Susam's Stories

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.

Read on website | #miscellaneous | #story

A4 Paper Stories

2026-01-06T00:00:00Z

I sometimes resort to a rather common measuring technique that is neither fast, nor accurate, nor recommended by any standards body and yet it hasn't failed me whenever I have had to use it. I will describe it here, though calling it a technique might be overselling it. Please do not use it for installing kitchen cabinets or anything that will stare back at you every day for the next ten years. It involves one tool: a sheet of A4 paper.

Like most sensible people with a reasonable sense of priorities, I do not carry a ruler with me wherever I go. Nevertheless, I often find myself needing to measure something at short notice, usually in situations where a certain amount of inaccuracy is entirely forgivable. When I cannot easily fetch a ruler, I end up doing what many people do and reach for the next best thing, which for me is a sheet of A4 paper, available in abundant supply where I live.

From photocopying night-sky charts to serving as a scratch pad for working through mathematical proofs, A4 paper has been a trusted companion since my childhood days. I use it often. If I am carrying a bag, there is almost always some A4 paper inside: perhaps a printed research paper or a mathematical problem I have worked on recently and need to chew on a bit more during my next train ride.

Dimensions

The dimensions of A4 paper are the solution to a simple, elegant problem. Imagine designing a sheet of paper such that, when you cut it in half parallel to its shorter side, both halves have exactly the same aspect ratio as the original. In other words, if the shorter side has length $ x $ and the longer side has length $ y , $ then \[ \frac{y}{x} = \frac{x}{y / 2} \] which gives us \[ \frac{y}{x} = \sqrt{2}. \] Test it out. Suppose we have $ y/x = \sqrt{2}. $ We cut the paper in half parallel to the shorter side to get two halves, each with shorter side $ x' = y / 2 = x \sqrt{2} / 2 = x / \sqrt{2} $ and longer side $ y' = x. $ Then indeed \[ \frac{y'}{x'} = \frac{x}{x / \sqrt{2}} = \sqrt{2}. \] In fact, we can keep cutting the halves like this and we'll keep getting even smaller sheets with the aspect ratio $ \sqrt{2} $ intact. To summarise, when a sheet of paper has the aspect ratio $ \sqrt{2}, $ bisecting it parallel to the shorter side leaves us with two halves that preserve the aspect ratio. A4 paper has this property.

But what are the exact dimensions of A4 and why is it called A4? What does 4 mean here? Like most good answers, this one too begins by considering the numbers $ 0 $ and $ 1. $ Let me elaborate.

Let us say we want to make a sheet of paper that is $ 1 \, \mathrm{m}^2 $ in area and has the aspect-ratio-preserving property that we just discussed. What should its dimensions be? We want \[ xy = 1 \, \mathrm{m}^2 \] subject to the condition \[ \frac{y}{x} = \sqrt{2}. \] Solving these two equations gives us \[ x^2 = \frac{1}{\sqrt{2}} \, \mathrm{m}^2 \] from which we obtain \[ x = \frac{1}{\sqrt[4]{2}} \, \mathrm{m}, \quad y = \sqrt[4]{2} \, \mathrm{m}. \] Up to three decimal places, this amounts to \[ x = 0.841 \, \mathrm{m}, \quad y = 1.189 \, \mathrm{m}. \] These are the dimensions of A0 paper. They are precisely the dimensions specified by the ISO standard for it. It is quite large to scribble mathematical solutions on, unless your goal is to make a spectacle of yourself and cause your friends and family to reassess your sanity. So we need something smaller that allows us to work in peace, without inviting commentary or concerns from passersby. We take the A0 paper of size \[ 84.1 \, \mathrm{cm} \times 118.9 \, \mathrm{cm} \] and bisect it to get A1 paper of size \[ 59.4 \, \mathrm{cm} \times 84.1 \, \mathrm{cm}. \] Then we bisect it again to get A2 paper with dimensions \[ 42.0 \, \mathrm{cm} \times 59.4 \, \mathrm{cm}. \] And once again to get A3 paper with dimensions \[ 29.7 \, \mathrm{cm} \times 42.0 \, \mathrm{cm}. \] And then once again to get A4 paper with dimensions \[ 21.0 \, \mathrm{cm} \times 29.7 \, \mathrm{cm}. \] There we have it. The dimensions of A4 paper. These numbers are etched in my memory like the multiplication table of $ 1. $ We can keep going further to get A5, A6, etc. We could, in theory, go all the way up to A$ \infty. $ Hold on, I think I hear someone heckle. What's that? Oh, we can't go all the way to A$ \infty? $ Something about atoms, was it? Hmm. Security! Where's security? Ah yes, thank you, sir. Please show this gentleman out, would you?

Sorry for the interruption, ladies and gentlemen. Phew! That fellow! Atoms? Honestly. We, the mathematically inclined, are not particularly concerned with such trivial limitations. We drink our tea from doughnuts. We are not going to let the size of atoms dictate matters, now are we?

So I was saying that we can bisect our paper like this and go all the way to A$ \infty. $ That reminds me. Last night I was at a bar in Hoxton and I saw an infinite number of mathematicians walk in. The first one asked, "Sorry to bother you, but would it be possible to have a sheet of A0 paper? I just need something to scribble a few equations on." The second one asked, "If you happen to have one spare, could I please have an A1 sheet?" The third one said, "An A2 would be perfectly fine for me, thank you." Before the fourth one could ask, the bartender disappeared into the back for a moment and emerged with two sheets of A0 paper and said, "Right. That should do it. Do know your limits and split these between yourselves."

In general, a sheet of A$ n $ paper has the dimensions \[ 2^{-(2n + 1)/4} \, \mathrm{m} \times 2^{-(2n - 1)/4} \, \mathrm{m}. \] If we plug in $ n = 4, $ we indeed get the dimensions of A4 paper: \[ 0.210 \, \mathrm{m} \times 0.297 \, \mathrm{m}. \]

Measuring Stuff

Let us now return to the business of measuring things. As I mentioned earlier, the dimensions of A4 are lodged firmly into my memory. Getting hold of a sheet of A4 paper is rarely a challenge where I live. I have accumulated a number of A4 paper stories over the years. Let me share a recent one. I was hanging out with a few folks of the nerd variety one afternoon when the conversation drifted, as it sometimes does, to a nearby computer monitor that happened to be turned off. At some point, someone confidently declared that the screen in front of us was 27 inches. That sounded plausible but we wanted to confirm it. So I reached for my trusted measuring instrument: an A4 sheet of paper. What followed was neither fast, nor especially precise, but it was more than adequate for settling the matter at hand.

I lined up the longer edge of the A4 sheet with the width of the monitor. One length. Then I repositioned it and measured a second length. The screen was still sticking out slightly at the end. By eye, drawing on an entirely unjustified confidence built from years of measuring things that never needed measuring, I estimated the remaining bit at about $ 1 \, \mathrm{cm}. $ That gives us a width of \[ 29.7 \, \mathrm{cm} + 29.7 \, \mathrm{cm} + 1.0 \, \mathrm{cm} = 60.4 \, \mathrm{cm}. \] Let us round that down to $ 60 \, \mathrm{cm}. $ For the height, I switched to the shorter edge. One full $ 21 \, \mathrm{cm} $ fit easily. For the remainder, I folded the paper parallel to the shorter side, producing an A5-sized rectangle with dimensions $ 14.8 \, \mathrm{cm} \times 21.0 \, \mathrm{cm}. $ Using the $ 14.8 \, \mathrm{cm} $ edge, I discovered that it overshot the top of the screen slightly. Again, by eye, I estimated the excess at around $ 2 \, \mathrm{cm}. $ That gives us \[ 21.0 \, \mathrm{cm} + 14.8 \, \mathrm{cm} -2.0 \, \mathrm{cm} = 33.8 \, \mathrm{cm}. \] Let us round this up to $ 34 \, \mathrm{cm}. $ The ratio $ 60 / 34 \approx 1.76 $ is quite close to $ 16/9, $ a popular aspect ratio of modern displays. At this point the measurements were looking good. So far, the paper had not embarrassed itself. Invoking the wisdom of the Pythagoreans, we can now estimate the diagonal as \[ \sqrt{(60 \, \mathrm{cm})^2 + (34 \, \mathrm{cm})^2} \approx 68.9 \,\mathrm{cm}. \] Finally, there is the small matter of units. One inch is $ 2.54 \, \mathrm{cm}, $ another figure that has embedded itself in my head. Dividing $ 68.9 $ by $ 2.54 $ gives us roughly $ 27.2 \, \mathrm{in}. $ So yes. It was indeed a $ 27 $-inch display. My elaborate exercise in showing off my A4 paper skills was now complete. Nobody said anything. A few people looked away in silence. I assumed they were reflecting. I am sure they were impressed deep down. Or perhaps... no, no. They were definitely impressed. I am sure.

Hold on. I think I hear another heckle. What is that? There are mobile phone apps that can measure things now? Really? Right. Security. Where's security?

Read on website | #absurd | #mathematics | #story

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

A Kid Who Could Read My Mind

2009-07-03T00:00:00Z

Last night, I found myself in the midst of a peculiar group of strangers. Among them was a mysterious kid who made an astonishing claim. He possessed the ability to read minds. With utter disbelief, I approached him. He covered both my ears with his palms and asked me to think of something. I thought of the number 5 and he immediately said aloud, '5'. I was astonished. How could the kid read my mind? To ensure that he did not simply get lucky, I thought of another number, 7. Again, he immediately declared, '7'. I thought of a few more numbers and he correctly announced all the numbers. I was completely bewildered. The people around were equally perplexed. They felt that both of us were working together to hoodwink others. An old man in the group who seemed intrigued by all this approached us and asked us to perform more experiments in a structured manner. He proposed that the numbers be generated using a pseudorandom number generator of a scientific calculator and written down on a paper. He asked me to silently read the numbers on each paper, then think of those numbers and see whether the kid could then read my mind.

While he was preparing a list of random numbers, I was trying to understand how on earth the kid could be reading my mind. He could not be have got lucky every single time. He seemed to have direct access to my mind. However that would be a miracle, maybe not in future, but it is a miracle at least today in the year 2009. I kept thinking, coming up with various hypotheses and discarding them. By sheer luck, I thought of a contraption involving one random number generator (RNG) connected to two displays. In this apparatus, the output of the RNG appeared immediately on the first display, but only after a delay of 1 second on the second display. Now it would seem that the first display was predicting the output of the second display. I continued thinking more in that direction. What could possibly be the source of the thoughts that was accessible to both?

After a little thought, it occurred to me that it could be my own mind. Probably the kid and the people I could see were simply creations of my mind. Perhaps, I was asleep and dreaming. I wanted to test the hypothesis. If it was only a dream and my mind was creating all the experience, I should have been able to read the minds of others. I had no clue how the experience of reading others' minds would feel like but I went ahead to test the hypothesis. I asked another person to be a volunteer. He agreed. I covered his ears with my palms and asked him to think of a number and suddenly the number "17" echoed in my mind. I was thrilled. I asked him whether "17" was the number. He confirmed that it was indeed the number he had thought of.

Unfortunately, I could not stay in this state of lucid dream for long. I woke up soon after realising that I was in a lucid dream. If you find lucid dreaming intriguing I recommend Waking Life to you, a very thought provoking movie that shows the experience of a man trapped in a persistent lucid dream state.

Read on website | #miscellaneous | #story

Combinatorics Coincidence

2008-09-14T00:00:00Z

Combinatorics for Fun

At my current workplace, there are several engineers who have an affinity for combinatorics. As a result, discussions about combinatorics problems often occur at the cafeteria. Probability theory is another popular topic of discussion. Of course, often combinatorics and probability theory go hand in hand.

Recurrence Relation

At the cafeteria one day, I joined in on a conversation about combinatorics problems. During the conversation, I happened to share the following problem:

For integers $ n \ge 1 $ and $ k \ge 1, $ \[ f_k(n) = \begin{cases} n & \text{if } k = 1, \\ \sum_{i=1}^n f_{k-1}(i) & \text{if } k \ge 2. \end{cases} \] Find a closed-form expression for $ f_k(n). $

Nested Loops

Soon after I shared the above problem, a colleague of mine shared this problem with me:

Consider the following pseudocode with k nested loops:

x = 0
for c₁ in 0 to (n - 1):
    for c₂ in 0 to (c₁ - 1):
        ...
            for c_k in 0 to (c_k-1 - 1):
                x = x + 1

What is the final value of x after the outermost loop terminates?

Coincidence

With one problem each, we went back to our desks. As I began solving the nested loops problem shared by my colleague, I realised that the solution to his problem led me to the recurrence relation in the problem I shared with him.

In the nested loops problem, if $ k = 1, $ the final value of $ x $ after the loop terminates is $ x = n. $ This is also the value of $ f_1(n). $

If $ k = 2, $ the inner loop with counter $ c_2 $ runs once when $ c_1 = 0, $ twice when $ c_1 = 1 $ and so on. When the loop terminates, $ x = 1 + 2 + \dots + n. $ Note that this series is same as $ f_2(n) = f_1(1) + f_1(2) + \dots + f_1(n). $

Extending this argument, we now see that for any $ k \ge 1, $ the final value of $ x $ is \[ f_k(n) = f_{k-1}(1) + f_{k-1}(2) + \dots + f_{k-1}(n). \] In other words, the solution to his nested loops problem is the solution to my recurrence relation problem. It was an interesting coincidence that the problems we shared with each other had the same solution.

Closed-Form Expression

The closed form expression for the recurrence relation is \[ f_k(n) = \binom{n + k - 1}{k}. \] It is quite easy to prove this using the principle of mathematical induction. Since we know that this is also the result of the nested loops problem, we can also arrive at this result by another way.

In the nested loops problem, the following inequalities are always met due to the loop conditions: \[ n - 1 \ge c_1 \ge c_2 \ge \dots \ge c_k \ge 0. \] The variables $ c_1, c_2, \dots, c_k $ take all possible arrangements of integer values that satisfy the above inequalities. If we find out how many such arrangements are there, we will know how many times the variable $ x $ is incremented.

Let us consider $ n - 1 $ similar balls and $ k $ similar sticks. For every possible permutation of these balls and sticks, if we count the number of balls to the right of the $ i $th stick where $ 1 \le i \le k, $ we get a number that the variable $ c_i $ holds in some iteration of the $ i $th loop. Therefore the variable $ c_i $ is represented as the number of balls lying on the right side of the $ i $th stick.

The above argument holds good because the number of balls on the right side of the first stick does not exceed $ n - 1, $ the number of balls on the right side of the second stick does not exceed the number of balls on the right side of the first stick and so on. Thus the inequalities mentioned earlier are always satisfied. Also, any set of valid values for $ c_1, c_2, \dots, c_k $ can be represented as an arrangement of these sticks and balls.

The number of permutations of $ n - 1 $ similar balls and $ k $ similar sticks is \[ \frac{(n + k - 1)!}{(n - 1)! \, k!} = \binom{n + k - 1}{k}. \] This closed-form expression is the solution to both the recurrence relation problem and the nested loops problem.

Read on website | #mathematics | #combinatorics | #puzzle | #story

Very Remote Debugging

2007-09-03T00:00:00Z

There is a wonderful story about a legendary Lisp debugging story in the second chapter of the book Practical Common Lisp by Peter Seibel. Quoting the story here:

An even more impressive instance of remote debugging occurred on NASA's 1998 Deep Space 1 mission. A half year after the space craft launched, a bit of Lisp code was going to control the spacecraft for two days while conducting a sequence of experiments. Unfortunately, a subtle race condition in the code had escaped detection during ground testing and was already in space. When the bug manifested in the wild--100 million miles away from Earth--the team was able to diagnose and fix the running code, allowing the experiments to complete. One of the programmers described it as follows:

Debugging a program running on a $100M piece of hardware that is 100 million miles away is an interesting experience. Having a read-eval-print loop running on the spacecraft proved invaluable in finding and fixing the problem.

The original source of this story is an article called Lisping at JPL written by Ron Garret in 2002. This story occurs in the section called 1994-1999 - Remote Agent of the article.