# Four Random Questions

For the past few months now I have been creating a few flags for various things, essentially building a collection of objects, individuals and similar such that I like or otherwise find value in. One of the things that I like to do, aside from designing and describing the flags, is to give a little background on both the thing and the flag that depicts the thing. For the latter, this is a simple case of highlighting proportions and collinear points on the flag of note, which is known as a description.

The entire system that I use to describe these flags are a subject of a lengthy article that I have taken a long time to finish and still haven’t finished, and so I won’t get into the details of all that in this one. However, the other thing is not related to flags at all, depending solely on the thing that the flag represents, and this is the topic that we will discuss here.

Like most people, the things I am interested in and like enough to express that interest is diverse and varied, so while half of them are vtubers, the other half has items as varied as fairy chess pieces, meteorological phenomena, random concepts from my conworld, tuning systems, and even individual numbers. As I continue to find things that I like and can turn into flags, that list would only get more and more varied and (as the cool kids say) chaotic. While having a diverse set of interests is a sign of good creative and mental health, writing a profile that I can reuse across a group of objects with vanishingly few things in common to compare with is bound to be a challenge, doubly so when whatever such that do exist will surely erode away as things from further afield get added.

And so, I decided to go for a horoscope-like approach, where each question probes a particular aspect in the simulated mind of an anthropomorphised version of the object – unless the object is already fully participating in some society as an individual, in which case the anthropomorphism is skipped – which form the titular Four Random Questions. They are very simple: given that the object knows all that I do:-

- What will it call the fairy chess piece combining the movement capabilities of a Rook and a Knight? (For the purposes of this question, castling is a King move and therefore not a factor)
- What is the sum of all positive integers, ?
- What is its favourite MTR station?
- What is its favourite musical intervals?

Each of these questions have multiple correct answers – with some delicate massaging of the semantics – but also answers that are uncontroversially incorrect, even with the most aggressive of creative reinterpretation. In this document, I will go through each question, explain why I would pick some values over another, and if necessary distinguish how the determination process differ between human-like objects and otherwise. For some questions, an explicit answer-by-answer explanation might be given; for others, a general process to finding an appropriate answer will be provided. We will finish out the article by outlining a few reasons why I chose these four questions, some possible incorrect interpretations that might arise, and a few sample values for things that I haven’t made a flag for yet and might not do.

# The Rook+Knight Question

What do you call a chess piece that has the combined powers of the Rook and the Knight?

The game of Chess, and its predecessors, have been around the Old World for over a thousand years. From its Indian origins in Chaturanga and Shatranj and heading westwards through the Middle East to its modern form – which is in this context typically called “FIDE Chess” to distinguish it from its variants ancient, mediaeval and modern – the basic premise of pieces moving around on a square grid, and a special member amongst them which must be defended at the cost of the game is a time-tested concept. Amongst all the variants, the oldest pieces have not changed at all, both in name and in nature. They are the Rook, the piece that moves any number of squares orthogonally; and the Knight, which moves one square orthogonally and then immediately one square diagonally outward, ignoring all pieces it may have passed through.

Now the thing about Chess and its predecessors is that although the basic premise is very simple, few people throughout history play it. Of those people, very few of them again think of variations on the that game, and by this point those that have the idea to do so have become sparse and isolated. Notably, this amount is not small, but widely separated, so the net effect is that any particular idea can easily be invented by any player but would not spread very far beyond him, resulting in the same idea being reinvented continuously through the years and countries.

The subject of this question is a classic demonstration of the result of this particular set of circumstances. The Rook and the Knight are ancient pieces, and the idea of a piece that combines their powers is fairly obvious to anyone who has played the game and wishes to make modifications to the game. This is evidenced by the large amount of names that have been given to this piece, which the Chess Variant Pages dutifully enumerates in games going back to the 14th century. We excerpt its table in Table 1.

Table 1: Selected names for the Rook + Knight Compound.

Name Year Game Notes ----------- -------- ----------------- --------------------- Champion 1617 Carrera's Chess Oldest attested name, also used with (many) other pieces War Machine c. 1700 Turkish Great Chess Name now more commonly associated with another piece Marshall 1840 The Sultan's Game Chancellor 1887 Chancellor Chess Princess c. 1900 – Primarily used by problem makers Nobleman 1980 The Knightliest Black Hole

In addition to all these names that are used within games, there are also more theoretical designations. Ralph Betza, in his “Funny Notation”, designates this particular piece with the notation “RN”; in his games, he tends to call the piece the Chancellor. There’s also clinical and neutral “Rook + Knight compound” which may be used if one does not feel like choosing a name.

The names given above, and many more, form a set of answers from which we can choose from. Because of my own preferences, if I cannot find a quality that that stands out, the default answer is Chancellor, though various fairly generic factors are in effect that would change the answer for good portion of objects.

Table 2: Selected responses to the question and associations thereof.

Princess

Helpful, straightforward, though somewhat prone to looking to the simplest answer possible

RN

Clinical, concise and values correctness over legibility; alternatively, is a nurse or other medicine-related or -adjacent field.

NR

Similar to the above, but without the medical-related pun; may also be chosen for other related objects.

Rook-and-Knight (or similar)

Simple, a little naive, but nevertheless accurate and germane; knows many things well but not to the point where speculating on what-ifs is desirable

Knook

Modern, hip, “in it” with the currently popular crowd

Duke

Contrarian, enterprising, favours the path not taken; interested in nobility

Marshall

Alternative, contrarian, though not as contrarian as the response “Duke”; interested in military and police; largely non-confrontational

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Table 2 shows a number of possible responses to the question, and some motivating qualities that would result in that response. There are a large number of other responses and why I would pick them, so this only shows a portion of what the full response and reasoning space would be, but it is a representative portion, so remains helpful.

Note that for this question, the number of possible responses is a fairly large number in a human sense, but there is only a finite number of choices and new answers are slow to emerge, though it is by no means closed-ended. This is in contrast to some other questions which have slightly more possible responses.

# The Sum of all Positive Integers Question

What is the value associated with the sum of all integers, 1 + 2 + 3 + 4 + ..., otherwise written as \(\sum_{i=0}^\infty i\)?

We take a sharp turn from games to fairly abstract mathematics, and ask a question that at first glance should be uncontroversial but ultimately can be interpreted to its limits to almost any response at all.

The primary conflict here is embodied in the exact wording of the question itself: we ask for a number associated with the expression in question, which may or may not be identical to simply asking for what that expression equals. Furthermore, the question as posed has a fairly obvious answer of infinity, which is the answer provided by default but is also one that would be the most intuitively obvious.

Regardless of whatever seems intuitively appealing, one can apply unconventional interpretation of the question to arrive at a different answer, some with formal mathematical backing, others more of a refusal to look at the question at all due to its mathematical nature – as it turns out, many people tend to dislike looking at numbers. Table 3 shows a number of sample responses and some reasons to motivate that response. The explanations are slightly longer, both to compensate for the lack of background presented here, and because each answer has a lot more justification for its selection.

Table 3: Selected response to the Sum of All Positive Integers question along with justifications as to why they are chosen

-1/12

This specific response has a vivid history around it, to the point where it is the primary reason why this is selected as a question to begin with.

First, the mathematical basis of this particular value can be traced back to various ways of “summing” the given expression. The sum does not converge, but specialised summation methods can be employed to force an answer anyway, and it is these results, from Ramanujan and similar, that result in the value minus one-twelfth.

This is also the solution that falls out of treating the infinite sum as a polynomial and performing a mathematical technique known as analytic continuation within the complex numbers, where it can be shown that the only way to “smoothly” extend the series to an arbitrary value is to associate it with the Riemann zeta function, where

\[ \zeta(s) = \sum_{i=0}^\infty \frac{1}{n^s}

= \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{2^s} + ... \]

and in particular

\[ \zeta(-1) = -\frac{1}{12}

justifying this particular claim. Indeed, while the sum does not converge, it would appear that if a finite answer is to be forced out of it, then there is no other answer but -1/12.

This response either shows some sort of understanding of the underlying mathematics, or that it’s a “cool answer” because someone in some particularly popular video said it once. Which is which is deliberately left ambiguous.

A small number thematically related to the object

More likely, the subject likely doesn’t care about maths and therefore has no real reason to care about either of the subject at hand. In this case, the answer is an opportunity to directly add a level of personality to the profile and thus to show a little bit more about the subject itself, ignoring the main substance of the question.

Whether this is because of a sense of vanity or because it is the default numerical answer due to a final lack of brainpower is also deliberately left ambiguous.

0

The number system has never been specified, and for some items that exist in a finite world, an infinite sum might not make much sense. This is particularly obvious in, for example, a chess piece, which primarily lives in an 8 × 8 board and has no reason to believe in the existence of numbers higher than however many moves it takes to finish a game – or, if we limit ourselves only the present moment, 8.

Given this particular approach to the world, the response might be to attempt the sum in a finite universe of sorts, where numbers “wrap around” after a certain highest number; addition may choose to respect this “wrap around” condition resulting in clock addition replacing regular addition, or it may simply “top out” at the highest number. Either way, one can reasonably deduce the number 0 from this approach, but other values are also reachable from this approach.

Does not exist, not a number, &c.

A mathematically reasonable approach is to see that the series does not converge, and therefore the straightforward answer is that there is no value such. This is a mathematically honest answer and so there’s not much assailable about it.

As a personality gauge, this response shows that the subject is similarly straightforward and honest, with an understanding of the world clear enough for all situations in the world that doesn’t require esoteric redefinitions of terms and similar.

Something that looks like an error, starting with E_, &c.

A more robotic approach, and so more likely to show up with mechanical objects or robotic characters but also may apply to someone who would look at the sum and attempt to calculate the value by simply doing the additions until time runs out or he gets tired.

-1/4, π, other unrelated quantity

Subject understands that this is a maths question but doesn’t really know so much and just randomly shouts amounts that might come close, at least thematically.

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Unlike the previous question, there is variance in the way responses are selected. Instead of having a fixed (though fairly large) list where there are obvious right or wrong answers, an unlimited number of answers can be accepted, yet there are still some responses that cannot be, such as a non-numerical non-sequitur.

Someone wise once said, “it turns out once throwing out the correct answer makes finding the best wrong answer tricky”. This statement is well-demonstrated by this question, as most of the responses are fairly unsatisfactory in a strict mathematical sense, but they still make a lot of sense as a personality test. As such, the descriptions above do not strictly test the object’s numerical ability but more its approach to numbers and quantities, which is just as well, as we had no guarantee that objects described have a capacity to do maths anyway.

# The Patron MTR Station Question

Of all the MTR stations past, present and perhaps the future, which one has the most thematic similarity to you?

This and the next question is typically styled as the subject’s “favourite”, though a careful wording would avoid using the word as some of the objects do not really have a capability to have a favourite.

One of the most enduring features of a public transportation system, to those that are fortunate enough to have one worth talking about, is that eventually one grows to associate places and features with its stations, even as the system becomes mundane, low-quality, even hated for other reasons. The stations become landmarks around which one navigates around, even if it is wrong – reality shifts to confer truth to what was once lies, and they move the centres of neighbourhoods to themselves, even if they are built out on the margins.

All that is to say, it’s not that much of a stretch to then consider what things can appear, behave or have similar histories to each of the stations in question. I happen to choose the MTR because that is the one I am most familiar with, but the same question can be asked with the network that links up places you are familiar with.

There are northwards of 100 stations that are valid answers, and even if the number of responses are finite and in fact the smallest possible values that can be given, an answer-by-answer guide will not be provided because it is not how the answers are generated. Instead, I nominate a station primarily by comparing the subject with a number of key qualities associated with the station, including but not limited to:-

- Appearance, like theme colour, shape (in terms of navigability and ease of comprehension) and size;
- Neighbouring areas, nearby facilities, business and connectivity with other trains and lines;
- History, changes of names, status like being closed; and
- Patronage, nature of users both in terms of how they use the station and what brings them to the station, as well as their numbers.

Such choices are typically justified in a case-by-case basis, and typically doesn’t require more than one or two sentences explaining a passing resemblance. However, if the object to be profiled warrants, I can still provide a longer explanation that reinforces the association.

# The Patron Musical Interval Question

Which musical interval wider than or equal to a unison has the greatest thematic similarity to you?

This question is very much like the previous question, but there are additional nuances that are available here not available in the other questions. There are also an infinite number of responses, and this is explicit rather than implicit.

First, let’s define what counts as an interval. When one hears a note, one hears a specific waveform. This waveform can be split up into a combination of sine waves of various frequencies, with most of the frequencies being integer (or near-integer) multiples of the lowest frequency in the list, known as the fundamental. The higher frequencies, called the overtones, contribute to the overall “texture” of the sound – called the timbre – and for our purposes we will neglect the exact composition of these and only be aware that they exist. With two or more notes, the ratio between the two fundamentals form the interval that forms the response to this question.

We have not specified a particular instrument in the statement above, and this is because the exact method one creates the tones is irrelevant. This means that any two fundamentals can be chosen, and thus any interval may be present, even if some instruments cannot produce some of them. This further implies that the the space of possible responses is expanded to basically any pair of frequencies, so the space of intervals is expanded considerably. We now have access to not only the classical intervals of old, we also have new exciting intervals that both add flavour to the existing ones, but also carve out new spaces of their own. We can capture specific nuances of subjects by appealing to the generic interval class and the emotions they are associated with, and use the exact expression to highlight certain nuances the subject may have.

All intervals are understood and evaluated in isolation, without considering nearby intervals that might change the flavour. Unlike chords (here understood as a combination of two or more intervals, requiring three or more distinct pitches), an interval by itself generally carries very little musical context by itself, and having to carry the context would cause the response to become too long and unsuitable for use. However, potential context is still considered, and cultural associations are the very basis from which I judge the intervals to begin with.

## Expression format

Unlike other questions, the possible responses to this question can be formatted in a number of different ways, and the way that the value is formatted is important to the overall response. These formats correspond to the many different ways an interval be written, each one being good with a certain type of interval and some not suited for others.

The methods I use to express a musical interval are as follows:-

- Basic fraction of the form m/n where both numerator and denominator are fractions. These are best to write intervals based on just intonation. This can also be written as a decimal fraction if that is necessary. It is always written in “linear” format, using a slash, rather than an upright format, with the two numbers in two rows. An integer can also be provided by explicitly writing 1 as the denominator.
- A number of steps in a specified scale, written m\n. To find the interval, go up m steps in the scale specified by n. If n is a number, assume the just octave 2/1 divided into n steps logarithmically.
- A bare cent interval, written as x¢. The unit for musical intervals is called the cent, and it is defined as 1¢ = 1\1200.
- A specialised interval notation known as a “monzo”. It is written as a row vector |a_p a_q a_r〉, where all a are integers and the indices are prime numbers. In some cases it is possible that both conditions are relaxed, so that a are merely rational numbers and the indices are a generalised subset of rational numbers that generate a unique factorisation. In any case, the value represented by the monzo is \(p^{a_p} q^{a_q} r^{a_r} \cdots\), that is, the product of powers of the indices by the entries.

Monzos are not transparent in their magnitude, so if a monzo is specified an approximation in cents must also be given. It is however helpful to use when expressing particularly elaborate just intonation intervals with the numerator and denominator in the tens of millions or even higher, which happens a lot when stacking just intonation intervals with each other. A value in cents might also be given for a fraction-based interval for similar reasons, though if neither the numerator nor denominator are large this will not be done.

While m\n = (am)\(an) for all a, so they represent the same interval in physical terms, it is conventional to not specify intervals in simplest terms, as it may be the case that it is because of neighbouring intervals or an interval’s role within a scale that forms the basis for its selection. In particular, given the role of twelve-tone equal temperament in today’s music, a representation where n = 12 is almost always preferred over theoretically simpler expressions, e.g. the tritone is by default written as 6\12, not 1\2.

In addition, to help with identifying the type of interval that is intended, a name is also provided. This is the usual “circle of fifths” name that an interval have, based on the diatonic heptatonic scale. In simpler terms, the first seven ordinal numbers are assigned to the conventional meanings of the do-re-mi, with “do” being the first, “re” being the second; and then additional modifiers such as “major”, “minor”, “augmented” and “diminished” are used to specify a particular flavour thereof. This is used even if the interval is not used in the heptatonic diatonic scale – in which case additional modifiers such as “neutral”, “supra-minor” or “super-major” or modifications on the ordinal number word itself such as “hemi-fourth” would be used. Additionally, the word “tritone” is used to refer to the interval roughly equal to half an octave, that is, the interval that which if you stack two of which on top of each other one gets an octave. All terminology, although specialist, is standard and can be used anywhere that specialises in tuning theory.

In most cases the interval selected will be between 1/1 and 2/1, inclusive; that is, between the unison and the octave. However, this is not a strict limitation, and if needed intervals from higher octaves can be selected. All intervals are ascending intervals; intervals between 0/1 and 1/1, not including either end, are excluded as they are descending intervals. All such intervals can be mapped back into increasing intervals.

## Response procedure

As discussed before the idea behind selecting an association is largely based on traditional prejudices with the seven pitches in the diatonic scale plus a few new associations with the xenharmonic (extra-diatonic) intervals. They are stereotypical: tritones are chaotic, slightly evil, perhaps to be avoided; fifths are smooth, easygoing, but more of a base to build more out of rather than something to be enjoyed by its own; major thirds are mellow, bright and optimistic; minor thirds are darker and melancholic; and so on and so forth.

One adds further nuances by selecting what specific second, third, &c. that has the additional nuance; for example, if we have something that we wish to express as “extraordinarily preppy and outgoing”, far more than a classical major third – which we can select between its just form 5/4 or its 12edo form 4\12 – we may elect to use a septimal major third, which is wider and perceived as even brighter than a major third, which may or may not be considered super-major: in this case we can again choose between the canonical just form 9/7, or any of a number of tempered interval such as 9\24, 4\11 and 3\8. If we do deem it super-major, then we annotate the amount with “SM3”; otherwise we just write “M3”, which is the standard symbol for the major third.

On occasion, an association based on a common name can be provided: the classical “wolf fifth” of quarter-comma meantone, which has the rather complicated value of |7 0 -11/4〉[1] exceeding the just fifth 3/2 by a significant amount, is easily associated with canine-related objects that are also unique from the others in the sense that it provides particular tension through its marked poor fit from the generally accepted common amount. There are, helpfully, other wolf fifths too, so even there we can have additional nuance as required.

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[1] A wolf fifth is formed by closing the gap that eleven narrower-than-just fifths, each detuned by one-quarter of the syntonic comma 81/80, has with seven just octaves with another interval that we label a fifth, or technically a diminished sixth. The interval between eleven just fifths and seven octaves is almost exactly another fifth, and is exactly that when the detuning is exactly one-twelfth of the syntonic comma, so detuning three times as much, multiplied with the effect of doing so eleven times, makes the wolf fifth much too wide, giving it its character.

# Discussion

Having spent all the effort explaining what the responses entail, we now turn to the more general questions that deal with this horoscope-like questionnaire. The primary points of concern is the selection of such questions, how responses correlate with each other, what to expect from the responses given and what the whole point of the exercise is.

## Choice of questions

These four questions are tuned so that I, specifically, can answer them, but it would be much harder to do so for someone else to answer them, not least the subjects themselves. This is done by selecting not just one but several deeply niche interests that I have a comfortable grip on solely due to my own upbringing.

That is to say, the questions are selected primarily so that even if someone contradicts me on one question, it would be fairly unlikely that he will also know enough to contradict the other three questions. As previously mentioned, the principal purpose is to provide a more-or-less harmless profile that is applicable to basically everything imaginable, so the values have to correspondingly meaningless and irrelevant. Thus the responses look specific but are actually vague to the point where any response can be justified, and so be “correct”, even if the subject claims otherwise. And because it’s without consequence, getting different answers for the same thing is not really something worth being worried about.

One final benefit to such questions is that it brings forth an impression of a world that has gone so bonkers that such things have become relevant, which is about as humorous as something that requires this kind of explanation can be. This effect compounds with each additional suite of answers I write, as it gives the appearance of quantity that helps with the appearance of a larger world that emphasises these questions.

In any case, there are other questions that are ready candidates for alternation. An easy one is to tap into knowledge of roads and travel, and relate objects to similar motorway junctions. Cities, by themselves, are too easily relatable and have stereotypes that are hard to work around with, but motorway junctions, or motorways themselves, are not known for having any sort of personality while still having many qualities to latch onto to make associations. Motorway junctions in particular come with categorisations already, and yet are still each unique because of their geographical context. I happen to prefer to use British motorway junctions[2] but other networks are of course entirely possible for others.

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[2] The Motorway Database has all the junctions in scope, though I only really pick the largest of these. Nevertheless, there is enough information here that selecting responses from here is fairly easy.

## Reading responses

Having said all that, how should a reader understand a given response?

The easiest way is to simply not read it. After all, it’s just flavour text, all peripheral, and does not affect the rest of the content. However, they do still relate to the main content, so there is still some value that the potential reader can extract from the values, even though it’s obscure and hidden.

First, it’s basically impossible to extract information from one entry; a number of entries must be read together to piece together some correlations. Additionally, some knowledge of the things that are being used as responses is also required. It does not have to be particularly deep; just a cursory knowledge of its basic properties is enough. Though, some more quantity-oriented questions will require additional skills, especially surrounding maths, to navigate properly.

From there, the path is to reconstruct the correspondences by leveraging knowledge of both the thing being compared to and the subject of the comparison; my own intuition is fairly one-dimensional and largely ties to one particular defining feature, name or quality, so it should not be excessively difficult to find them out. With some help, the correspondences can be sketched out, and this can be used to see what surprises future responses have.

## Conclusion

Association games are always fun, and is at the core of this activity amongst many others. The idea behind choosing these four questions to associate arbitrary objects to is to make use of the ability to associate things with other things that look completely unrelated. While the entire exercise is mostly extraneous, and can be safely ignored, it does provide an amount of flavour that would otherwise be explained more dryly – and more importantly, in a way that could be erroneous and misleading.

The exact choice of questions is largely down to what I know well, and I have documented part of the reasoning behind each question and how to interpret them here. Some questions can require more in-depth explanations than others, especially if they rely on quantities. Regardless, the associations are free enough that there is basically no risk of getting it wrong – much like horoscopes, most answers can be justified one way or another, though some may be closer shaves than others. Even direct claims otherwise cannot hurt the truthfulness of the original association.

And so now there is a new way to create “flavour text” for arbitrary objects in various contexts. The result looks like a profile or a quick information box that are popular in encyclopedias and elsewhere in highlighting facts in a concise manner, but the entries are whimsical and irreverent, like they come from another world where things have different priorities, increasing appeal of the thing it appears in.