The Four-Colorability of the World: Addendum on Planarity

After presenting my conclusions in my previous post on the four-colorability of the world to several friends and friends of friends, we pondered the question of whether the map of the world’s countries is planar. It is in fact not: consider the induced subgraph of (that is, consider the connections among) Turkey, Georgia, Armenia, Azerbaijan, Iran, Turkmenistan, Kazakhstan, and Russia.

Trick-Taking Card Games

Consider one round of a four-player trick-taking card game with no trumps and no breakages (for instance, if hearts did not need to be broken in Hearts). (That is, players must follow suit of the first card played in the trick unless they can’t.) How many of the 52! orderings of cards represent a legal order in which the cards could be played in the round?

To provide examples, any ordering that starts

6♣,K♣,Q♢,10♣,A♣,J♣,3♣,K♢,…

is illegal, as [6♣,K♣,Q♢,10♣] constitutes the first trick, and thus the player who played the K♣ starts the next trick, and thus the player after them has broken the rules as they clearly have at least one club (the J♣ they played this trick) that they did not play last trick (where they played a Q♢), but

2♣,2♢,2♡,2♠,3♣,3♢,3♡,3♠,4♣,4♢,4♡,4♠,…,A♣,A♢,A♡,A♠

is, as it turns out in this case the four parties were each dealt all the cards of one suit, and thus subsequent players will never be able to follow suit of the first player.

Here are some ideas for extensions: have a trump suit, have a breakage rule, have both, solve this for Napoleon.

How I Generate My Passwords, and Why, Fuck You Apple

What is much more problematic than using a weak password is using the same password in multiple places.

There’s a gigantic issue with using a new password for each site that you go to, though: you have to memorize that many passwords. You could use a scheme where you use the same base password and then append the name of the site to the end, but that’s really not more secure than just using the same password in each site.

You could decide to use one of those password managers that exist out there. Personally, I find myself unable to believe that I could just trust some external entity with all my passwords and have an expectation they’re all alright. Oh wait, I’m not even paranoid.

Here’s how I generate my passwords. This scheme is actually memorable, and actually produces significantly different passwords.

Here’s all you memorize to cover all of your websites you care about your password in: a string, a hierarchy, and a function.

(Also, there are definitely places you don’t care about your password. Some of them send you your password in plaintext. Don’t even bother making a secure password in those cases.)

1. The String

Any string of length that’s at least some moderately large number.

-sP~m*KsjO04

, for example, is an excellent string. It doesn’t matter that you don’t understand any particular patterns in the string, because there’s only 12 characters in it, and you can easily memorize just one string of 12 random characters.

Continue reading “How I Generate My Passwords, and Why, Fuck You Apple”

Death to Coinstar

A lot of people consider big banks evil. Why do people consider big banks evil?

Many like to express their frustrations towards financial institutions as places that don’t actually give material good to the world, and make money off of money, or often phrased as “making money off of nothing” or “selling a nonexistent product”. (Like churches.) Another issue people often take is the fact that banks make money off of people with unfortunate or exploitable financial situations.

If you hate what there is in this world that makes money off of nothing, then there’s another institution you might not have thought much about that exhibits this banklike property: Coinstar. Like banks, Coinstar makes money off of your money and gives you nothing else in return. Coinstar is as evil as big banks.

As evil? Pardon me; my apologies to big banks. Banks do give you something: they give you a relatively secure place to put your money and saves you a lot of the inconvenience of dealing with large amounts of money, and they allow you to take out loans so you have the money to do things that would hopefully be nice things to have but for which you for now don’t have the financial resources for but will more likely later, although how much of a nice time banks have with these loans is much of why people hate them. Coinstar actually literally just makes you pay a surcharge to be able to use money you already had.

But wait, doesn’t Coinstar make your coins more useful and less heavy to carry? Yes, what you’re buying with Coinstar is convenience, but this is convenience you almost certainly don’t need, because the denominations of change that actually make you go to a coinstar machine—pennies and nickels—are denominations of money for which the major issue with the pecuniary worth you are throwing away by using the Coinstar machine is not even the Coinstar surcharge but the worth of your time traveling to the machine to get rid of them. (Face it, if all you had to deal with was quarters, you wouldn’t use a Coinstar machine. Quarters are actually still useful in this age, though maybe not for long. They’re still not ridiculously inconvenient at laundry and vending machines.)

If you make federal minimum wage in the United States ($7.25/hour), then you have lost more than a penny’s worth of time if you took 5 seconds to deal with a penny. You’d think that given this the United States would’ve just gotten rid of the penny by now. Hahahaha. Guess why they haven’t?

Continue reading “Death to Coinstar”

Buttons

You have a button. If you press the button, you have a 50% chance to win $500 and a 50% chance to lose $200. You don’t get to press the button more than once. Do you want to press the button?

You have a button. If you press the button, your friend has a 50% chance to win $500 and a 50% chance to lose $200. You don’t get to press the button more than once, and you don’t get to consult with your friend ahead of time about your friend’s opinion on what you should do. Do you want to press the button?

B is considering murdering C, and currently has a 50% chance of deciding to so. You have a button. If you press the button, with a 50% chance, B’s chance of deciding to murder C goes up to 80%, and with a 50% chance, B’s chance of deciding to murder C goes down to 5%. (Suppose, say, the button causes a video to be shown to B that alters B’s impression of C.) Do you want to press the button? If you press the button, and C ends up murdered, how much are you to blame for the fact that C is dead? If you do not press the button, and C ends up murdered, how much are you to blame for the fact that C is dead?

B is considering murdering C, and currently has a 50% chance of deciding to so. You have a button. If you press the button, with a 20% chance, B’s chance of deciding to murder C goes up to 100%, and with an 80% chance, B’s chance of deciding to murder C goes down to 0%. Do you want to press the button? If you press the button, and C ends up murdered, how much are you to blame for the fact that C is dead?

B is considering murdering C, and currently has a 2% chance of deciding to so. You have a button. If you press the button, with a 1% chance, B’s chance of deciding to murder C goes up to 100%, and with an 99% chance, B’s chance of deciding to murder C goes down to 0%. Do you want to press the button? If you press the button, and C ends up murdered, how much are you to blame for the fact that C is dead?

B is known to have a mental illness and is considering murdering C, and currently has a 50% chance of deciding to so. You have a button. If you press the button, with a 50% chance, B’s chance of deciding to murder C goes up to 80%, and with a 50% chance, B’s chance of deciding to murder C goes down to 5%. Do you want to press the button? If you press the button, and C ends up murdered, how much are you to blame for the fact that C is dead?

B is considering murdering C, and currently has a 50% chance of deciding to so. You have two buttons. If you press the first button, with a 50% chance, B’s chance of deciding to murder C goes up to 80%, and with a 50% chance, B’s chance of deciding to murder C goes down to 5%. If you press the second button, you defer this exact same choice to someone else, and you do not know who it is who is next presented this choice. Will you press either button, and if so, which one? If you press the first button, and C ends up murdered, how much are you to blame for the fact that C is dead? If you press the second button, and C ends up murdered, how much are you to blame for the fact that C is dead?

B is considering murdering C, and currently has a 50% chance of deciding to so. You have two buttons. If you press the first button, with a 50% chance, B’s chance of deciding to murder C goes up to 80%, and with a 50% chance, B’s chance of deciding to murder C goes down to 5%. If you press the second button, with a 50% chance, B’s chance of deciding to murder C goes up to 70%, and with a 50% chance, B’s chance of deciding to murder C goes down to 1%. If you press the first button, and C ends up murdered, how much are you to blame for the fact that C is dead? If you press the second button, and C ends up murdered, how much are you to blame for the fact that C is dead? If you do not press either button, and C ends up murdered, how much are you to blame for the fact that C is dead?

Ring Species, Dialect Continuums, and Behavioral Policy

= is transitive. ≈ is not.

≈ is also not well-defined for general usage, but notice that no matter what error ε>0 one chooses for which one designates two quantities to be approximately equal (≈) if and only if their positive difference is less than ε, one can find quantities a, b, and c such that a≈b and b≈c, but a≉c. (For instance, use b=a+0.8ε and c=a+1.6ε. Note that we are assuming that the domain of quantities has a largest and smallest element more than ε apart, but this quite goes without saying considering otherwise the ≈ operator is useless.)

The important point here is that the sum of insignificant changes can be significant.

In particular, when one is trying to categorize a set of items into categories, it is possible that any sort of meaningful classification upon a particular quality can be defied.

Consider the typical definition of a species, that is, a set of individuals that can interbreed with each other. Interbreeding is something that can occur between organisms of slightly different genomes (thankfully, for the sake of life existing), but the genomes still need to be similar enough for two organisms to be able to produce fertile offspring. As such one could imagine that individual A can successfully breed with individual B, who can successfully breed with individual C, who can successfully breed with individual D, and so on, but, say, individuals A and H cannot successfully interbreed. This phenomenon can of course be generalized to not just individuals but groups of individuals, where no members of one group can successfully interbreed with members of another group. Thus, by the interbreeding definition of a species, these are clearly not in the same species. And yet, every step along the way we find individuals that can interbreed with each other, all the way from A to H, so each should be in the same species as the previous one. What gives?

It turns out this is not just a hypothetical. There are actual cases of this occurrence found on Earth, in what are known as ring species. It is an occurrence that fundamentally challenges the concept of a species itself, an instance where there is very clearly no acceptable line to draw to divide the individuals into members of different species. It appears necessary to accept that sometimes there is neither a line nor an equivalence, but rather a gradual continuum in species membership.

It also turns out this is not just a problem with species. Let’s turn to linguistics. What is a dialect? Let’s use a definition of a dialect as a version of a language with possible slight differences in phonological, morphological, and grammatical specification, such that these differences are small enough that what is spoken by people of different dialects of the same language is mutually intelligible, that is, people can readily understand what the other is conveying despite the differences in their speech. Well, then what constitutes a language? It turns out there’s several cases of what we consider different languages that are mutually intelligible, like with Norwegian and Swedish, or with Czech and Slovak. But also, there exist languages like Chinese for which there are so-considered dialects that are mutually unintelligible but for which there exist a set of intermediate dialects that are mutually intelligible with the next dialect in the chain, stretching the entire span of linguistic change between these dialects.

(Actually, mutual intelligibility gets even weirder. Mutual intelligibility is not only intransitive but also asymmetric. All sorts of weird relationships between languages that are different flavors of mutual intelligibility arise.)

Continue reading “Ring Species, Dialect Continuums, and Behavioral Policy”

Is the World Map Four-Colorable?

Is it possible to color a map of the world’s countries using four colors such that no two countries that border each other have the same color?

If by this point in the post you have already reached the answer “Yes, use the Four-Color Theorem”,* hold your horses, because the Four-Color Theorem applies to a division of the plane into contiguous regions, and an assertion that countries are contiguous regions is extremely nontrivially untrue: tons of countries own islands, and many countries own exclaves, like the USA’s Alaska and Russia’s Kaliningrad Oblast, completely disconnected from the main part of the country.

(*And if at this point reading the paragraph you reached the objection “commas are supposed to go inside quotation marks”,* you should note that for one, only Americans do this, and second, the reason Americans do this is because back-in-the-day it was considered of aesthetic value to put tiny pieces of punctuation inside the quotation marks, due to how typography back then came out. In the modern world, this issue is nonexistent. Like with slavery and imperial units, this is another place where the United States has decided that the fact that we have been doing it is more important than any advantage in reason, so if you think that tradition itself is a dumb reason to back a decision, that is, if you think the world should have nice things, the choice to make is to place commas and periods outside quotation marks when it makes the punctuation of the sentence make logical sense.)

In fact, if you perform an image search for four-colored world maps, chances are you’ll be hit with a first page of exclusively incorrect maps.

google_image_search_four_color_map

yahoo_image_search_four_colors

duckduckgo_image_search_four_colors

Maps 1 through 5 color France and the Netherlands the same color, whereas these two countries do in fact border each other, that is, on a Caribbean island. On top of this, Map 2 is not an accurate map of the world: it shows Czechoslovakia undivided and Eritrea unseparated from Ethiopia, implying that the time in history the map corresponds to is far before Montenegro separated from Serbia, which the map also claims. Map 4 seems to be an attempt at four-coloring the world while also considering bodies of water as entities, given that the Caspian Sea is colored a different color from the World Ocean; it is quite a respectable attempt, but it once again fails the above criterion.

Map 6 correctly decides to color France and the Netherlands differently, but fails to color Turkey and Azerbaijan differently: the two countries share one of the shortest borders in the world because Azerbaijan includes the Naxçıvan exclave (which, incidentally, Map 2 and Map 3 also fail to separate). Maps 7, 8, and 9 also fail at at least the France-Netherlands situation.

With the aid of the map charting tools of mapchart.net, I drew up an actual four-color world map.

actual_four_color_map

(Notice that I chose colors for the UK, Argentina, and Chile such that Antarctica resolves correctly no matter which of them you listen to—if you decide to recognize claims in Antarctica as legitimate and favor the Argentine lines over the British lines, the Argentine claim is still separated from the Norwegian claim by 5 degrees of longitude.)

This shows that the world can be colored using four colors right?

…not necessarily. This is one of many possible conceptions of the political borders of the world. This map by itself doesn’t show that the map of the world is four-colorable, regardless of which country’s diplomatic recognitions you follow. In fact, this map of the world is actually the world in not a single UN-recognized country’s eyes, because every single country of the world recognizes the People’s Republic of China xor the Republic of China as being the one true China.

So can the map become un-four-colorable depending on how border disputes resolve?

Continue reading “Is the World Map Four-Colorable?”