by Joseph DeVincentis
Answer:
AEROGEN
We can tackle this puzzle in stages.
These clues just set up the ground rules for how emissions standards work in the puzzle. (There are emissions standards in real life that do something similar, but with less variability among standards and not being limited to integers.)
- Nobody really agrees how bad these gases are for the environment, so there are five different standards regarding emissions of the gases. Each company is regulated by a different one of the five standards.
- Each company's total emissions score is calculated as the sum, over all five gases, of the emitted amount of that gas times the rating for that gas.
These clues help us figure out the mapping of companies to standards:
- Each company's name has at least two different letters in common with the name of the standard that governs it.
- This means All Chem is not regulated by GREEN or OZONE, Electribe is not regulated by GASES or OZONE, Fuel Plus is not regulated by GREEN or OZONE, Heat Co is not regulated by GREEN, and Pharm Us is not regulated by CLEAN, GREEN, or OZONE. This clue alone tells us Electribe is regulated by GREEN and Heat Co by OZONE.
- CLEAN and GASES regulate two companies with the same number of letters in their names.
- Since only All Chem, Fuel Plus, and Pharm Us are left, the two companies must be All Chem and Pharm Us. But we previously learned Pharm Us isn't regulated by CLEAN, so it must be regulated by GASES, and All Chem by CLEAN. By process of elimination, Fuel Plus is regulated by SCORE.
These clues are nowhere near adequate to help us figure out the ratings of all the gases in all the standards, nor do the rest of the clues provide this:
- Each standard rates each of the five gases using an integer factor from 1 to 26.
- Each company emitted an integer number of units of each gas from 1 to 26.
- Energon is the only gas with five different ratings in the five different standards.
- A total of three different gases are rated as 5 by at least one standard, and every standard rates at least one gas at 5.
- Kryptonite has the least average rating across all the standards.
- Aether is rated 19 in the SCORE standard, the greatest rating it gets in any of the standards.
But, along with the A to Z comment in the flavor text, they suggest that the five letters of each standard name translate via A=1, ..., Z=26 into the ratings for the five gases, and that we should read off the grid of gases emitted by each company as letters. With that understanding, we can fill in everything for the standards.
These clues help us match all the total emissions scores with companies:
- The total emissions scores were 202, 304, 520, 627, and 771 in some order.
- Heat Co was able to keep its emissions of the gas with the greatest rating in its standard down to a single unit, but had double-digit emissions of every other gas.
- Heat Co's score is 26 for Energon, and at least 10*(15+15+14+5) for the other gases, making its total emissions score at least 516. But since there is no way to raise that minimum by just 4 (it can only increase in increments of 5, 14, or 15), its score couldn't be 520.
- No company had a digit in its total emissions score matching the number of letters in its name.
- So Heat Co's score couldn't be 627, and must have been 771.
- All Chem and Pharm Us didn't score 627 either.
- Electribe didn't have the least total emissions score, but it emitted the least total amount of the five gases, just 27 units, less than half of that in the two gases with the greatest ratings in Electribe's standard.
- Those two gases with the greatest ratings are Energon with 18 and Tachyon with 14. The other three have 7, 5, and 5. The greatest possible total emissions score it could have under this restriction is 12*18 + 1*14 + 12*7 + 1*5 + 1*5 = 324. But since it didn't have the least total emissions score, it must have had the second least, 304.
- The only company left whose score could have been 627 is Fuel Plus.
- All Chem had a greater total emissions score than Pharm Us.
- By process of elimination, All Chem scored 520 and Pharm Us 202.
These clues help us figure out each company's emissions of individual gases:
- Heat Co was able to keep its emissions of the gas with the greatest rating in its standard down to a single unit, but had double-digit emissions of every other gas.
- As noted from this clue earlier, its score was 516 plus increments of 5, 14, and 15. Since we now know its score was 771, the additional units of gas contributed 255 to its score. So it must have had 0, 5, 10, or 15 of those increments of 14, corresponding to total emissions of 10, 15, 20, or 25 units of Kryptonite.
- All Chem was the only company to emit more than 19 units of a single gas.
- So Heat Co emitted 10 or 15 units of Kryptonite.
- Fuel Plus was the only company to emit the same number of units of any gas as there are letters in the gas's name.
- So Heat Co could not have emitted 10 units of Kryptonite, and must have emitted 15.
- Tachyon emissions accounted for 70 times as much of All Chem's total emissions score as did Aether emissions.
- Aether is rated at 3 for All Chem, and Tachyon at 14, so for each unit of Aether emitted, contributing 3 to its score, it must have emitted 15 units of Tachyon, contributing 210. But no company emitted more than 26 units of any gas, so it must have emitted 1 unit of Aether and 15 Tachyon.
- Electribe didn't have the least total emissions score, but it emitted the least total amount of the five gases, just 27 units, less than half of that in the two gases with the greatest ratings in Electribe's standard.
- If it emitted only 11 units of the top two gases, it could have had a total emissions score of no more than 10*18 + 1*14 + 14*7 + 1*5 + 1*5 = 302. If it emitted 12 units of the top two combined, it would have an odd total score. So it emitted 13 units of Energon and Tachyon combined, and 14 total of the others.
- All Chem emitted as much Tachyon as it did Aether and Kryptonite combined. Electribe emitted as much Energon as it emitted Flubber and Kryptonite combined. Heat Co emitted as much Kryptonite as it did Energon and Tachyon combined.
- We already know All Chem emitted 1 Aether and 15 Tachyon, so it must have emitted 14 Kryptonite. These gases contribute 1*3 + 14*1 + 15*14 = 227 of its 520, leaving 293. Energon is rated 12 and Flubber 5 for them, so possible emissions of these gases (subject to the constraints that it emitted more than 19 units of some gas but not more than 26 of any) are 14 Energon + 25 Flubber and 24 Energon and 1 Flubber.
- Combined with the previous work on Electribe, we can write its emissions as N Energon, 13-N Tachyon, N Flubber and Kryptonite combined (which are both rated 5 in its standard), and 14-N Aether, and write an equation: (14-N)*7 + N*18 + N*5 + (13-N)*14 = 280+2N = 304. So N=12 and Electribe emitted 2 Aether, 12 Energon, 12 total Flubber and Kryptonite, and 1 Tachyon.
- We already know Heat Co emitted 1 Energon and 15 Kryptonite, so it emitted 14 Tachyon. These three gases account for 1*26 + 15*14 + 14*5 = 306 of its total emissions score of 771. Aether and Flubber are both rated at 15 in its standard, so the other 465 from its score represents 31 total units of these gases emitted. Since it didn't emit more than 19 of any one gas, it emitted at least 12 of each.
- The only company which emitted only 1 unit each of two different gases was Fuel Plus.
- This invalidates the second All Chem solution where it emitted 1 Aether and 1 Flubber, so it emitted 14 Energon and 25 Flubber.
- Fuel Plus emitted twice as much Flubber as it emitted Tachyon.
- If Tachyon was one of the gases it emitted 1 unit of, then it emitted 2 Flubber, and these gases account for 35 in the total emissions score. If the other 1 is Aether or Kryptonite, then the most it could score from it and the other two gases is (19+3)*19+18*1 = 436, nowhere near enough to reach 627, so this case can't be.
- If Tachyon and Energon are the 1s, there's still 2 Flubber, and these three gases account for 38. It could have emitted up to 19 each of Aether (scoring 19 each) and Kryptonite (scoring 18 each). But since the remaining 589 score is 19*31, this case can only work if it emitted 19 units of Kryptonite scoring 18 each and 13 units of Aether scoring 19 each.
- Otherwise, the two 1s are among Aether, Energon, and Kryptonite.
- Fuel Plus was the only company to emit the same number of units of any gas as there are letters in the gas's name.
- This invalidates the 1 Tachyon, 1 Energon solution above.
- If Aether and Kryptonite are the 1s, then these accounted for 37 of the total emissions score, and the other gases accounted for less than (3+15+5)*19 = 437, not enough.
- So Fuel Plus emitted 1 unit of Energon, and 1 of either Aether or Kryptonite.
- If the other of Aether or Kryptonite is the one it emitted as many units of as the letters in the gas's name, then these gases and Energon accounted for at most 18*10 + 19*1 + 3*1 = 202. It could have emitted up to 18 units of Flubber and 9 of Tachyon to meet the "twice as much" clue, which gives another 15*18 + 5*9 = 315, not enough.
- It emitted an even amount of Flubber, so it must have emitted 7 units of Tachyon and 14 of Flubber to meet these two clues. This makes 15*14 + 5*7 + 3 = 248 toward the emissions score from Flubber, Tachyon, and Energon.
- If it emitted 1 Aether, then Aether and Kryptonite can only make up 19*1 + 18*19 = 361, which combined with the 248 makes only 609.
- So Fuel Plus emitted 19 Aether, 1 Energon, 14 Flubber, 1 Kryptonite, and 7 Tachyon to just reach 627.
- For each gas, there was at least one company that emitted only one unit of the gas.
- We know Fuel Plus emitted 1 of Energon and Kryptonite (and was the only one to emit 1 of more than one gas), All Chem emitted 1 Aether, Electribe 1 Tachyon, and Heat Co 1 Energon. This leaves Pharm Us as the only one who could have emitted 1 Flubber, so it did.
- Pharm Us emitted as much Energon as it did of the other four gases combined, and it emitted just as much Aether as it did Tachyon.
- Suppose Pharm Us emitted 7 units each of Aether and Tachyon. Then those gases contribute 7*7+19*7 = 182 to its total emissions score. There's also a score of 19 for one unit of Flubber emitted, and no matter what else we choose, the total exceeds its score of 202. So it must have emitted less Aether and Tachyon.
- Suppose it emitted 6 Aether and 6 Tachyon. Now those gases contribute only 156, 1 unit of Flubber adds 19, the minimum 2 units of Kryptonite adds 10, and 6+6+1+2 = 15 Energon from this clue, adding another 15 to the score. This makes 200, but if we add just one unit of Kryptonite, the score is too large. So this also doesn't work.
- Suppose instead it emitted only 4 Aether and 4 Tachyon. Then those gases contribute 104, 1 unit of Flubber adds 19, and the maximum 19 Energon adds 19. By this clue, these numbers allow 10 units of Kryptonite emissions, which contribute 50. This is only 192, not enough.
- So Pharm Us must have emitted exactly 5 units each of Aether and Tachyon. These and the 1 unit of Flubber account for 149 of the total emissions score, and to make it come out to 202 while obeying the clue about Energon, we need 7 units of Kryptonite and 18 Energon.
- Two companies emitted the same amount of Kryptonite.
- From earlier work we know All Chem emitted 14 Kryptonite, Fuel Plus emitted 1, Heat Co 15, Pharm Us 7, and Electribe emitted 12 total of Flubber and Kryptonite. Since Electribe emitted 1 Tachyon and another clue tells us they didn't emit only 1 unit of another gas, the repeated amount couldn't be 1, so it must be 7. Electribe emitted 7 Kryptonite and 5 Flubber.
- One company emitted exactly as much Aether as two other companies did combined. The same company emitted exactly as much Flubber as two other companies did combined (not necessarily the same two companies).
- Now we've determined all the emissions amounts except for allocating 31 units of gas emitted by Heat Co between Aether and Flubber, with a minimum of 12 and maximum of 19 of each gas. The other Aether emission amounts are 1, 2, 5, and 19, so the only way to satisfy the Aether part of this clue is if the 19 (emitted by Fuel Plus) is the sum of 1+18, 2+17, or 5+14, the larger amount in each case being emitted by Heat Co.
- Fuel Plus emitted 14 units of Flubber, and the other companies' emissions are 1, 5, and 25. The only way to make the sum required for this clue to work is 1+13, with Heat Co emitting 13 units of Flubber. So Heat Co emitted 18 units of Aether.
And finally we have filled in the whole grid:
| CLEAN | GASES | GREEN | OZONE | SCORE | Aether | Energon | Flubber | Kryptonite | Tachyon | Total Score |
All Chem | O | X | X | X | X | 1 | 14 | 25 | 14 | 15 | 520 |
Electribe | X | X | O | X | X | 2 | 12 | 5 | 7 | 1 | 304 |
Fuel Plus | X | X | X | X | O | 19 | 1 | 14 | 1 | 7 | 627 |
Heat Co | X | X | X | O | X | 18 | 1 | 13 | 15 | 14 | 771 |
Pharm Us | X | O | X | X | X | 5 | 18 | 1 | 7 | 5 | 202 |
Aether | 3 | 7 | 7 | 15 | 19 |
Energon | 12 | 1 | 18 | 26 | 3 |
Flubber | 5 | 19 | 5 | 15 | 15 |
Kryptonite | 1 | 5 | 5 | 14 | 18 |
Tachyon | 14 | 19 | 14 | 5 | 5 |
Then the numbers in the company-vs.-gas grid (1, 14, 25, ...) translate into letters spelling ANY NOBLE GAS ANAGRAM ONE RAGE. Those last seven letters have few anagrams; if we have trouble finding the correct answer, onelook.com (suggested by "take just one look" in the flavor text) can help. One of those anagrams, AEROGEN, is defined as "any noble gas." AEROGEN is the answer.