Note that the naval signal flags in the examples in the first three sections spell SEMAPHORE. Indeed, each section is a different mechanic for cluing a letter in semaphore. The examples in the first three section should confirm that this is the right idea, but solvers are on their own to figure out the precise mechanic in the last three sections. Here are the different mechanics:
Read the hour hand as the direction of the signaller's left-hand flag (right to the viewer) and the minute hand as the direction of the signaller's right-hand flag.
Read the coordinates as points on the unit circle, giving first the signaller's left-hand flag then the signaller's right-hand flag.
Read the locations of the indicated wedges in Wheel of Fortune, giving first the signaller's left-hand flag then the signaller's right-hand flag. The wheel is given in the orientation shown on the Wikipedia page for Wheel of Fortune (as of 2019):
(The examples provide enough information to orient the wheel.)
Treating Kansas as the base point, read the direction of the indicated state, giving first the signaller's left-hand flag then the signaller's right-hand flag.
Treating the letter X on a standard Mac keyboard as the base point, read the direction of the indicated key, giving first the signaller's left-hand flag then the signaller's right-hand flag.
"Starburst display" is another name for the standard 14-segment display. Carrying out the indicated set operations on lit segments results in two lit segments, yielding a semaphore letter. The letters are displayed as shown on the Wikipedia page for the Fourteen-segment display (as of 2019):
(Note that, in every mechanic where relevant, the signaller's left hand is clued, then the signaller's right hand.)
The non-example lines are labeled from 1 to 27. Put them in order and read the semaphore letters:
| Number | Clue | Mechanic | Directions | Semaphore Letter |
|---|---|---|---|---|
| 1 | | We're not in Kansas anymore | E-NE | W |
| 2 | altZ | X marks the spot | SW-W | H |
| 3 | N∩Y ∪ W∩Z | Starburst display | SW-NW | I |
| 4 | | We're not in Kansas anymore | N-NW | T |
| 5 | D⌘ | X marks the spot | NE-S | E |
| 6 | K ∩ (R - C∩F) | Starburst display | SE-W | S |
| 7 | | We're not in Kansas anymore | NE-W | Q |
| 8 | DA | X marks the spot | NE-NW | U |
| 9 | T∩Y ∪ W∩Z | Starburst display | S-SW | A |
| 10 | 3:45 | Look at the time! | E-W | R |
| 11 | (√2/2, √2/2) (0, -1) | Going in circles | NE-S | E |
| 12 | Free Play $500 | Wheel of Fortune | SW-NW | I |
| 13 | | We're not in Kansas anymore | SE-SW | N |
| 14 | ⌘Z | X marks the spot | S-W | B |
| 15 | K∩Z ∪ W∩Z | Starburst display | NE-SW | L |
| 16 | 1:55 | Look at the time! | NE-NW | U |
| 17 | (√2/2, √2/2) (0, -1) | Going in circles | NE-S | E |
| 18 | $500 $800 | Wheel of Fortune | NW-W | O |
| 19 | | We're not in Kansas anymore | SE-SW | N |
| 20 | ⌘alt | X marks the spot | S-SW | A |
| 21 | S∩X ∪ T∩Y | Starburst display | S-NW | C |
| 22 | 11:45 | Look at the time! | NW-W | O |
| 23 | (1, 0) (-√2/2, -√2/2) | Going in circles | E-SW | M |
| 24 | $2500 $800 | Wheel of Fortune | N-W | P |
| 25 | | We're not in Kansas anymore | S-SW | A |
| 26 | Z | X marks the spot | SE-W | S |
| 27 | K ∩ (R - C∩F) | Starburst display | SE-W | S |
This yields the phrase WHITE SQUARE IN BLUE ON A COMPASS. A white square in blue is the naval signal flag for P:
The semaphore representation of P has the signaller's left hand straight up and right hand straight out to the viewer's left. On a compass, this would be NORTH WEST, which is the answer to the puzzle.