naming large numbers. To see how they differ, here is a table showing what each name corresponds to in the two systems.

Number	Long Scale	Short Scale
\( 10^6 \)	Million	Million
\( 10^9 \)	Milliard	Billion
\( 10^{12} \)	Billion	Trillion
\( 10^{15} \)	Billiard	Quadrillion
\( 10^{18} \)	Trillion	Quintillion
\( 10^{21} \)	Trilliard	Sextillion
\( 10^{24} \)	Quadrillion	Septillion
\( 10^{27} \)	Quadrilliard	Octillion
\( 10^{30} \)	Quintillion	Nonillion
\( 10^{33} \)	Quintilliard	Decillion

Number

Long Scale

Short Scale

\( 10^6 \)

Million

\( 10^9 \)

Milliard

Billion

\( 10^{12} \)

Billion

Trillion

\( 10^{15} \)

Billiard

Quadrillion

\( 10^{18} \)

Trillion

Quintillion

\( 10^{21} \)

Trilliard

Sextillion

\( 10^{24} \)

Quadrillion

Septillion

\( 10^{27} \)

Quadrilliard

Octillion

\( 10^{30} \)

Quintillion

Nonillion

\( 10^{33} \)

Quintilliard

Decillion

As the table shows, the two systems agree on the value of a million, but diverge immediately after. From that point on, the same names represent increasingly different magnitudes.

Making Sense of the Short Scale

I used to quite like the long scale names because the prefix in each name is determined by how many times we multiply a million by itself. Each prefix reflects the value of \( n \) in the number \( 10^{6n}, \) where \( n \) is a positive integer. For example, if \( n = 3, \) we get \( 10^{18}, \) which in the long scale is called a trillion. But in the short scale, the same number is called a quintillion. How can we make sense of it?

One way to interpret the short scale is to count how many additional thousands are multiplied onto one thousand. A thousand is, of course, just one thousand. Multiply it by another thousand and we get a million. Multiply that by another thousand and we get a billion. One more multiplication gives us a trillion and so on. I do not find this quite as neat as the long scale but it works.

In effect, the prefix in the short scale corresponds to \( n \) of the integer \( 1000 \times 1000^n, \) where \( n \) is a positive integer. Stated more succinctly, it represents \( n \) of \( 10^{3(n + 1)}. \) So for example, if \( n = 5, \) then \[ 10^{3(n + 1)} = 10^{18}, \] which is called a quintillion in the short scale. The quinti- prefix still represents five but now it is tied to the number of additional thousands we multiplied to the first thousand. It is not quite the elegant pattern I grew up with but this is the convention that dominates today. In the world of technology, computing and finance, as well as in most English-speaking contexts, the short scale has prevailed.

Conclusion

Today, of course, I use only the short scale, because that is the convention adopted by the part of the world relevant to me. I no longer use the long scale. But it remains in memory, simply because I spent a large part of my early life with it. It still lives in some corner of my heart, partly out of nostalgia and partly due to its elegance.

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ASCII PC

2025-04-15T00:00:00Z

An ASCII art depiction of a PC resting on a computer case, with a classic Model M keyboard placed in front.

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Associations

2024-12-07T00:00:00Z

A record of some associations between various numbers, sounds and colours that occur in my mind.

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Equal Length Words

2024-10-14T00:00:00Z

I have an unusual hobby. I collect related words of equal length. This page lists my collection of related words that are all the same length and connected to one another in some way. This particular fascination with related words of equal length originated as a result of computer programming. While developing software, I often need to name variables, functions and other elements in a way that reflect the concepts I am working with. Occasionally, I end up choosing names for related elements that are the same length, which becomes visually striking when these names align neatly in the text editor.

Each group of words is presented in monospaced font so that the equal length of the words in each group is visually evident. Blank spaces separate the groups.

allow
block

begin
pause
start

beverage
doughnut

black
white

bottom
middle
prefix
suffix

broadcast
syndicate

cancel
resume

client
server

correct
mistake

create
delete
remove
update

discovery
traversal
wandering

error
issue
right
wrong

fail
okay
pass

failure
success

head
tail

index
value

inner
outer

input
radio
range
reset

look
peek

password
username

permit
forbid

pull
push

Note that some of the word groups here have been contributed by visitors to this page. Please visit the comments section to see the original contributions.

In case you are wondering why I would need words like beverage or doughnut in computer programming, I use them in my little game called Nerd Quiz to display a doughnut or beverage symbol (view source) depending on the points scored.

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Thurston's Paean

2023-07-18T00:00:00Z

I recently came across a beautiful and thoughtful answer on MathOverflow by the late mathematician William Thurston. A brief background about him from the Wikipedia article about him:

William Paul Thurston (October 30, 1946 – August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.

Thurston was a professor of mathematics at Princeton University, University of California at Davis and Cornell University. He was also a director of the Mathematical Sciences Research Institute.

MathOverflow makes all answers posted to the website available under a Creative Commons licence. In particular, all answers posted before 08 Apr 2011 (UTC) are available under the terms of the Creative Commons Attribution-ShareAlike 2.5 Generic (CC BY-SA 2.5) licence. Thurston wrote the answer I am about to share on 30 Oct 2010. Due to the licence terms, this post too is available under the terms of the same licence.

Thurston posted his answer while replying to a MathOverflow question: What's a mathematician to do?. The question enquires about how an ordinary mathematician can contribute to mathematics. Thurston's answer from mathoverflow.net/a/44213 is reproduced below:

It's not mathematics that you need to contribute to. It's deeper than that: how might you contribute to humanity, and even deeper, to the well-being of the world, by pursuing mathematics? Such a question is not possible to answer in a purely intellectual way, because the effects of our actions go far beyond our understanding. We are deeply social and deeply instinctual animals, so much that our well-being depends on many things we do that are hard to explain in an intellectual way. That is why you do well to follow your heart and your passion. Bare reason is likely to lead you astray. None of us are smart and wise enough to figure it out intellectually.

The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding.

The world does not suffer from an oversupply of clarity and understanding (to put it mildly). How and whether specific mathematics might lead to improving the world (whatever that means) is usually impossible to tease out, but mathematics collectively is extremely important.

I think of mathematics as having a large component of psychology, because of its strong dependence on human minds. Dehumanized mathematics would be more like computer code, which is very different. Mathematical ideas, even simple ideas, are often hard to transplant from mind to mind. There are many ideas in mathematics that may be hard to get, but are easy once you get them. Because of this, mathematical understanding does not expand in a monotone direction. Our understanding frequently deteriorates as well. There are several obvious mechanisms of decay. The experts in a subject retire and die, or simply move on to other subjects and forget. Mathematics is commonly explained and recorded in symbolic and concrete forms that are easy to communicate, rather than in conceptual forms that are easy to understand once communicated. Translation in the direction conceptual -> concrete and symbolic is much easier than translation in the reverse direction, and symbolic forms often replaces the conceptual forms of understanding. And mathematical conventions and taken-for-granted knowledge change, so older texts may become hard to understand.

In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary. Revolutionary change does matter, but revolutions are few, and they are not self-sustaining --- they depend very heavily on the community of mathematicians.

In the comments to the answer, one of the commenters was Suresh Venkatasubramanian who was a professor in the School of Computing at the University of Utah back then. He is now a professor of Computer Science and Data Science at Brown University. In his comment, Suresh proposed that this answer be called Thurston's Paean. Here is his complete comment:

This seems like an ideal counterpoint to Hardy's Lament. I'm calling it Thurston's Paean :). Seems poignant now that he has passed.

Thurston's answer does appear to be a perfect complement to Hardy's lament in the 1940 essay A Mathematician's Apology. While Hardy's lament is remarkably beautiful and introspective, it may also feel a little depressing at places. Thurston's post on the other hand is full of hope and purpose that goes beyond the actual work of doing mathematics. Indeed Thurston's Paean is a befitting title for his answer.

Read on website | #mathematics | #miscellaneous | #quote

No Beam From the Eye

2021-10-30T00:00:00Z

Light Through the Ages: Ancient Greece to Maxwell is a very fascinating article written by J J O'Connor and E F Robertson in 2002 that takes us through a journey of how our understanding of light has evolved over the last few millennia. Here is an excerpt from the article:

The biggest breakthrough in ancient times was made by al-Haytham around 1000 AD. He argued that sight is due only to light entering the eye from an outside source and there is no beam from the eye itself.

The article then goes on to say:

He gave a number of arguments to support this claim, the most persuasive being the camera obscura, or pinhole camera. Here light passes through a pinhole shining on a screen where an inverted image is observed. Anyone visiting Edinburgh in Scotland should go to see the camera obscura there near the top of the Royal Mile and marvel at just how effective the camera obscura is in this enjoyable tourist attraction.

By the way, there is a camera obscura in the London too at Royal Observatory, Greenwich which offers pretty good view of the surroundings that includes Greenwich Park, the University of Greenwich, the River Thames, etc. Here is a picture I took the last time I was there: Camera Obscura, Royal Observatory, Greenwich.

Read on website | #miscellaneous

Dark Night Skies

2020-12-17T00:00:00Z

Having grown up in a small town in the 1990s, I was fortunate to experience pretty dark skies at night. The combination of a clear night sky and a few books on amateur astronomy from our school's library fuelled my desire to understand the universe and our place in it.

I made photocopies of two star charts (one for each hemisphere) from a book I found in our school library and began using the northern hemisphere chart to study the night sky. The winter sky was and still is my favourite. Spotting the easily recognisable Orion constellation and knowing that its second-brightest star, with its distinctly reddish hue, is a red supergiant that could explode at any moment was quite fascinating.

The 1990s were also a great time to grow up as an astronomy enthusiast. Two bright comets graced our skies in 1996 and 1997. First, Comet Hyakutake appeared in March 1996, followed by Comet Hale-Bopp, which became spectacularly bright in April 1997.

Now, 23 years later, as an adult living in a severely light-polluted city, the recent Geminids meteor shower was rather disappointing. After midnight, my wife and I stepped outside to watch it. Under a truly dark sky, we could have seen two to four meteors streaking across the sky every minute. But instead, the entire sky glowed orange and we managed to spot only one or two meteors every five to ten minutes.

I think it's very unfortunate that children growing up in cities today don't have the opportunity to experience the beauty of the night sky the way I did during my childhood. A clear, dark night sky can be a powerful source of inspiration and wonder for young minds.

Read on website | #miscellaneous

Microcentury

2020-07-17T00:00:00Z

Optimal Lecture Time

I recently found this interesting paragraph from an article titled Ten Lessons I Wish I Had Been Taught that is based on a talk presented by Gian-Carlo Rota in Apr 1996:

Running overtime is the one unforgivable error a lecturer can make. After fifty minutes (one microcentury as von Neumann used to say) everybody's attention will turn elsewhere even if we are trying to prove the Riemann hypothesis. One minute overtime can destroy the best of lectures.

That's fine advice. In fact, the whole article is full of good advice like this. Although it was written primarily for mathematicians, a lot of what is said in the article applies quite well to professionals in other fields too.

The excerpt I have quoted above got me thinking about exactly how long a microcentury is. It couldn't be exactly 50 minutes, could it?

Wiktionary on Microcentury

The English Wiktionary entry for microcentury (as of revision 59316064 on 7 May 2020) mentions:

A time period of a millionth of a century, equal to 52 minutes and 34 seconds.

Not a standard unit of measurement and used mostly humorously to denote the maximum length of a lecture.

This looks incorrect to me. This is based on the oversimplified assumption that a century contains 36500 days, that is, it assumes that a century is a span of 100 years where each year has exactly 365 days. If a century were to have exactly 36500 days, then indeed it would have 3 153 600 000 seconds and one millionth of it would be 3153.6 seconds which is equivalent to 52 minutes 33.6 seconds. This looks consistent with the Wiktionary entry. However, an actual century on the calendar does not have exactly 36500 days. Some years are leap years, so the actual number of days in a century is more than that.

Assumptions

Let us find out how long a microcentury is as accurately as possible. We will count the leap years. We will ignore leap seconds because they are irregularly spaced and unpredictable. We will also ignore the following gap between 2 Sep 1752 and 14 Sep 1752 when the British Empire switched from the Julian calendar to the Gregorian calendar:

$ cal 9 1752
   September 1752
Su Mo Tu We Th Fr Sa
       1  2 14 15 16
17 18 19 20 21 22 23
24 25 26 27 28 29 30

The above output can be obtained by running the cal command as shown above on a Unix or Linux system. Ignoring this gap is equivalent to assuming that we are working with the Gregorian calender since the year 1 AD.

We will call a year that is a multiple of 100 to be a centurial year. Further, we will not debate whether a centurial year begins a new century or ends one, that is, we don't care whether the current century runs from 2001 to 2100 or if it runs from 2000 to 2099. The computation presented in the next section works equally well for any span of 100 years.

Computation

Any span of 100 years contains exactly one centurial year, that is, a year that is a multiple of 100. A centurial year is a leap year if and only if it is also a multiple of 400. Apart from the centurial year, a century contains 24 occurrences of years that are multiples of 4 and these are all leap years. From these facts, we can conclude that a span of 100 years contains:

Exactly 25 leap years if the centurial year within the span is a multiple of 400.
Exactly 24 leap years, otherwise.

Therefore a century has either 36524 days or 36525 days. In other words, a century has either 3 155 673 600 seconds or 3 155 760 000 seconds. Here is a quick demonstration of this with a simple Python program:

#!/usr/bin/env python3

import datetime

for year in range(1, 2400, 100):
    delta = datetime.date(year + 100, 1, 1) - datetime.date(year, 1, 1)
    print('{:04}-{:04}: {} d = {} s'
          .format(year, year + 99, delta.days, delta.total_seconds()))

Here is the output of this program:

0001-0100: 36524 d = 3155673600.0 s
0101-0200: 36524 d = 3155673600.0 s
0201-0300: 36524 d = 3155673600.0 s
0301-0400: 36525 d = 3155760000.0 s
0401-0500: 36524 d = 3155673600.0 s
0501-0600: 36524 d = 3155673600.0 s
0601-0700: 36524 d = 3155673600.0 s
0701-0800: 36525 d = 3155760000.0 s
0801-0900: 36524 d = 3155673600.0 s
0901-1000: 36524 d = 3155673600.0 s
1001-1100: 36524 d = 3155673600.0 s
1101-1200: 36525 d = 3155760000.0 s
1201-1300: 36524 d = 3155673600.0 s
1301-1400: 36524 d = 3155673600.0 s
1401-1500: 36524 d = 3155673600.0 s
1501-1600: 36525 d = 3155760000.0 s
1601-1700: 36524 d = 3155673600.0 s
1701-1800: 36524 d = 3155673600.0 s
1801-1900: 36524 d = 3155673600.0 s
1901-2000: 36525 d = 3155760000.0 s
2001-2100: 36524 d = 3155673600.0 s
2101-2200: 36524 d = 3155673600.0 s
2201-2300: 36524 d = 3155673600.0 s
2301-2400: 36525 d = 3155760000.0 s

Thus one millionth of a century has 3155.6736 or 3155.7600 seconds, that is 52 minutes 35.6736 seconds or 52 minutes 35.7600 seconds.

Conclusion

If we round off the number of seconds in a microcentury to one decimal place, we can say that a microcentury has 52 minutes 35.7 seconds.

Read on website | #miscellaneous | #quote

Global Palindrome Day

2020-02-02T00:00:00Z

Date Formats

There are three popular date formats followed worldwide: big-endian, little-endian and middle-endian. Here is how today's date looks like in these formats if the year is written out in full:

Endianness	Format	Date
Big-endian	YYYY-MM-DD	2020-02-02
Little-endian	DD-MM-YYYY	02-02-2020
Middle-endian	MM-DD-YYYY	02-02-2020

Endianness

Format

Date

Big-endian

YYYY-MM-DD

2020-02-02

Little-endian

DD-MM-YYYY

02-02-2020

Middle-endian

MM-DD-YYYY

02-02-2020

The ISO 8601 standard specifies the big-endian date format. The big-endian format is popular in Canada, China, Japan, Lithuania, etc. The little-endian format is used by the majority of the world including countries like Brazil, Egypt, Germany, India, Italy, United Kingdom, etc. The middle-endian date format is primarily used in the Philippines and the United States.

The preferred choice of delimiter between the date components (day, month and year) varies from country to country. In this post, we use hyphen (i.e. "-") as the delimiter.

Palindromes

Today's date is a palindrome in all three date formats, i.e. it reads the same backward as forward in all three formats. It is globally palindromic. Note that we ignore the hyphens or any other delimiters while deciding if a date is palindrome or not. In fact, in general, punctuation, capitalisation and spaces are usually ignored while deciding if a given text is palindrome or not.

Let me define global palindrome day to be a day when its date is a palindrome in all three date formats (YYYY-MM-DD, DD-MM-YYYY and MM-DD-YYYY). Then today is a global palindrome day!

There are only eleven such global palindrome days between the years 1000 and 9999. Here is a list that enumerates all of them:

1010-01-01
1111-11-11
2020-02-02
2121-12-12
3030-03-03
4040-04-04
5050-05-05
6060-06-06
7070-07-07
8080-08-08
9090-09-09

The previous global palindrome day was over 908 years ago! The next such day is over 101 years away!

Today is special in another way. It is day 33 (a palindrome) of the current year and 333 (also a palindrome) days remain in the year.

Ambigrams

As a bonus, today is also a global strobogrammatic day when viewed on a seven-segment display, i.e. for all three date formats, today's date when shown, without the delimiters, on a seven-segment display appears the same when rotated by 180°. A strobogrammatic sequence of digits like this is also a type of ambigram. Here is how today's date looks in YYYYMMDD, DDMMYYYY and MMDDYYYY formats on a seven-segment display:

In the list of global palindrome days provided in the previous section, six of the eleven days are global strobogrammatic days too. Only the ones that belong to the years 3030, 4040, 6060, 7070 and 9090 are not strobogrammatic on a seven-segment display; the rest all are.

Read on website | #miscellaneous

Good Blessings

2016-12-11T00:00:00Z

Back in 2009, I had completed a year working for RSA Security. I was then looking for more intellectually stimulating work where I could work on projects involving mathematics and algorithms. While I had talked to senior leaders of the organisation about my aspirations, nothing concrete had materialised for several weeks. Then something interesting happened. One fine Monday morning, as I walked into the office and began to settle at my desk, the head of our research and development division came to my desk looking for me. He said, "Burt is here. He wants to talk to you."

There was no Burt in that office as far as I knew, so I became a little confused. I was led into a conference room where I could meet him. Mildly disoriented, I opened the conference room door. Then I saw Burt, the person I was about to meet. As soon as I saw him, I immediately recognised who he was. He was Dr Burt Kaliski, the Chief Scientist of RSA Laboratories. He had flown in that morning to meet certain teams in our office. During his visit, he had carved out some time from his schedule to meet me.

I had never met or talked to Dr Kaliski before. But I was very well aware of his work and his accomplishments many years before joining RSA. At a previous job, while implementing digital signatures for a banking product, I had spent a lot of time reading the RFCs for PKCS #1, PKCS #5 and PKCS #7, which were all authored by Dr Kaliski. It was an honour to meet the person whose work had taught me a lot and helped me begin my career as a software developer in the area of information security.

He began the meeting by explaining that he happened to learn about my wish to work on more challenging projects. He enquired about the type of projects I desired to work on. He listened to my thoughts patiently. He offered some advice and then wished me the best for my future. Little did I know back then that the short 30 minute interaction I had with him would end up changing the trajectory of my career.

Soon after the meeting, I followed his advice, talked to several teams, found out what they were working on and finally decided to join a team that worked on a network security product. It turned out to be the best decision of my career! While working on that product, I learnt to design and implement efficient algorithms for petabytes of data. Some parts of the project relied heavily on probability theory, so it was a joy to be able to take a subject that was a hobby until then and apply it to solve actual real-world problems. Working on that product offered the intellectual challenges that my younger self yearned for back then. I think the most interesting experience there was my work on parser generators. It involved writing algorithms in C++ to parse a formally specified language which in turn was used to generate in-memory FSMs to parse and analyse network packets and events. Parsing one language and generating in-memory parsers for another language proved to be intellectually demanding as well as satisfying. I have fond memories of my time there working on parser generators alongside remarkably skilled engineers from whom I learnt a lot.

All of that happened seven years ago. A couple of months ago, while recounting that 30 minute meeting with Dr Kaliski, I could not recall if I had ever thanked him properly for his kindness and generosity. Seven years ago, there was no compelling reason for him to meet me. I was then just a regular software engineer who was hired into the company a year ago. We had never talked before. But he took the time to offer me help, suggestions and advice on how I could make a good career. Unsure if I had expressed my gratitude to him for how he helped me sculpt my career, I wrote to him recently to thank him and explain how that one meeting had a large and positive impact on my life.

Dr Kaliski replied a few days later and he expressed his happiness to know that the steps I took in the subsequent years worked out well for me. Before ending his message, he wrote a little heart-warming note that I am going to remember forever. Quoting it verbatim here:

“One of my goals is to be able to provide encouragement to others who are developing their careers, just as others have invested in mine, passing good blessings from one generation to another.”

Thank you, Burt! I will do likewise.

Read on website | #miscellaneous | #story | #quote

Susam's Miscellaneous Pages

Five Minutes of Prime Time

The Problem of Pedagogy in Advanced Mathematics

Touch Typing Number Keys

Nerd Quiz #4

Deep Blue: Chess vs Programming

Three Inverse Laws of AI

Introduction

Contents

Pitfalls

Inverse Laws of Robotics

Non-Anthropomorphism

Non-Deference

Non-Abdication of Responsibility

Conclusion

Circular Recursive Negating Acronyms

Nerd Quiz #3

Nerd Quiz #2

Nerd Quiz #1

Million Times Million

Growing Up With the Long Scale

Same Names, Different Numbers

Making Sense of the Short Scale

Conclusion

ASCII PC

Associations

Equal Length Words

Thurston's Paean

No Beam From the Eye

Dark Night Skies

Microcentury

Optimal Lecture Time

Wiktionary on Microcentury

Assumptions

Computation

Conclusion

Global Palindrome Day

Date Formats

Palindromes

Ambigrams

Good Blessings