Solution: Binary Choice

Answer: RAIL/ALARM

The first step of this puzzle, as hinted by the flavortext "It's impossible to satisfy everyone with just one thing," is to satisfy all the given criteria in each category with two things. This is indeed possible for each category, and in fact, it's possible with a unique pair of things that don't both satisfy any single criterion. Below is some example logic for how to get each pair in each part, though other logic is possible. A common technique is to notice that things satisfying A can't satisfy either B or C, so the thing satisfying B and the thing satisfying C must both be the other thing, letting the search for that thing narrow to things satisfying both B and C.

Looking for this list of songs, it's easy to find this official list, which is very inconvenient to search through due to its pagination. After looking for a more convenient list (and making sure not to use an old 2004-top-500 list), one can find this list which is accurate, on a single page, and much more convenient to search through.

Searching "last" on this page to find songs with the word "Last" in their title, we find only "At Last" and "Last Nite." These are at positions 155 and 115, so neither has a rank with all its digits divisible by 3, and neither is a single word. So we need the song with a rank with all its digits divisible by 3 and the song with a single-word title to both be the other song. Furthermore, neither of these songs is from the '70s, as can be seem from the "Year" column in the linked list. (The song "At Last" was actually originally from 1941; Etta James recorded an arrangement of it in 1960. But both these years are too early.)

We note there are only 31 ranks with all their digits divisible by 3 on the list, so it's not too bad to look through them all for songs from the '70s with single-word titles. Doing this, we find 9 (Dreams), 63 (Jolene), and 396 (Alison). (If we're concerned about making a year mistake, researching all the single-word songs that are options reveals that their "Year" column is correct, or close enough to not make a difference.)

Now we need to check for RIAA awards given to the five remaining possible songs (At Last, Last Nite, Dreams, Jolene, Alison), which we can do by searching for the songs at this official site (or looking on Wikipedia). We can find that all these songs got some award except "Alison," which did not; all the results for "Alison" are from Alison Krauss. (Note: Searching "Dreams" gives too many results to check through easily, but searching the author "Fleetwood Mac" shows that indeed Dreams got a gold award.) So one of the song titles we need is Alison.

Finally, Alison is from the album My Aim Is True, At Last (the Etta James version) is from the album At Last!, and Last Nite is from the album Is This It. Given these options, it's clear that we're meant to ignore the punctuation and say that At Last is from an album with the same title as the song. We've sort of implicitly checked that there's no given criteron both Alison and At Last satisfy, so we can say:

Bulbapedia is extremely helpful here, if it's operational. At the time of writing this solution, however (a few weeks before hunt), it was suffering outages of various sorts. If it's down, Serebii.net has similar information.

Looking for "Fire" in the list of Psychic-type Pokémon (or for "Psychic" in the list of Fire-type Pokémon), we see only three that are both types, which are Victini (494, Psychic/Fire, Darmanitan (555, Fire/Psychic in Zen Mode, sometimes), and Delphox (655, Fire/Psychic). None of these numbers are squares. Also, in this list of Pokémon by what level they start to evolve at, the 31 section has 5 Pokémon, none of which are in the three that are both Fire-type and Psychic-type and only one, Drilbur, of which has a square National Pokédex number (529). So we need to take Drilbur. Drilbur weighs 18.7 pounds, while Victini weighs 8.8 pounds, Darmanitan weighs 204.8 pounds or more depending on its mode, and Delphox weighs 86.0 pounds. Thus, the only option remaining for a Pokémon weighing under 10 pounds is Victini, so our two pokemon must be Drilbur and Victini.

There are some facts we haven't looked at yet. Looking at information about the Pokémon, or looking directly at this picture of Drilbur and this picture of Victini, will show that Drilbur has stripes and white claws, while Victini has neither. (Victini has claws, but they're orange as shown here, or red for shiny Victini.) Finally, Drilbur's Bulbapedia page says that "By spinning its body, it can dig straight through the ground at a speed of 30 mph." is a common Pokédex entry for Drilbur, while Victini's page says nothing remotely related to Victini rotating.

The only states bordering the Mississippi river to their east (for more than several-mile stretches where the river bends) are Minnesota, Iowa, Missouri, Arkansas, and Louisiana. The only states with the northern mockingbird as their state bird are Arkansas, Florida, Mississippi, Tennessee, and Texas. Looking at this list of states by elevation and sorting by elevation, we see that neither the Mississippi-to-east states or the northern-mockingbird states include any state with a peak more than 2 miles high. So we conclude that we need a state bordering the Mississippi river to that state's east and with the northern mockingbird as its state bird, which restricts us to Arkansas.

Looking at this list of states by admission order, we see that Arkansas is at position 25, which is not twice a prime number. Comparing the nine positions which are twice prime numbers (4, 6, 10, 14, 22, 26, 34, 38, 46) to the previously-linked list of states by elevation, we see that the only overlap is Colorado. So our two states are Arkansas and Colorado.

Now let's finish up data collection. Colorado is not a Mississippi-to-east state or a northern-mockingbird state. Arkansas is in the Central Time Zone, while Colorado is not (it's in the Mountain Time Zone). From this page of state's number of representatives over time, we see that Colorado has gone from 1 to 7 (soon to be 8) one at a time, while Arkansas has gone down from 7 to 6 and from 6 to 4 in the past. Colorado has three cities with over 300,000 people (Denver, Colorado Springs, Aurora), while Arkansas has none (its largest city is Little Rock with only about 200,000 people). Finally, Colorado has area just over 100,000 square miles while Arkansas has area that's significantly less.

A good place to start is alpha decay to an element named after a planet. Alpha decay reduces atomic number by 2, and this list of elements named after astronomical objects tells us that three elements (tellurium, uranium, neptunium) are straightforwardly named after planets, one (mercury) is sort of named after a planet, and two (cerium, plutonium) are named after dwarf planets. To hedge our bets, we consider all six, and plan to get angry at the puzzle writer if it isn't one of the first three. The elements two after these (in atomic number order) are xenon, neodymium, lead, plutonium, americium, and curium. Looking at them one-by-one on Wikipedia, or looking (for the ones there) at this site which doesn't obviously seem reputable and doesn't contain anything past neptunium, but does seem to agree with Wikipedia, we see that none of these six elements have heat of fusion over 30 kJ/mol. Also, looking at this wikipedia page about metalloids, we see that none of these six elements are metalloids. So we need a metalloid that has heat of fusion over 30 kJ/mol. By looking at Wikipedia or sorting by heat of fusion on the above site (which does contain all the metalloids), we see that the only options are carbon, silicon, boron, and germanium.

We check this list of elements named after places on Earth to see which of our remaining elements are named after parts of Europe; we find only germanium. So germanium is one of the elements we're looking for. Its atomic number is 32, which is not divisible by 5, so we look at the atomic numbers of our options for the other elements. They're 54, 60, 82, 94, 95, and 96, respectively, so we only keep 60, neodymium, and 95, americium. However, we need an element with atomic mass over 200 amu, and neodymium's atomic mass isn't big enough, so the second element must be americium.

Now let's finish up data collection. Germanium has a stable isotope; in fact, as this page about germanium's isotopes says, four of its five naturally occurring isotopes are stable. As we can see from this page about americium's isotopes, all of them are radioactive and unstable. Looking at this element discovery timeline tells us that americium was discovered after 1900, while germanium was discovered before. Germanium's atomic symbol is Ge, which is not Scrabble-legal; americium's atomic symbol is Am, which unsurprisingly is Scrabble-legal. Finally, americium decays into neptunium, which is definitely named after a planet, so we do not get to yell at the puzzle writer.

It is key here to use either the UN's main list or Wikipedia's list. Most other lists, like Worldometer's list or this document on the UN website, are outdated (one example of how is that they have Swaziland rather than Eswatini, though this one has Burma for Myanmar instead).

That being handled, we note from this page about octahedral numbers that they grow quickly enough for there not to be many options there. The only options are 1, 6, 19, 44, 85, and 146 (after that they get too big). Translating these to countries, we get Afghanistan, Antigua and Barbuda (note that Antigua and Barbuda is a single country), Benin, Cyprus, Jamaica, and Saint Lucia.

A scan of this list of countries by area will show that none of the above countries are bigger than Chile, and the only country bigger than Chile that is a single island is (arguably) Australia. (One might consider Indonesia, but Indonesia is an island chain with many big islands and even more little ones; it can't be accurately called "a single island.") However, if we use Australia, then the other country needs to be in Africa, be on our above octahedral-number list, and have Arabic as its sole official language. The only country in Africa on our octahedral-number list is Benin, and the only official language of Benin is French, with Arabic not commonly spoken. So we can't use Australia, and thus the country that's a single island is on our octahedral-number list.

We find that of the countries on our octahedral-number list, Afghanistan and Benin are not islands, while Antigua and Barbuda, though a single country, are two islands of roughly equal size. Our remaining options are Cyprus, Jamaica, and Saint Lucia. If you zoom in, all of these have small islands of their coasts (yes, even Cyprus; look just off the coast of Famagusta) but there's one clearly main island in all three. However, none of these countries are in Africa. Also, for "Borders an ocean, but not a sea" to make sense, we must be using a definition of sea which is something along the lines of "things called seas" rather than just "any body of water" or "any part of an ocean." Given this, Cyprus borders the Mediterranean Sea, while Jamaica and Saint Lucia border the Carribbean Sea, and so the other country must be the one that borders an ocean, but not a sea. Finally, none of Cyprus, Jamaica, and Saint Lucia even have Arabic as an official language, so the other country must satisfy that clue too. (Türkiye recognizes part of Cyprus as a separate country, but even that part, like Türkiye, speaks Turkish, not Arabic.)

We now look again at the list of countries by area from above, this time looking for sufficiently large African countries that border oceans but not seas. This is narrowed down significantly by Africa bordering both the Mediterranean Sea and the Red Sea. We end up with Democratic Republic of the Congo, Angola, South Africa, Mauritania, Tanzania, Nigeria, Namibia, and Mozambique. We find that Mauritania does have Arabic as its sole official language, though it has other recognized national languages; of the others, Tanzania has regions where Arabic is commonly spoken, but has official languages of Swahili and English (not Arabic), and the other six don't have many Arabic speakers at all. So one of our countries is Mauritania.

We notice that Mauritania is not within 300 miles of another continent's mainland via measuing distance on Google Maps. (It is within 300 miles of the Canary Islands, which are part of Spain, but of course that's not mainland Europe.) So we must pick one of Cyprus, Jamaica, and Saint Lucia which is within 300 miles of another continent's mainland. Cyprus is within 300 miles of both mainland Asia and mainland Africa, so it counts whether we consider it part of Europe or part of Asia. Saint Lucia is within 300 miles of mainland South America. However, Jamaica is not within 300 miles of the mainland at all. So we've eliminated Jamaica. Finally, Mauritania is not entirely west of 10°W, so the other country must be not entirely west of 10°W. Cyprus is not not entirely west of 10°W; in fact, it's entirely east of 10°W. So our other country is Saint Lucia.

Now we finish data collection. Mauritania is partially a desert; much of its territory is in the Sahara. Saint Lucia is not (in fact, it has a tropical rainforest climate). One might suspect the mountains of Saint Lucia lead to a rain shadow effect on one side but they seem not to be tall enough for that; though I couldn't find precipitation maps for the whole island, Wikipedia says "Average annual rainfall ranges from 1,300 mm (51.2 in) on the coast to 3,810 mm (150 in) in the mountain rainforests," and searching "Saint Lucia desert" gives nothing. Mauritania uses green in its flag, and Saint Lucia does not. Finally, Mauritania definitely borders the Atlantic Ocean, and checking for seas it borders gives nothing more sea-like than the Bay of Arguin, which is a bay and not a sea unless we consider all bays of oceans to be seas, which hopefully at this point it is clear that we should not do.

Whew, we're finally done with finding pairs! Now what do we do? One thing we might have noticed while finding pairs (in addition to the fact that there are no criteria both satisfy, as mentioned above) is that each pair's members are the same length as each other, which is also the number of criteria in that section. This length matching is also clued by the extraction table at the bottom of the puzzle, which we can put our pairs in (using the column corresponding to whether they satisfy the first criterion). The fact that there are no criteria both satisfy and this length matching can both help us find pairs somewhat more easily than the above logic.

However, the key next step is to use the text below the extraction table, "When you compare the truths you find to others' positions, you'll see they only differ by a bit." Since this is a puzzle about binary, we expect the bit to be a literal bit difference in two numbers, or other bitstrings, one of which is "the truths you find" and the other of which is "others' positions." We seem to have found various truths; marking the truths as 1 and the falsehoods as 0, we get bitstrings, as shown in the table below:

This doesn't seem very helpful, but that's because we still don't have the positions. (However, as we found in testsolving, teams not having the positions can still stare for a while at something like this table, perhaps with additional columns for, e.g., the letters at positions given by the 1 bits.) The positions are just the things described as "positions" in the first clue of each category. That is, they're the positions of the pair of items found in the given category order. We add those to the table, and also their binary representations in case those are more helpful:

Now, we can finally see something; in the right half (the 1 half) of the table, each truth bitstring can have one bit flipped to become a binary position. However, the binary positions in the 0 half seem too short to do this. The key is that they need to be padded. Specifically, they need to be padded to the number of bits needed to represent any member of the category (i.e., there are 50 states, so some need 6 bits, so we pad 11001 to 011001), or equivalently (due to the puzzle's design) to the same number of bits as the 1 position. (To solve the puzzle, it's necessary to figure out that the 0 positions need to be padded or otherwise assumed to start with zeros, but it's possible to skip figuring out why the exact amount of padding each needs was chosen.) This being done, each 0 truth bitstring can have one bit flipped to become a 0 binary position. Which bits these are (from the truth bitstring) is noted in the below table.

Before we continue, we should perhaps explain this a bit more. Each row has two positions and two bits, and matches up with another row. The positions, when padded as described above, are binary complements, like the truths (where one position is 0, the other is 1, and vice versa). This manifests itself in each pair of positions summing to one less than a power of 2. Each binary-complement pair of positions is one bit off from a binary-complement pair of truths, forming a cycle (the cycle order isn't used, though.) This manifests itself as the two flipped bits in each row being in the same place. The lengths of the positions, as said before, are as long as needed to fit every member of the set, requiring one set to have between 32 and 63 members (US states), another between 64 and 127 (elements), another between 128 and 255 (UN members), another between 256 and 512 (songs), and the last between 512 and 1023 (Pokémon). Finally, each thing with first bit 0 in its truths has first bit 0 in its position, and vice versa. The fact that the bit flips are 2nd/3rd/4th/5th/6th bits is a coincidence.

Anyway, we now have these "Xth of Y" things. We can use them to index into the answer in their half of the row (which always matches in length). It turns out we've now fully used the truths and positions, so we delete those.

We still haven't used the "0 number" and "1 number" columns. These columns, together, have each of the numbers 1 through 10 exactly once, and so give an ordering for the letters we just got. Putting the letters in this ordering, we get SUBMITRAIL. We reparse this as SUBMIT RAIL and thus submit the answer RAIL.

Sinful extraction

This puzzle's sin is wrath. Its sinful text is "You're wrathful about something, but you're not sure what. Use your WRATH to find initial clues, then use a bit of information from each clue to determine which half to look in." We look at the clues, to try to find something matching WRATH, and we notice that each of the five sections above the extraction table has one clue starting with the corresponding letter of WRATH (reading in order on the puzzle page).

• Was written in the '70s
• Reaches speeds of over 15 mph by rotating
• Always (in every US census) had its number of representatives increase by 1 or stay the same
• Turns into an element named after a planet with just 1 alpha decay
• Has a position which is an octahedral number, in the current (2022) official list of UN member states sorted alphabetically

(This is hinted by "initial," though it's used slightly differently in the sinful text, implying that the clues are initial rather than that we want to take their initials.)

Now we need to "use a bit of information from each clue." Indeed, each clue has either a single 0 or a single 1 (sometimes hidden as a digit in a longer number), and no clue has both. The extraction table has two columns of pair members called "First bit 0" and "First bit 1," so this 0 and 1 (representing "First bit 0" and "First bit 1") tells us which column ("half" of the extraction table) to look in and thus which word to take, as given below:

• Was written in the '70s (0, ATLAST)
• Reaches speeds of over 15 mph by rotating (1, DRILBUR)
• Always (in every US census) had its number of representatives increase by 1 or stay the same (1, COLORADO)
• Turns into an element named after a planet with just 1 alpha decay (1, AMERICIUM)
• Has a position which is an octahedral number, in the current (2022) official list of UN member states sorted alphabetically (0, MAURITANIA)

(The member of each pair for which these clues are true is always the "First bit 1" thing, though this is a coincidence.)

We're almost done. We just need to use the position of the clue in the list of clues to index into the member of the pair, as done below:

• Was written in the '70s (0, ATLAST, 4th = A)
• Reaches speeds of over 15 mph by rotating (1, DRILBUR, 4th = L)
• Always (in every US census) had its number of representatives increase by 1 or stay the same (1, COLORADO, 6th = A)
• Turns into an element named after a planet with just 1 alpha decay (1, AMERICIUM, 4th = R)
• Has a position which is an octahedral number, in the current (2022) official list of UN member states sorted alphabetically (0, MAURITANIA, 1st = M)

Reading these letters gives us the answer ALARM, which (if we're woken up by it) is indeed something we might be wrathful about (though the "You're wrathful about something, but you're not sure what." was mostly added for the rest of the sinful text to make more sense in the surface reading).

Author’s Notes

This puzzle came from a discussion with another team member about what you can do with a bunch of lists of bits, for a (different) puzzle idea in very early stages (by which I mean that it had about 5 minutes of discussion/thought at the time, and was abandoned about 5 minutes after that). All puzzle discussions can lead somewhere.

During development of this puzzle, we had to make the extraction (and the two-items-per-section initial step, which was initially somewhat hard to get) significantly clearer and make the puzzle significantly cleaner. I'm quite glad that the idea is in such a clean form now.