What xkcd Means

Question

What xkcd Means

^{This is a variety cryptic inspired by the comic "What xkcd Means" (linked above) and the work of Ucaoimhu. It is also an entry into the 35th Fortnightly Topic Challenge: Restricted Title 1.}

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It means saving a few seconds at a long red light via elaborate and questionably legal maneuvers.

To save a few seconds in a questionably-legal way, about a third of this grid's clues have a missing letter. (Adding the letter back will always make a word.) Reading these letters in order will give you a clue to WORD 1.

It means having someone call your cell phone to figure out where it is.

Another third (or so) of these clues have a word transmitted over the phone. If the original transmission has n rings, take the nth letter of the clue (if 0, take the last letter). These letters, in order, will give a clue to WORD 2.

It means calling the Ackermann function with Graham's number as the arguments just to horrify mathematicians.

Several of these clues will horrify people interested in numbers (and this puzzle). If the horrifying number is n, take the nth letter of the clue; this will give you WORD 3.

It means instinctively constructing rules for which floor tiles it's okay to step on and then walking funny ever after.

There are several shaded cells in this grid. Some of these cells have multiple letters in them; crossing clues' enumerations will hide this by counting them as a single letter. Some of the shaded cells will spell words, which are all members of a certain category. The rest of the shaded cells should be interpreted as abbreviations for members of this category.

Now we turn to the remaining clues, which all have an extra definition for a different word added on to one side. (One of the words you get this way will be a variant spelling, and another will be a shortened form of a word.) These extra words can be arranged into a word square in some order, where all words going across come from Across clues and all words going down come from Down clues.

To find the source of a set of rules for which floor tiles it's okay to step on, look at the letters of the regular portion of the clue that are closest to the extra definition.

Find a path from the top left to the bottom right, then take the corresponding cell in the word square for every third step to get WORD 4.

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You should now have found four words. Put these together and use this to find an 8-letter name; this name doesn't appear relevant, but what would xkcd be without a healthy dose of science? Use a famous scientific result to transform this name into a word describing xkcd's humor.

Oh, I almost forgot - there are two extra letters in each part. (In part 3, they are the last two, and in part 4, they are hidden differently than the other three pairs.) Put them to get a seemingly-related word that doesn't describe xkcd's humor.

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ACROSS
3. Analyse unusually direct favor (4)
6. Umpy concealed within ashy perfume (5)
9. Thousand and one lei totalled (5)
10. Refugee sent back, then second in line is moved to the front? It's a curse (6)
12. Orally polls for location of banks in the underworld? (4)
14. Pounds in dark energy source, hiding answer at the shore (7)
16. Sine dividing two pi, I hear, for character of Faust (3, 2)
18. At last, one of European country gets rid of virus (3)
19. Come in and don't align rebel (5)
20. Mortal peasant holding money from Rajasthan, given away for free (6)
24. Would Peru discharge? (5)
26. Develop containment for recipe, safeguard joint (6)
28. Liars are smart, pushing capital to the other extreme (5)
30. Times: 120 seconds after Egyptian god makes a cheer (3)
33. Non-IRL gay pair awkwardly makes beehives (5)
34. Otherwise, sew a vein for tax crime (7)
35. Adequate enormous .png file, maybe? (4)
37. Large BLT swallowed by Scottish river flood (6)
38. White House's soldier looking back by restroom for a bit (5)
39. Superhero lost extraterrestrial amateur actor Harris (5)
40. Old myth of a sugar-addled ditching (4)

DOWN
1. Idol thumbs-ups chefs without company (3)
2. Ancient tongue's note audible twice (5)
4. Revived voiceover again, making 'buoy' into 'sign of love' (3)
5. Untidy torte for track event (4)
6. Tug on that man, Ash, hiding (5)
7. Shot major annoyance with bread (4)
8. "Trust" is a race without a foot covering (4)
11. Mowed, by the sound of it, an estate (5)
13. Group of snowmen's sure thin after eating The Thing while upset (5)
15. Reject pivoted to gain real notoriety (4)
16. Vital element besieged by brewed optical infection (4)
17. Old before the Queen, ultimately (3)
21. Order engineered parts for handle (5)
22. Inquisitive MC, maybe, in the Big Apple (4)
23. Knight's partner: king with rheum in church (5)
24. Oddly plumbs small places for pits (4)
25. Covers fro and hears a solver's cry (3)
27. Dynasty is source of food to the audience (5)
29. Lot on the record for abbey boss (5)
30. Type of colorblindness relating to Nu's gender (3-1)
31. Aura formed by killer computer and lad, obviously (4)
32. Dear gets accomplished (4)
35. God's breakfast food (3)
36. On the way without ale in tooth (3)

Answer 1

Here is the filled grid:

Explanation of "transmitted over the phone" and "horrifying people interested in numbers" for clues, but not yet for WORD 2/WORD 3 extraction:

"Transmitted over the phone" is a homophone indicator. In each clue, one word has been replaced with a homophone; the 'original transmission' word must be used for the clue to be parsed.
"Horrifying people interested in numbers" is an indicator that a number has either been removed from the clue, or are doing double duty in each clue, depending on your perspective. In each clue, either the number of the clue should be read as the beginning of the full clue, or the enumeration should be read as the end of the full clue.

Clues, answers, and explanations:
ACROSS

3. Analyse unusually direct favor (4)

This is a tile clue. The extra definition is "Analyse"
CREDIT = favor; (DIRECT)*

6. Umpy concealed within ashy perfume (5)

This is a missing letter clue. The actual clue uses "Jumpy" instead of "Umpy"
HYPER = jumpy; hidden in "asHY PERfume"

9. Thousand and one lei totalled (5)

This is a phone clue. The actual clue uses "lay" instead of "lei"
KAPUT = totalled; K+A+PUT

10. Refugee sent back, then second in line is moved to the front? It's a curse (6)

This is a missing letter clue. The actual clue uses "course" instead of "curse"
REGIME = course; EMIGRE< then [R<->E]GIME

12. Orally polls for location of banks in the underworld? (4)

This is a phone clue. The actual clue uses "poles" instead of "polls"
STYX = location of banks in the underworld (as in river banks and the River Styx); STICKS homophone

14. Pounds in dark energy source, hiding answer at the shore (7)

This is a horrifying numbers clue. The actual clue begins with "14 pounds..."
COASTAL = at the shore; COAL containing ST+A (short for STone=14 pounds and Answer, respectively)

16. Sine dividing two pi, I hear, for character of Faust (3, 2)

This is a phone clue. The actual clue uses "sign" instead of "sine"
PINKIE PIE = character of (Lauren) Faust; INK (sign) in PIE PIE (twice homophone of PI)

18. At last, one of European country gets rid of virus (3)

This is a missing letter clue. The actual clue uses "least" instead of "last"
ANY = at least one; GERMANY minus GERM

19. Come in and don't align rebel (5)

This is a missing letter clue. The actual clue uses "malign" instead of "align"
ENTER = come in; DISSENTER minus DISS

20. Mortal peasant holding money from Rajasthan, given away for free (6)

This is a tile clue. The extra definition is "given away for free"
PERSON = mortal; PEON containing RS. (short for Rupees)

24. Would Peru discharge? (5)

This is a phone clue. The actual clue uses "wood" instead of "would"
PECAN = wood; PE (country code of Peru) + CAN (as in, remove from a job)

26. Develop containment for recipe, safeguard joint (6)

This is a tile clue. The extra definition is "joint"
ENSURE = safeguard; ENSUE (develop) containing R(ecipe)

28. Liars are smart, pushing capital to the other extreme (5)

This is a phone clue. The actual clue uses "Lyres" instead of "Liars"
HARPS = lyres; SHARP with first-to-last shift

30. Times: 120 seconds after Egyptian god makes a cheer (3)

This is a horrifying numbers clue. The actual clue begins with "30 times..."
RAH = a cheer; 30 times 120 seconds = 3600 seconds = H(our), after RA

33. Non-IRL gay pair awkwardly makes beehives (5)

This is a missing letter clue. The actual clue uses "Non-girl" instead of "Non-IRL"
APIARY = beehives; (GAY PAIR minus G)*

34. Otherwise, sew a vein for tax crime (7)

This is a phone clue. The actual clue uses "so" instead of "sew"
EVASION = tax crime; (SO A VEIN)*

35. Adequate enormous .png file, maybe? (4)

This is a tile clue. The extra definition is "Adequate".
EPIC = enormous; a .png can be said to be an E-PIC (using E- as a prefix indicating digital technology)

37. Large BLT swallowed by Scottish river flood (6)

This is a missing letter clue. The actual clue uses "bolt" instead of "BLT"
DELUGE = flood; LUG (large bolt) swallowed by DEE (Scottish river)

38. White House's soldier looking back by restroom for a bit (5)

This is a missing letter clue. The actual clue uses "Brit" instead of "bit"
IGLOO = white house; GI< + LOO

39. Superhero lost extraterrestrial amateur actor Harris (5)

This is a tile clue. The extra definition is "actor Harris"
GREENHORN = amateur; GREEN HORNET minus E.T.

40. Old myth of a sugar-addled ditching (4)

This is a horrifying numbers clue. The actual clue ends with "...ditching 4"
SAGA = old myth; (OF A SUGAR)* minus FOUR

DOWN

1. Idol thumbs-ups chefs without company (3)

This is a tile clue. The extra definition is "Idol"
OKS = thumbs-ups; COOKS minus CO

2. Ancient tongue's note audible twice (5)

This is a horrifying numbers clue. The actual clue ends with "...twice 5"
LATIN = ancient tongue; LA (note) + (TEN homophone)

4. Revived voiceover again, making 'buoy' into 'sign of love' (3)

This is a phone clue. The actual clue uses "boy" instead of "buoy"
REDUX = revived; REDUB (voiceover again) exchanging B for X

5. Untidy torte for track event (4)

This is a phone clue. The actual clue uses "tort" instead of "torte"
TROT = track event; (TORT)*

6. Tug on that man, Ash, hiding (5)

This is a horrifying numbers clue. The actual clue ends with "...hiding 5"
HEAVE = tug on; HE (that man) + AE (ash ligature) containing V

7. Shot major annoyance with bread (4)

This is a missing letter clue. The actual clue uses "Short" instead of "Shot"
PITA double def. for 'bread' and as short for "pain in the a**"

8. "Trust" is a race without a foot covering (4)

This is a tile clue. The extra definition is "foot covering"
RELY = trust; RELAY minus A

11. Mowed, by the sound of it, an estate (5)

This is a phone clue. The actual clue uses "mode" instead of "mowed"
MANOR = an estate; homophone of MANNER (mode)

13. Group of snowmen's sure thin after eating The Thing while upset (5)

This is a missing letter clue. The actual clue uses "thing" instead of "thin"
YETIS = group of snowmen; YES (sure thing!) eats IT<

15. Reject pivoted to gain real notoriety (4)

This is a tile clue. The extra definition is "notoriety"
SPURN = reject; SPUN containing R (real)

16. Vital element besieged by brewed optical infection (4)

This is a phone clue. The actual clue uses "brood" instead of "brewed"
PINKEYE = optical infection; KEY (vital element) surrounded by PINE (brood)

17. Old before the Queen, ultimately (3)

This is a horrifying numbers clue. The actual clue ends with "... ultimately 3"
ERE = old before; ER (Elizabeth Regina) + (thre)E

21. Order engineered parts for handle (5)

This is a tile clue. The extra definition is "order"
STRAP = handle; (PARTS)*

22. Inquisitive MC, maybe, in the Big Apple (4)

This is a missing letter clue. The actual clue uses "Mac" instead of "MC"
NOSY = inquisitive; OS (Mac, maybe) in NY

23. Knight's partner: king with rheum in church (5)

This is a phone clue. The actual clue uses "room" instead of "rheum"
KNAVE = Knight's partner; K + NAVE

24. Oddly plumbs small places for pits (4)

This is a missing letter clue. The actual clue uses "pints" instead of "pits"
PUBS = places for pints; odd letters of PlUmBs + S (small)

25. Covers fro and hears a solver's cry (3)

This is a missing letter clue. The actual clue uses "from" instead of "fro"
AHA = solver's cry; first letters from "A[nd] H[ears] A"

27. Dynasty is source of food to the audience (5)

This is a missing letter clue. The actual clue uses "flood" instead of "food"
REIGN = dynasty; RAIN homophone

29. Lot on the record for abbey boss (5)

This is a missing letter clue. The actual clue uses "Blot" instead of "Lot"
PRIOR double def; abbey boss, and blot on one's (criminal) record

30. Type of colorblindness relating to Nu's gender (3-1)

This is a missing letter clue. The actual clue uses "nuts" instead of "Nu's"
RED-GREEN = type of colorblindness; RE + (GENDER)*

31. Aura formed by killer computer and lad, obviously (4)

This is a missing letter clue. The actual clue uses "lead" instead of "lad"
HALO = aura; killer computer = HAL + O(bviously)

32. Dear gets accomplished (4)

This is a phone clue. The actual clue uses "deer" instead of "dear"
DOES double def.

35. God's breakfast food (3)

This is a missing letter clue. The actual clue uses "Goad's" instead of "God's"
EGG double def.

36. On the way without ale in tooth (3)

This is a missing letter clue. The actual clue uses "male" instead of "ale"
COG = tooth; COMING minus M(ale) IN

Extracting WORD 1

In order, the missing letters spell JOEMGORRGANMLBTEAM. This gives the clue "JOE MORGAN MLB TEAM" for WORD 1=REDS, with two extra letters GR

Extracting WORD 2

The 'number of rings in the original transmission' is the number of "O"s in the original word. In order, the extracted letters spell DOTOEORUMIND. This gives the clue "DOTE OR MIND" for WORD 2=CARE, with two extra letters OU

Extracting WORD 3

In order, the extracted letters spell NAMEND. The text instructs us then that WORD 3=NAME and the two extra letters are the last two letters, ND

Extracting WORD 4: Shaded cells category

The shaded tiles with full words are "RED", "PINK", and "GREEN"; this suggests that the category is 'colours'.

Extracting WORD 4: Extra definitions word square

There are nine clues with extra definitions. In order, they are:
3A: "Analyse" = SIFT
20A: "given away for free" = COMP(lementary)
26A: "joint" = KNEE
35A: "Adequate" = OKAY
39A: "actor Harris" = ED
1D: "Idol" = IKON (var.)
8D: "foot covering" = SOCK
15D: "notoriety" = FAME
21D: "order" = TYPE
These arrange into the word square (with Across clues across and Down clues down)
SIFT
OKAY
COMP
KNEE
with extra two letters ED from the unused clue.

Extracting WORD 4: Finding the set of rules, the path, and the final WORD 4

The letters from the regular portion of the clue that are closest to the extra definition, in order, spell UNDERTALE.
In the game Undertale, there is a puzzle involving multi-colour floor tiles, and rules for how to traverse a maze of them. http://undertale.wikia.com/wiki/Multicolor_Tile_Puzzle

Starting in the top left, we are on Orange, which immediately sets our flavour to Orange. We can't go right to RED, so we go down to PINK. We can't go down again to Red, so we go right to PInk. We can't go up to RED, and we can't get through Blue while our flavour is still Orange, so we go right; we slip over the PUrple tile (setting our flavour to Lemon), onto the Yellow tile, which immediately send us back over the PUrple tile to the PInk. Now we can go down to the Blue without being sent back, so we do. From here, we go right to PInk, down to Green, and right to Green to end in the bottom-right corner.
Overall, our moves were DRRRLLDRDR. On the word square, we trace the path SOKAYAKOMEE; starting with the first letter and taking every third letter thereafter, we get WORD 4=SAKE.

Putting it all together:

Putting the WORDs together, we get REDS CARE NAME SAKE or "Red Scare namesake", a clue to Sen. Joseph MCCARTHY. Now, we use the well-known equation $E=mc^2$ to transform this name into a word describing xkcd's humour: EARTHY.
The two extra letters from each part, in order, spell out a word that doesn't describe xkcd's humour: GROUNDED.
Two seemingly connected words, but given how racy and space-y xkcd can be sometimes, I wholeheartedly agree: it's EARTHY but not GROUNDED!