There was once a deeply influential white paper from Satoshi Nakamoto. The ideas introduced there would go on to energize the cryptocurrency community. A brief summary of that white paper is presented below.
Money can be stored on your computer. In fact, any computer file can be used as a unit of currency. For instance, you could make a Word file called “1 dollar.docx” and treat it as a real dollar bill. The problem is that no one would want to use it. That is because you can get away with making as many “1 dollar” documents as you wish.
If we used a trusted third party, like a bank, then the above problem with duplicating money would not exist because banks were widely regarded as being reliable.
So, could we find a way to store money on computers that people would want to use and that didn’t rely on a third party such as a bank? This had been a difficult problem for some time. What Satoshi did was to give a solution to this in a whitepaper that introduced Bitcoin to the world.
Overview of Whitepaper
In the whitepaper, Satoshi described how to use certain data structures to represent past transactions. These transactions are stored into ledgers. Satoshi then proposed to broadcast these ledgers widely and to make them harder to fabricate.
To broadcast these ledgers widely, you need to transmit them so that they can be stored across entire networks of computers. To make fabrication impractical, Satoshi: (1) stacked stored transactions on top of one another; (2) had these networks favor ledgers with the most transactions; and (3) made the process of such stacking very computationally intensive.
The idea is that if a group of crooks tried to fabricate some transaction, they would have difficulties because their ledger would be short on transaction history compared to other, well established, ledgers in the network.
Proof of Work
Satoshi made the process of such stacking computationally intensive by introducing a protocol known as Proof of Work. It involves pitting computers in the network against one another in various computing contests.
Proof of Work ensures that transactions in the ledger worked on by the largest number of computers on the network are recognized faster than transactions in the other ledgers. Therefore, ledgers with the longest transaction history are secure from fraudulent groups.
In the whitepaper, Satoshi also introduced systems of incentives, including one for participants in the network to support Proof of Work. This is useful because with incentives, enough people are incentivized to improve the reliability of the network.
At some point, Satoshi left without a trace. To this day, no one knows who this individual was. It is not even known if Satoshi represented one person. We are now left with an interesting idea, a cryptocurrency market that is exploring these ideas and their implications.
You may have heard of the binary numeral system before, but what exactly is it? You may have come across it in some classes, and the binary system is used extensively in computers. You may also have heard that this system is related to our normal numeral system. What is all the commotion about? I will help demystify this system in this post.
A very short answer would be that our normal system is based on ten, while the binary numeral system is based on two. I will explain at length what that means and what it entails.
Our counting system consists of two parts. The first part is a set of digits ranging from 0 to 9. They say this corresponds to the fact that we have ten fingers on our hands, which are also called digits.
The second part is a system for positioning these digits to describe numbers. When we want to write numbers larger than nine, we position the digits further to the left. When we want to write numbers smaller than one, we shift them to the right.
To transition to the binary numeral system, we first have to reduce the total number of available digits from ten to two. We do this by only considering the digits 0 and 1. With these two symbols, we can write 0 to mean zero and 1 to mean one as in our normal system.
Before taking the next step, we should make an observation about our normal system. The ones place is for multiples of one, the tens place is for multiples of ten, the hundreds place is for ten times ten, the thousands place is for ten times ten times ten, and so on. In addition, the tenths place is for multiples of one tenth, the hundredths place is for one tenth of one tenth, and so on.
To transition to the binary numeral system, we should replace all the tens (multiply by ten) and tenths (multiply by one tenth) described above with “twos” and “halves”, respectively.
The ones place should be as before, but the tens place should be replaced with a “twos place”, which corresponds to two. For example, 10 in the binary numeral system represents two because the digit 1 here is in the “twos place”. We can go further. Since the hundreds place corresponds to ten times ten, what replaces the hundreds place should correspond to two times two. So, the hundreds place should be replaced with a “fours place” as four is two times two. For the thousands place, we note that it corresponds to ten times ten times ten. So what replaces that should correspond to two times two times two. That is, the hundreds place should be replaced with an “eights place” as eight is two times two times two.
Recall that the tenths place corresponds to one tenth. So as we want to replace ten with two, the appropriate replacement here would be one half. That is, we should replace the tenths place with a “halves place”. For instance, 0.1 in the binary numeral system represents one half because the digit 1 here is in the “halves place”. What about the hundredths place? Since the hundredths place corresponds to one tenth or one tenths, we should replace that with one half of one half. So as one half of one half is one quarter, we should replace the hundreds place with a “quarters place”.
Lastly, to calculate what a number in the binary numeral system is, we add the various “places” occupied by 1 together. If 101 were written in the binary numeral system, what number would that equal to? Note that the “fours place” and the “ones place” are occupied by 1. So to calculate 101, we would need to add four and one, which is five. In summary, 101 is five.
If 1.1 were written in the binary numeral system, we can read it like the following. Note that the “ones place” and the “halves place” are occupied by 1. So to calculate 1.1, we would need to add one and one half, which is one and a half. What if 0.11 were written in the binary numeral system? Note that the “halves place” and the “quarters place” are occupied by 1. So to calculate 0.11, we would need to add one half and one quarter, which is three quarters. In summary, 1.1 is one and a half and 0.11 is three quarters.
This could make for good fun. For instance, someone who is sixty years old could write 111,100 as their age. Why is that? Let us consider the places. We discussed the “ones place”, the “twos place”, and the “fours place”. Going further to the left, we can speak of the “eights place”, the “sixteens place”, and the “thirty-twos place”. How come? Because eight is two to the power of three, sixteen is two to the power of four, and thirty-two is two to the power of five.
Looking at 111,100, we notice that the “thirty-twos place”, the “sixteens place”, the “eights place”, and the “fours place” are all occupied by 1. So to calculate 111,100, we add thirty-two, sixteen, eight, and four together to obtain sixty.
So if someone writes 111,100 as their age on their birthday cake, don’t be fooled. They are not actually over a hundred thousand years old; they are much younger!
The binary numeral system presents an alternative that is much like our system but, at the same time, is quite different. Hopefully, this post can further clarify what this interesting numeral system is all about. You may wonder about how we obtained 111,100 above. I may describe how to do this in more detail in another post.
A well-known conjecture in mathematics, known as the Sunflower Conjecture, was introduced and investigated by Erdős and Rado in 1960. Since then, many mathematicians have worked to understand its mysteries. However, the conjecture remains unsolved to this day.
A sunflower brings to mind a particular image: Petals are neatly arranged, fanning out from a circular mass. We can simplify this imagery by drawing a series of loops, each representing one petal.
In this simplistic visualization, a sunflower can be made up of many such loops in such a way that you are reminded of a child’s drawing.
Now, let’s imagine that each of these loops contains some pebbles. If a pebble is contained in more than one of a sunflower’s loops, it will automatically be in its center, assuming that the loops intersect at the center.
In this way, we have effectively described a sunflower as a collection of loops.
However, loops don’t always form a sunflower. Loops can be arranged in such a way that they do not form a sunflower. For example, if there are three loops but only two of them intersect, then those three loops do not form a sunflower.
Something interesting happens here. Imagine a collection of loops and assume that each of them contains, say, seven pebbles. If I have many such loops, will a sunflower always appear? It turns out that if this collection has 130 loops, then no matter how you arrange these loops, you can always find a sunflower with at least three petals. And if you increase the number of these loops further, you will eventually find sunflowers with even larger numbers of petals.
In 1960, Erdős and Rado proved that this always happens.
In this so-called Sunflower Conjecture, they conjectured that the growth rates of these numbers, such as 130 mentioned above, are at most exponential functions on the number of pebbles in each loop. For example, if the Sunflower Conjecture were true for sunflowers with three petals, then we can find some number relating to the aforementioned growth rate, perhaps an extremely large number such as 1,000,000,000, that will enable us to make interesting conclusions.
If I have, say, 1,000,000,000^2000 (means to the power of 2000) loops with 2000 pebbles in each loop, then among all of those loops, I can always find at least three of them that form a sunflower with at least three petals. I can also make the same conclusion when there are 1,000,000,000^7 loops with seven pebbles in each loop. In the above, I stated that 130 such loops are enough, but there are no problems here as 130 is much smaller than 1,000,000,000^7.
What makes this nice is not that 1,000,000,000^7 is much larger than 130, it is that I can find a single number, such as 1,000,000,000, which captures all pebble sizes (from one to infinity) for sunflowers with three petals.
Fundamentally, we are interested in determining the smallest number that captures all the pebble sizes and whether such a number exists.
Deepening this mystery is that no one knows if such numbers even exist.
*
Let us take a deeper look into some of the numbers involved.
Recall our earlier example involving 130, with seven pebbles in each loop.
If we increase the number of pebbles in each loop to eight, then it turns out that you will need 258 loops to have the same result. That is, if there are 258 loops with eight pebbles in each loop, you can always find three loops that form a sunflower with three petals.
By how much did the number grow when transitioning from seven pebbles to eight pebbles? We calculate that 258/130 is about 1.98, which is nearly two. What happens if we go further? Further increasing the number of pebbles per group to nine gives 514 as our answer. Comparing this number with the previous one, we obtain 514/258, which is about 1.99.
The sequence 1.98, 1.99 may seem to approach 2, but that turns out to be false. If we were to extend the above sequence of ratios by making further calculations, we would eventually begin to find that many of these ratios are at least 3.16. This is a consequence of a result by Abott, Hansen, and Sauer. This is a far cry from numbers like 1,000,000,000. But no one knows whether these ratios will stop growing.
*
Our discussion of sunflowers is indicative of the fact that there are very difficult questions relating to this that can be asked. The Sunflower Conjecture may open the door to many mysteries further ahead.
Does the simplicity of a language affect one’s thinking? In other words, does having a simple language lead to clear and simple thoughts? I imagine that an automated train only requires a small set of directives, such as when to stop at a station. On the other hand, people have a large vocabulary to express themselves fully. Realistically, a small set of directives is deficient for people, but can we simplify our vocabulary and still be expressive?
In 2001, Sonja Lang created a language known as Toki Pona to explore simplicity and language. Currently, it consists of 137 words. By contrast, English has around 170,000 words. Lang wanted a language that would simplify her thoughts, and a good number of people took to it.
There are quite a few YouTube videos on Toki Pona; one even has over a million views. A Discord group, referred to as a cultural center by Sonja, is dedicated to this language. There are also various blogs that discuss it.
Certain interesting features have emerged; one is how the words in Toki Pona are very polysemous, a term that refers to words with multiple interpretations. Such words are considered very “broad” and heavily dependent on context, which help when you have a small lexicon.
Here is a taste of Toki Pona. When you need to go to the bathroom, you can say, “mi wile tawa tomo telo.” The first word, “mi,” means “I” in this context. The last two words, “tomo telo,” translate to bathroom. The word “tomo” is a broad term that can mean a room, building, or a related structure, and “telo” roughly means something related to water. Here, the word “telo” describes the type of “tomo” that is being discussed. The words “wile tawa” translate to a desire to go there. Specifically, “wile” expresses an obligation or desire, and “tawa,” in this case, expresses the action of going somewhere. Putting it all together, we get this message: I + desire to go + bathroom.
Another example is “mi moku.” This can be taken to mean “I am food,” “I am eating,” “I am drinking,” and so on. As you can see, these words are highly context-dependent.
The broadness of words pervades this language. As you can see below, four English words on the left correspond to a single Toki Pona word on the right. In other words, despite their differences in meaning, all four of the English words below would be called “lili” in Toki Pona. This table is part of a Swadesh list, a standardized system for comparing languages to one another.
small
lili
short
lili
narrow
lili
thin
lili
Comparison of English words to a Toki Pona word.
Someone could say this broadness was limiting. Someone else could say that this tested the idea of language in interesting ways—such as how clear an idea can be when using imprecise words.
Toki Pona words are broad and have no exact meaning, so they are harder to pin down. A single word could be used instead of a few dozen. For instance, “lili” signifies something that is “smallish.” The word “lili” alone would be far less clear without context. Online dictionaries written on Toki Pona don’t have many entries, but many of these entries are long and describe several possible interpretations. It is polysemy at its finest.
One good thing about these broad words is that they help clarify and focus your thinking by removing inessential words. This could be especially helpful if you want to simplify your thoughts.
What if you were not interested in seven different ways to express something? Maybe you want to filter out the noise. If there were fewer words to choose from, then only the most important ones remain. Toki Pona appears to leave bare the essentials of ideas since it lacks the words to add anything else. With the words that do remain, you can focus on the simpler forms of the ideas you are conveying.
A financial statement could just be a piece of paper talking about money, a government could be seen as a large group of people, and a furnace could be thought of as a warm box. It may seem pointless to carry out this exercise, but taking this a bit further, you can imagine the following situation. Let’s say that someone, somewhere, began speaking of things in this odd way over a period of time. At first, it may seem a bit silly, but over time, the influence of this habit may grow on them, and as a result, thoughts that were originally more elaborate might become simplified.
This offers a glimpse into modes of thinking I never considered. What if language was more fluid and malleable than we think? Ideas and language are often linked together at the hip. The linkage between them is like a bridge. We also seem to treat language as something static. But perhaps, by modifying this bridge, we can alter our thoughts and perceptions through an altered lens when formulating thoughts and speech.
I speculate that thinking laterally in this way may lead to unusual results. You may never look at a financial statement or a furnace in the same way. Many things that may seem important in the short run may now appear inessential. (For example, a furnace is just a warm box.) In addition, how one communicates may be altered. On the one hand, it would be harder to be precise. On the other hand, essential points in many conversations may become more apparent. One can only guess. What I can say is that Toki Pona is very thought-provoking. Who knows where such inventions will take us.
I sometimes have a bias for minimalism, the kind that encourages one to forego practicality and focus purely on form and aesthetics. In a language like Toki Pona, it manifests as a language of few words. This type of minimalism may be interpreted as reducing elaborate and convoluted thinking.
It is said that Toki Pona is a Taoist language.
Minimalism reminds me of another experience I had. For a short while, I dabbled in Scheme, a programming language known for having relatively few features.
Programming languages are notorious for having many features. There is a very practical reason for this, but it can become bothersome. Proponents of Scheme espoused the idea that, conceptually, many programming notions can be reduced to a few elegant ones that can be used to express everything you would need. The problem, it seems, was that such an idea is rather impractical in practice. Using such a minimalistic framework appears to make certain day-to-day tasks more cumbersome. Moreover, such philosophical considerations are likely irrelevant to many practical programming tasks.
There is a parallel here with Toki Pona. It is said that Toki Pona tests the idea of maximizing the power of words with a relatively small dictionary.
Despite these practical limitations, minimalistic ideas such as those found in Toki Pona and in Scheme hold a certain appeal. Instead of focusing on the tasks they are supposed to help complete, they focus on certain ideas and explore the potential behind them. It is as if they are encouraging us to use them as an art form, not simply as a means to an end.
Somewhat ironically, one reason I wanted to learn Toki Pona was due to laziness. I wanted to exercise the language part of my brain while minimizing the amount of time I needed to learn the essential vocabulary. It indeed exercised my language muscles, and it turned out to be harder than it looked despite the small number of words.
I found that the grammar of this language was very different from English. The word order is very rigid; some say it is to compensate for the broad words. There are also particles, which are glue words that indicate grammatical relationships between the components of a sentence, a notion absent in English but important in Toki Pona. For instance, the sentence “mi moku e telo” can be taken to mean “I drink water.” This sentence contains a particle, the word “e”. What the particle does is indicate that the word “moku” is acting on the word “telo.” So, “moku e telo” means “drink water.”
It was interesting to learn how to put different words together and imagine what a second language learner may experience. There are just too many questions. What exactly makes an invention like Toki Pona tick? How can they change our perspectives on what we normally do? If you find this language intriguing, I suggest looking it up; there are several learning resources out there.
WordPress started as a blogging tool in 2004, and has since evolved into a website builder used by at least 40% of all websites today. WordPress comes in two forms. The original tool, WordPress.org, is open source and allows maximum customizability. The commercial tool, WordPress.com, is not as customizable but is easier to use. Both are widely used today. In the present post, I will review the original tool, WordPress.org.
Using WordPress.org feels like operating the control room of a large ship. When adding content to a WordPress website, you first enter the URL for your site and then add an extra extension to it. Next, it will prompt you for your login credentials and, after entering them, you are presented with an editable form of your site. In this editable form, there is a dashboard. With this dashboard, you can create pages and make blog posts, install plugins, and improve your SEO (short for Search Engine Optimization), which helps make a website appealing to search engines.
I had some rough material for tutoring calculus that I wrote in LaTeX (a typesetting program wildly popular with academics). With some powerful plugins, I was able to turn that material into a polished website. One of these plugins enabled me to use LaTeX. Another of these plugins arranged my website according to a preset layout and provided a search bar to make navigating my website easier.
Some setup is required to use WordPress.org; this requirement can be quite challenging. I had to learn about hosting services and domain names before setting up my calculus tutorial website. WordPress.org does not have a hosting service, so I had to find one that supported WordPress. I ended up choosing DreamHost as my hosting service and the domain name came free. However, not all hosting services give you a free domain name. In such a case, you must use domain name registrars to get one.
My experience without WordPress was very different. Before I used WordPress, I built another website from scratch using HTML, CSS, and JS, which took around a month. I controlled almost every detail of the site layout, from how buttons were placed to how to control page transitions. I even wrote JS code to add scrollbars that generated text.
But, compared to using WordPress, adding content to the non-WordPress website was also very time consuming. I would watch myself add more and more code to my CSS file as I made my website layout more responsive to different screen sizes. I would repeatedly experiment with and tweak my files to make the page transitions between pages as smooth as possible.
There was also the task of displaying the math formulas. My non-WordPress website is a math tutorial website for developers. So I wanted to use LaTeX. To achieve this, I wrote HTML and JS code, with the help of MathJax tools and some MathJax documentation, to properly display and render these math formulas and make them responsive to different screen sizes.
And I had to host my non-WordPress website. To do this, I used GitHub pages, a free hosting service that works very well with static non-WordPress sites. Using this service requires using a GitHub repository, so I used Git techniques whenever I needed to make changes of any kind to my website.
Having made the above comparison, I find that WordPress is an efficient tool with many very useful standardized features. I think WordPress is great for users. Compared to using HTML, CSS, and JS, using wordpress.org gives users more opportunities to focus on writing content while retaining control over the website’s general setup (such as choosing a hosting service).
OpenAI’s ChatGPT has been receiving a lot of attention lately. Microsoft even invested some USD 10 billion in this AI tool. After watching a YouTube video about OpenAI’s free ChatGPT preview, I quickly created an OpenAI account and started testing it out, suspecting that they may soon put up a paywall.
ChatGPT is quite peculiar. On the one hand, ChatGPT appears to be a one-of-a-kind coding tool; on the other hand, meaningful thought seems to be absent when it comes to noncoding-related matters.
As a coding tool, I have these thoughts. I would gaze at its answers to my prompts as it generated code for various tasks I gave it. It was blazing fast and effortless. One small quirk about this tool, which I think OpenAI will eventually iron out, is that sometimes it cannot tweak its own code.
I wanted to write code that would plot graphs with certain specifications. I also wanted the graph itself to depend on user inputs. After communicating my instructions to ChatGPT and having a short back-and-forth conversation between myself and this AI (with me giving feedback), it generated the code (which worked).
The quirk occurred when I asked it to tweak the program to allow for more flexibility with user input. This should have been relatively easy. However, it generated long and convoluted code that didn’t work.
It seems likely that setbacks such as the above quirk are only temporary; after all, this is only a research preview, and an upgraded version may be coming out sometime this year. This tool seems to be more limited with more complex coding tasks. But, after seeing its current progress, my impression is that such issues will also be largely resolved over time.
Later on, I wanted to test ChatGPT for its ability to self-reflect. I asked ChatGPT to write a review for itself. After a long pause—perhaps I should dismiss this pause as network lag—it printed a series of facts in paragraph form. But the paragraph looked as if it were written by someone clumsily trying to promote him/herself. My impression was that it was reading off a list of factoids. I also asked it to generate a longer response in its review. And it did more or less the same, but with more paragraphs and, I think, more facts.
I then asked for its thoughts on Nietzsche’s philosophy. In its several-paragraph response, it wrote about what could be the major perspectives of his philosophy and known implications of his work. It seems to read like a nice Wikipedia article. Specifically, ChatGPT scraped data from the internet to use as training data, and it seems likely that it simply collected thoughts from others on this philosopher. Perhaps this machine’s response could serve as a study tool.
Overall, looking at this tool makes me feel I am much older than I am and that the world is shifting beneath my feet. It would be interesting to see what tools such as ChatGPT evolve into in the future.
To me, pure math is quite peculiar. While precise and structured, it is, at the same time, fluid and expressive. It is almost as if pure math lives within the tension between being structured and being expressive.
If a theory is structured to the point of being concrete and rigid, then it becomes an entirely different pursuit, such as in chess, a game in which the possible actions that one can take are set in stone and where calculation and memorization become critical skills. For me, such theories would end up being restrictive and rigid.
On the other hand, if a theory is so fluid that there is no structure present, then anything goes, and we could make up whatever theory we wanted. That would, in a sense, be an ultimate expression of creativity. However, such theories would then be just speculations. On this point, I recall a set theory book mentioning that useful theories restrict the number of such speculations.
The fact that pure math is, in a sense, both structured and expressive allows me to explore my imagination and complexity in ways that I think are unique to this subject.
I feel inspired when I gaze at images of liminal spaces or otherworldly pictures of imaginary realms. To channel such feelings in meaningful ways, I seek challenging and stimulating problems in pure math. Through these challenges, I sometimes feel as if I am transported into these images.
I think these imageries lend themselves well to pure math since there are theories and problems that take place in settings that can be thought of as imaginary mindscapes. Such settings can also far exceed, or be far removed from, the practical limits of everyday life. For instance, there are results in pure math that involve quantities so large that, outside of pure math, they will probably only be found in works of fiction.
Something that I find mysterious is that complexity in pure math lies somewhere between a board game and real life. In a board game, everything is fixed on an eight by eight grid. Anything that needs to be expressed in that domain has to be through that board. On the other hand, there is no such restriction in math. Topics of discourse can be discussed, and things are not so set in stone. However, in math, there appear to be limits on how far you can go with complexity. In pure math, complexity is largely revealed by proofs. Beyond pure math, and more generally, in life, it is impossible to prove things. Perhaps that impossibility is because our reality is too complex to draw definitive conclusions. Pure math is simplified, when compared to our reality, to the extent that proofs are possible. However, this relative simplicity belies the enormous complexity within this subject. As a result, mathematical proofs can be powerful and insightful.
A world within itself is how I see pure mathematics after all these years. It is a world that is both structured and expressive, as well as imaginative and complex. This world is endless. Engaging with this subject can be compelling, from writing precise arguments to exploring my imagination.
I’m not particularly good at chess, although I’ve been playing it occasionally, to the point of joining chess clubs during my university years. The intense atmosphere that was sometimes present during chess games was thrilling. And I remember having fun moving oversized and glossy plastic pieces over cheap sheets of plastic that made up the chessboards.
Maneuvering the pieces requires a lot of care and attention on the player’s part. The idea is that you have to corral your pieces together so that they become a fully functional and working unit. But they could only move in certain directions and certain ways, so doing the above was like reading a roadmap at times. There were also arbitrary rules that one had to remember, such as when to castle, the en passant rule, and the stalemate rule.
The types of pieces on the chessboard were also important. You had knights that were shaped like horses. They moved in rather odd ways, but they were very powerful and could be unpredictable in their movements. Being short-ranged pieces, they could splinter the opposing army and provide critical reinforcements to your pieces.
There were also bishops that were donned with the headgear found on actual catholic bishops. As long-range pieces, they reinforced your army when you went in for a decisive attack. However, it is not easy to properly position these pieces and maximize their attacking potential.
And there were the rest of the pieces, with varying functionalities and purposes, such as the queen, the most powerful piece on the board, the rook that functions as a defensive resource, the pawns that represent the foot-soldiers, and the king that was the piece that had to be protected at all times.
The arrangements and configurations of the pieces were important, and people spent years playing games to fully utilize the resulting patterns. Sometimes, the formation of the pawns could make or break your game. At other times, knowing when to attack the opposing army could change the tide of the game, particularly when the opposing side’s king was vulnerable.
Like many large board games, this game comes in phases (the opening, middle, and endgame). A lot of memorization is needed to reach the higher heights of this game, especially nowadays, with games being deeply influenced by computer analysis. And to be a good player, being able to visualize and calculate with the pieces is also very necessary.
Sometimes, when looking at this game, I feel like I am looking at a snow globe. Within the globe is a liquid, and when shaken, small pieces of plastic float around, recreating the effect of a snowy day. The chess board is a little like that. It has sixty-four squares arranged into a grid. Given that this grid is limited in size— there are only so many squares to place your pieces on—the board reminds me of a snow globe ornament.
You could get lost playing games in that dimension, within this snow globe-like board, and with all of its idiosyncrasies. Thinking about this takes me back to the university chess clubs I went to, where I heard the clacking of chess pieces and played all those stimulating games.
