On the Two Parts of Empirical Knowledge

There’s two parts to looking at the world. The looking and the world. Most fields of inquiry fix their way of looking and go out in search of the world. Philosophy (at least some of it) instead turns to the way we’re looking at it.
Take for example the role of acetylcholine in the brain. If you ask “Why is that ACh there?” the organic chemist will answer with some sort of mechanical explanation. There’s some mechanism that created an ACh molecule and put it where it is. On the other hand, if you asked a molecular biologist, she would give some sort of purposive explanation. There’s ACh doing the stuff it does being the beings with the ACh doing what it does were able to reproduce. Here we see two scientists answering the same question with two different but compatible answers. Both look to the world, gather their evidence, and draw conclusions about the world. And both keep their ways of looking at the world more or less fixed throughout.
This isn’t meant as an insult to the scientists! Fixing a method of investigation is just how we get a science going. Until we have a concrete system of generating questions (or problems) and an established method of answering (or solving) them, we just don’t have a science. Once we do, though, we apparently get quite a bit of use out of it. The tricky part is figuring out which systems of generating questions and which methods of answering them are the good ones. This is where I see philosophy fitting in.
I take my work on consciousness in particular to be serving this role to neuroscience and psychology, for example. The two fields have very effective ways of investigating nervous systems and mental/behavioral structures. I think that they don’t yet have a great way of investigating subjective conscious experience itself yet (which isn’t a super unpopular view). Don’t get me wrong: I don’t deny the current best empirical data people have collected. My point is not that we have no information from our current perspective, but rather that with a fundamental reconfiguration of our understanding of what consciousness is, and with this reconfiguration a new vocabulary, calculus, etc., we can see it much more clearly.
It takes all kinds. Some people are excellent at taking the blueprints and paving the roads. Some people are great at taking the beaten paths and continuing to build. And some of us see some value in taking yet-undiscovered approaches to the same material. Thus there is in fact not a conflict here but rather two parts of the same larger enterprise.

TAing Intro Philosophy Pt. 1

Yesterday I ran three sections for the first week of spring quarter. This time around I’m TAing intro to philosophy, and some of the information in my slides seems like it might turn out to be useful. So I’m going to give blogging it a go. Since the audience is all of you on the web rather than just my class, I’ll be re-orienting the material accordingly.
Since the section met before any course content was on the table, I only had the course expectations and introductions to work with. Still, every section took the whole time and had to be cut off. First I went over course expectations, then did an icebreaker, then gave tips, then moved to discussion. In retrospect, I might have swapped the first two since the introduction activity got the class more active.

Course Expectations

I break the learning of philosophy, like foreign languages, into four parts: reading, writing, listening, and speaking. To facilitate learning to read philosophical texts, students are expected to do all of the readings listed on the syllabus. This is graded via exams. There will also be writing assignments to evaluate students’ ability to write philosophy. To aid this, there will be practice assignments in section that will contribute to their participation grades.
For listening, learning philosophy and interpreting complex speech will be developed through lectures and conversations. Attendance of lectures and sections will be graded. Putting philosophy into words is often the first way of making sense of things. As such, all students are expected to speak in discussion.
Of course, things happen. I’ve noticed that some students are horrifically overstressed. I understand that a lot of my students have several other classes, jobs, family obligations, and illnesses/disabilities. I take a moment to acknowledge this and encourage students to ask for help early, justified by the fact that I can be way more helpful with more time.
Following this, I moved onto communication standards. In email: I email back within two business days. I try to extend the same courtesy, though, and always give my students two days’ notice. In discussion, disagreement is actively encouraged. Nonetheless, civility is required at all times. We’re here to talk about concepts, reasons, ideas, etc. Not about the person talking. I draw a comparison to dodgeball. Normally, it’s not okay to whip balls at people. Just like how it’s not okay to tell people their deeply held ethical or religious beliefs are wrong. But in gym class, you can throw a ball at someone, and in philosophy class you can throw a rhetorical ball at someone. Even in gym, though, you can’t whip the ball at someone’s face, and in philosophy class you can’t just insult people.
I do call on people at random. Or in particular if someone seems like they have something useful to say. I have three reasons behind this. One, it forces all students to participate. Two, often the quietest students know things but are shy. Part of my goal is to train confidence in speaking to a group. Three, research indicates members of certain demographic groups tend to dominate conversations. I can use my position of power to amplify voices that might otherwise be silenced.
On a different note, yes, everything is a matter of perspective. You should have a perspective. When asked for perspective, I ask that my students please give it. It’s okay to be unsure or change your mind. Taking a position lets us play around with the ideas and see what works.

Introduction Activity

The activity begins by having everyone choose a partner. Then they, obeying a slide on the wall, collect the following information about their partners and become ready to share it:
  • Name
  • Major(s)
  • Year
  • Where you’re from
  • A major non-academic interest
They share this info about their partners. Now I’m 15-20 minutes in and everyone has spoken once. I also got to share all of this information about myself, so they know a little about me. But now I want to get some philosophical thinking going, so I ask them to  find an issue on which they and their partners disagree. In addition, they explain the reason for the disagreement. For example, I think we shouldn’t blame people. Lots of people disagree. But the reason I think we shouldn’t blame people is because people never have the ability to do other than they do, and blame is useless. Someone might disagree with either of those two, and whichever they disagree with is the more basic reason behind our disagreement about blame. The students got into pairs for another ten minutes and shared. Interestingly, in two or three of the three sections, students spontaneously added their input to the disagreements of others. (Lots of death penalty and abortion. Amusingly, one “trivial” disagreement from each class, including James vs. Jordan and whether pineapple goes on pizza.)

Tips & Tricks

I’ve been made aware of the fact that a lot of UC Riverside students haven’t really been taught how to effectively navigate college coursework, nor philosophy coursework in particular. So I offer the following hints on day one:
Reading: Ideally, read three times. First, skim to get the general “plot”. Second, read to understand. Third, read to see which points you can press up against and which parts from the start pay off in the end. (Less ideally, at least read the section headings and first and last sentence of each paragraph.)
Writing: Start early! I’ll provide more writing guidance as we approach the papers, but ideally, you should write down your thoughts in response to the readings to practice putting them in concrete form.
Listening: Paying attention can be hard. Use note-taking to your advantage. Too much: Writing down every word. Too little: Not writing down anything. Just right: Putting down the stuff you want to remember later but get out of your mental space for now.
Speaking: Philosophy is hard. You may want to have your thoughts mapped out on paper to guide you when speaking in class.

Why Are We Here?

Finally, I try to motivate the class. Most of the students are in the class because they were told to be. But I think the questions themselves are interesting. Who/what we are matters. What matters matters. And for a lot of people, the (in)divine nature of the universe matters.

Also, the skills themselves are incredibly useful. In an increasingly divided world, being able to read or listen to someone and thoroughly understand what they believe and why they believe it is important. Then, being able to come up with your own views, subject them to rational scrutiny, and then articulate them in spoken or written language clearly and interestingly enough for someone else to bother consuming is also, I take it, a rather desirable skill. I invite my students to partake in the project of developing these skills and answering these questions. I hope they accept.

The slides are available here: Phil_1_S_1 (1)

Welcome back, piracy

I saw this image on Facebook:

Image may contain: text

I’m not quite sure whether the network execs aren’t thinking this sorta thing through or just assume everyone forgot how to acquire things for free. Piracy rates plummeted when Netflix, Spotify, etc. got big because, sure, you could download things one by one for free, but then you have to remember to do it and manage a library and have storage space and all those annoyances. It’s easier to drop $7 a month to just have everything you want or might want in one place, ready to go whenever, and already managed in the cloud.

I get why they’re doing this: They all want their own streaming services now. Except that kills the benefits. Now it’s $7 or whatever per service, which would quickly let prices approach the old cable range, not in one place, requiring switching services depending on what you want, and managed with irrelevant borders. (Music has the added drawback of not being able to shuffle everything. Just whatever is on what you’re using at the moment.)

At that point, well, piracy is looking a lot nicer. (I’ve seen quite a few people at least see themselves as justified enough if they buy one service and then steal the rest. “I’m already paying for Netflix. I’ll just steal whatever HBO refuses to put on.”) Maybe some estimates have the gain from the people jumping on board the new services outweighing whatever loss there is from not collecting from existing services. Regardless, theft technology has gotten a lot better over the past several years, so perhaps this time we won’t see legal trolls trying to ruin people’s lives as much this time around.

How sure you need to be depends on what you’re doing (Or: As usual I think social media discussions are missing a more fundamental disagreement, this time about a SCOTUS nominee)

Brett Kavanaugh is being considered for a position on the Supreme Court of the United States. He’s also been accused of sexual assault. This information has been all over social media lately, and there seems to be, as there often is, a fundamental disagreement behind the arguments. On the surface, we see basic support versus opposition of the man. There are some straightforward statements of believe in Kavanaugh or else the women who have accused him, mostly prominently including Dr. Christine Ford. There is also quite the range of more general claims. For instance, some people are saying that you should always believe someone accusing someone of sexual assault. Others are taking the opportunity to speak up about what they take to be a worrying trend of false accusations. But these generalities are harder to grasp, so let’s look at the particular case at hand.

In this particular case, besides the basic disagreement about facts, there’s a prior disagreement about how sure either way you need to be to claim to believe in one side or the other. Or, more straightforwardly, to take one side or the other. What actions or consequences are at stake on a belief one way or the other change how easily we’ll take a side or make a belief claim.

A common way philosophers model sure-ness is by using what are called “credence levels,” numbers between zero and one that represent how confident one is in the truth of a statement.  I’m probably 99% sure it’s not going to snow in Phoenix tomorrow morning. Maybe even more sure than that. So I have a credence level of .99 for that. I think I have three decks of cards in my closet, but I’m not super sure. I wouldn’t even bet on even odds. My credence level is maybe .3.

These credence levels are also nifty for expressing how sure of something to be to act a certain way. For instance, in the US, a guilty verdict requires “proof of guilt beyond a reasonable doubt.” This is often expressed as requiring a credence level for the guilt of .99 or higher. But in a civil case, the standard is just believing the verdict is more likely than not. That is, .51 or higher. You can pick all kinds of cases. If you really hate rain, then maybe you only need a .1 credence level that it’ll rain tomorrow to bring an umbrella.

But before you can pick a requisite credence level in a given belief for a given action, some sort of goal is required. Or multiple goals. We have competing values that push us in either direction. I don’t want to get all wet walking to work, but I also don’t want to needlessly carry around an umbrella. I don’t care that much about staying dry, but carrying an extra umbrella annoys me a fair bit. So I need probably a .9 credence level that it’s going to rain. We want to have a functioning justice system, but we really don’t want to punish the innocent. Better a hundred guilty people go free than one innocent person get locked up. So we need that credence level of .99.

The difference in rhetoric of Kavanaugh’s supporters and detractors is revealing of entirely different focuses. The opposition is generally mostly focused on keeping a rapist out of the Supreme Court. For that end, you don’t seem to need to be all that sure. You might even, as I do, flip the proof of guilt beyond a reasonable doubt around to proof of innocence beyond a reasonable doubt. The cost of being wrong is much higher if he’s guilty. Meanwhile even if he’s innocent, a replacement can easily be found. Given the plethora of options and importance of the position, we should be really, really sure that we get someone really, really good. If there’s any reason at all to suspect a Supreme court nominee is a rapist, then we should just move on to the next option. So my opposition to Kavanaugh in this regard only requires a credence level higher than .01 for his guilt. (It is in fact higher than that, though I haven’t spent a lot of time fine-tuning my position. The credence level is far enough above .01 that not much farther thought is needed.)

On the other side, Kavanaugh’s supporters focus on a few things. Shouting “innocent until proven guilty” is one route, suggesting a demand for a .99 credence level for his guilt being needed to deny him the job. But digging a bit into it, there’s more of a focus on some notion of justice. The actual consequences are secondary to the importance of doing the right thing. This would move the bar probably at least to .51. Either Kavanaugh is deserving of the position or he is undeserving, but that fact has nothing to do with the actual results of him getting it. The question comes more down to “Is he a good guy?” as a quasi-factual question about his character.

Some do appeal to some notion of “ruining his life” being a bad thing, to which the standard response is that not being on the Supreme Court does not constitute one’s life being ruined. The standard response to that is that his reputation is being destroyed. I’d be really surprised if anyone hinged their judgement of him on how the Senate votes. The damage, deserved or not, is done. But this does bring up the presence of various actions to take or beliefs to have based on the credence level of a single statement. For example, while I only need a .01 credence level that someone is a rapist to say that they shouldn’t be a major government official, I do still think .99 is morally required to incarcerate someone. Given the former is the context usually at hand, .01 is the bar used to determine what to say I believe. In most of our lives, we have other contexts. If a friend shared a story about having been assaulted, my role would be to comfort, be confided in, or something along those lines. So the bar is pretty low for me to believe. Even if the evidence I have would seem a little suspicious, I don’t need to be very sure at all. On the other hand, if someone I don’t know accused a close friend, I’d probably need more convincing. (It’s a bit harder to pin this case down, though. I have plenty of evidence already built up leading me to believe that my friend wouldn’t do such a thing. So maybe I wouldn’t need that high a credence level in my friend’s guilt, but I would need a lot of evidence to get the credence level even to a medium level.)

These middle cases suggest to me that there’s some reasonable room for needing different amounts of convincing given different conditions. Whether you’re seeking to find someone good enough for a Supreme Court position, send someone to prison, support a friend, or achieve some notion of justice will determine how sure you need to be to take one position rather than the other. And how sure you need to be can vary from one extreme to the other. Given this fact, perhaps the prior questions need a bit more attention. We have the facts as they’ve been presented. Throwing them back and forth appears to be rather unconvincing. But maybe the sureness levels can be moved. If I think about it, I’m not super sure that .01 is the right bar to deny someone a spot on the Supreme Court. I could probably be convinced that I need to be more or less sure. Some people may also be able to be convinced on what the important values at hand are. This is often the route I take. I don’t think that I can convince people to believe Ford if they don’t already. I do think that I can convince a few people that the bar for being on the Supreme Court ought to be really high and so even if you think that Kavanaugh is probably innocent, you should still support moving on to someone who isn’t even accused.

There’s some back and forth from there. Usually the first defense is that a really corrupt political group might just get people to block everyone Trump appoints. I have responses to it, mostly appealing to the growing incredulity and subsequent increased scrutiny. But the point is that the badness of a rapist on the Court is much harder to deny, the badness is much harder to counterbalance, and even rather strong supporters of Kavanaugh are likely to have some non-zero level of suspicion. So this is the argument to have. It has another nice feature, too: The open acknowledgement of different contexts and aims requiring different levels of credence allows us to be more clearly supportive of people who have been assaulted and are in the current discussion being made afraid of coming forward for fear of widespread disbelief. It also satisfies some of the worries about the potency of false accusations. Sexual assault, among other crimes against others, is unfortunately often very difficult to be super sure either way what happened. But, we do at least have a fair bit of middle ground to work with. You need to be pretty damn sure someone did something wrong to lock them up. You don’t need to be so sure to just not give them a bunch of power. And you don’t need to be so sure to give someone some support when they’ve been hurt.

A better-or-worse ethical space

Often when I see ethical categories they fall into a rather neat binary of good and evil, right and wrong, moral and immoral, or some other duo. (I’ll just use “good” and “bad” from here on out for simplicity.) Sometimes the binary will be complicated by expanding into obligatory, permissible, forbidden, and superogatory. (That is, stuff you must do, stuff you can do, stuff you must not do, and stuff that’s really nice if you do but you don’t have to.)

You can also combine these two for eight total options (or, more likely seven actual options—I doubt there’s sense to be made of bad superogatory actions). The good obligatory actions and bad forbidden options are obvious. But then we also have necessary evils, that is, actions that are bad, but because every other option is worse, they are nonetheless the only acceptable option. And we have some actions that may be considered good at least in some significant aspects but because of some overriding factor are forbidden. For instance, one might believe that stealing to serve the needy is good but the law overrides the goodness. (One may argue that the overriding factor just makes it bad, but there is at least the conceptual space for the argument to be had.)

We also often see some understanding of things being more or less good or evil. While one might condemn both jaywalking and murder, murder is worse.  This is already apparent in the above description of a necessary evil being the least bad option.

An element I don’t see played with as often is the location or even presence of the middle line, so to speak. Some theories even look bad because the dividing line between good and bad. For instance, if one takes a consequentialist theory to say that the only good action is that which maximizes whatever the good consequences are and every other action is bad, then the theory seems pretty ridiculous. There are plenty of good choices that could have been better. That particular ridiculousness is found only in the misplaced middle line. Perhaps the line is somewhere else, with a multitude of good and bad choices to make.

Or, more radically, maybe there is no middle line. Maybe for any two choices one can be better or worse (or of the same valence) as the other. This seems particularly intuitive to me because the goodness and badness of choices, outcomes, and everything else does seem to be relative to some sort of standpoint. Any neutral line seems like little more than arbitrary, especially if inaction is properly recognized as itself a choice.

The other aspect of value space I think we need to question more is its boundedness. I’ve encountered some people who think that there is a cap on how good things can be and everything short of that is badness. The opposite can exist as well, and I’m somewhat inclined to it: there’s an absolute value minimum, and everything is building goodness on top of that. Of course, there’s also space for having both maximum and minimum value as well as value being unbounded on either side. What domain is being modeled will make a difference. For finite choices, obviously there are bounds. For total states of affairs, the bounds are much less obvious, if they exist at all.

Freedom of speech does not end at the First Amendment

Someone said or did something controversial. Then, private companies decided not to let the person use their platforms anymore. (Or they deleted a few posts or whatever.) Not too long after, the person (or their followers) make an appeal to freedom of speech. Something like “[Company] is violating [person]’s free speech!” with “and that’s bad” implicit at the end. In response, people who don’t like what was said come around with the revelation that the First Amendment to the US Constitution only applies to the US government. Private companies (and other countries) have the legal right to silence people or disallow them from using their platform.

Yes, that is true. The First Amendment, for example, does not say anything about the legal status of me deleting your comments on my blog, for example. Or WordPress deleting my blog posts on their website. Yet, in either case, freedom of speech is being violated. This is possible because free speech (or “speech that is neither restricted nor silenced”) is something we can understanding independently of the First Amendment. That the First Amendment makes reference to freedom of speech should make this rather apparent.

There are of course several positions one could hold regarding free speech:

  1. Free speech is good without exception: This is often the value touted by people trying to defend the right to say anything, anytime, anywhere. An equivalent phrasing is “All restrictions on speech are bad.” If we want to be less extreme, there are two ways to make exceptions:
  2. Everyone’s having access to free speech is good without exception: This makes room for private entities and such to restrict speech so long as the option to speak freely exists somewhere. Often this value is working in the background when people who are silenced or restricted are told they can go speak freely in their own space, or in a publicly owned space.
  3. Free speech is good with some content exception: That is, it’s bad to restrict speech based on location, time, speaker, etc., but there are some things that ought not be said, and silencing speech of that kind is fine or even good. Of course, with this value we get the further question of what speech is bad enough to restrict.

Unfortunately often, people talk past each other because they don’t agree on this initial question. Or, someone will disingenuously take up one of these for the sake of not having to defend her side in a later question.

I take it most people, at least in the US, have a position that is a combination of 2 and 3. But, if you endorse position 3, then you have to make material distinctions. You cannot just say “Well, everyone has the right (in the sense of “should be allowed”—not necessarily legal right) to say anything,” but rather “In general, everyone has the right to say anything, and this instance is not exception because….” But if the instance is particularly appalling, then what comes after “because” can be rather unpleasant. For instance, to defend someone spouting white supremacist nonsense, you have to defend that nonsense in particular against the reasons why it ought to be silenced. That’s usually going to look like you think the nonsense is not egregious and harmful falsehoods. So, appealing to position 1 instead is very tempting.

One caveat worth noting is that a good does not have to be completely overriding. For example, someone might actually hold position 1  but also believe that the good of free speech can be easily overcome. Maybe it is good to let anyone say anything, but keeping the peace is a greater good, so when the two clash, keeping the peace wins. Thus, positions 2 and 3 are not just position 1 plus recognizing sometimes there are greater goods. Rather, someone who holds position 2 actually does not think it’s particularly good to have setting-unrestricted speech. And someone who holds position 3 thinks that there are some things that are not at all good to allow people to say. (Again, the allowing can be done by a government or someone else. And if allowing it is bad, one might believe that everyone has a duty to silence it.)

I, for instance, hold position 3. In general, people being able to speak without restriction is good. But, there are some things that are bad enough that they aren’t worth allowing their expression. Perhaps it’s not the role of the state to make the restrictions, but the speech should somehow be disallowed. Yet, I also think there are some goods that can outweigh unrestricted speech. Sometimes privacy, for example, demands some speech restrictions. I shouldn’t be allowed to barge into your personal space to speak, and I shouldn’t be allowed to reveal all of your personal information to the world.

The other caveat worth mentioning is that these three positions are not exhaustive. The most obvious omissions are positions that don’t hold freedom of speech as a value at all. To address these positions requires stepping further back to investigate whether freedom of speech, in general, is good. But, in the original context I described, both sides agree that freedom of speech has some positive value.

We can pose a series of questions, then, to isolate disagreement and allow for more fruitful conversation:

  1. Is freedom of speech, at least in general, good? (If yes, go to 2)
  2. Are there exceptions to its goodness that include the situation at hand? (If no, go to 3)
  3. Is there some other good that outweighs the goodness of freedom of speech in te situation at hand?


The Learn Fun Facts blog posted an interesting fun fact about strings. Law’s fun fact is that 8+9+1+89+91=198. Why is this a fun fact? Because the left side of that equation takes every sequence of digits from 891 (except the whole thing) and adds them together to get the digits in reverse order. Contrary to Law’s statement, though, this is not the only three digit number that does this. At the end of the post, he challenges us to figure out if this happens for any five digit combinations. He suggests using a computer to do it, but I wanted to see if basic algebra and a little cleverness could do it. So first I set out on the 3-digit problem to see how the process works.

The 3-Digit Problem

So, what do we know to start with? We’re working with three digits. Let’s call them x, y, and z. So for 891, x=8, y=9, and z=1. To make them match up with 891, we note that 100x+10y+z=891. To reverse the digits, just reverse the coefficients: x+10y+100z=198.

Now, to figure out that it’s 8, 9, and 1, we can’t assume that from the start. But we are working on making the “substrings” (the sequences of digits within the number) equal x+10y+100z. How do we express the sum of substrings? x+y+z+10x+y+10y+z. If we clean that up a bit, we have 11x+12y+2z. Thus we are trying to find out when this is true:


Well, if you just have three variables and one equation, you’re going to get a lot of possible solution. But wait, x, y, and z all have to be single digits. So we know that 0≤x9, 0≤y≤9, and 0≤z≤9. And since we want a three digit number, 1≤x. Since we want to be able to flip it into another three digit number, 1≤z. And since they’re digits, we know that they’re all whole numbers. Okay, now we have some stuff to work with.

If we take our equation from before and subtract all the stuff on the right from both sides we get:


Then divide everything by 2:


We want to isolate one variable to work with, and that z is being subtracted right now, so let’s try moving it back to the right:


Well that’s quite the disparity in coefficients! And here’s where the magic is. Since 1≤z, the smallest thing 49z can be is 49. Since we know 1≤x≤9, we also know 5≤x≤45. And since y≤9, the biggest thing 5x+y can be is 54. But since 5x+y=49z, that means the biggest thing 49z can be is 54. And since z is a whole number, we can see that 49*2=98 which is too big. So z must be 1.

If we plug z=1 into our equation, we have:


Now we’re down to two variables. If we subtract y from both sides we’ll be able to get some nice bounds on 5x, so let’s do that:


Since 1≤y≤9, 40≤49-y≤48. And since 5x=49-y, 40≤5x≤48. But we know x has to be a whole number. So the only options for x are 8 and 9. So if we plug in either option:

(5*8 or 5*9)=49-y

Which is to say

(40 or 45)=49-y

So if we subtract the two options from each side:

0=(4 or 9)-y

Add y to each side:

y=4 or 9

And notice that the y=4 goes with x=9, and y=9 goes with x=8. So now all three variables are solved for. Either x=8, y=9, and z=1, or x=9, y=4, and z=1. These correspond to 891 and 941.

So remember that the fun thing about 891 is that 8+9+1+89+91=198. So now let’s look at 941. We can see 9+4+1+94+41=149. Neat! So there are in fact two three-digit numbers with this property.

Can it be done without the reversal?

On the Learn Fun Facts post, Jack Shalom asks in a comment whether there are any three-digit numbers whose substrings add up to the number itself. The answer is no, and here is the proof.

We start again with 1≤x≤9, 0≤y9, and 0≤z≤9. (We’re not reversing it, so z being 0 would be fine.) Now the equation we want to figure out what makes true is:


Which simplifies to:


If we shuffle the ys and zs to the right and xs to the left we get:


Since 1≤x, 89≤89x. But since y≤9 and z≤9, the biggest thing 2y+z can be is 2*9+9=27. So this system has no solution.

The 5-Digit Problem

The 5-digit version is obviously trickier. Because it’s so cumbersome to write and read the process of defining everything again, I’ll skip to the equation. This time I use a through e as digits for the number abcde since they’re easier to tell apart than some of the later letters.:


Which can be rewritten as:


Solving this would not be super interesting. First you would isolate e, find out it has to be 1 or 2, and continue from there in much the same way was with the 3-digit problem. So I won’t spend more time on that. However, the general case could be fun. Where did 1111, 1222, 233, 34, and 4 come from? Well, each digit gets a 1 from the single digit strings. Then the first four get a ten from the double digit strings and the last four get a one. Then the first three get a 100 from the triple digit strings, the second through fourth get a ten, and the last three get a one. Finally, the first two get a 1000 for the four digit strings, the second and third get a 100, the third and fourth get a 10, and the last two get a 1. So a got one of each, b got to double up on everything except hundreds, c was excluded from getting a thousand but got two hundreds, three tens, and three ones, and so on.

This pattern could rather easily be adapted to any length of number. I hypothesize that with such generated numbers there’s some way to generate the solutions with single digits, but that will take more work.