Figure 2 (Switches) Commentary (Compared to Fig.1 Above) New window.
In puzzle books they sometimes have two pictures for you to look for the small differences between them. I hope you like those. The switches and most of the cables in FIG.2 below are the same as in FIG.1 above that you just got intimately acquainted with. But most connections are different. Take a look at where the big blue wirenut was. Instead of nine wires there are only three there now. What has happened? For one thing (again starting from left to right) one cable-p is now designated as cable-d, which means it is a 3-wire cable bringing two circuits ("double") into our box, with their one shared neutral. If you follow the new added red circuit, you see that it only ends up involving switch-13 and cable-f. We will assume the rest of the switches and cables still run on the black circuit (whether or not their connections to the black circuit happen in our big box).
13- Cable-f and switch-13 are still doing what they always did -- back when it was switch-3 above. But the hot must have its own connection -- one of three new wirenuts -- to keep it separate from the black circuit's hots, which are 240 volts at odds with this red hot.
11- Even basic switch-1 is now connected differently as switch-11. Its lower terminal is not only getting a hot black from the wirenut as before, it is providing a new place for second-position cable-p's black to get connection to hotness -- rather than at the wirenut. Since putting two wires under one screw terminal makes an unreliable connection, let's assume the diagram means that one of those two blacks is actually attached by poking into a hole-terminal on the back side of the switch. Switch-11 also has two cable-L's that it turns on now. (The cable in that position had functioned as a p; now it is an L.) Their blacks are using a pigtail from one of the new wirenuts, to get their connection to switch-11. Instead of a pigtail, I suppose they could have both connected directly to the top switch terminal in the fashion shown at the lower terminal.
12- Switch-12 and cable-h are functioning as always, but their white wire is red part-way. This painting or taping is a current Code requirement, so that the white is not mistaken for a neutral. But to keep from assuming it is part of a 3-wire cable, a troubleshooter will want to doubt such reds, or even blacks, by looking closely at the color where the cable comes into the box. Just as you might see that someone had dyed their hair by looking at the roots.
14 & 15- Speaking of whites that are not neutral, look how cable-m no longer has a neutral. Now its white is doing something else. In fact, it is doing what switch-12's "reddish" white does, but for two switches. The white is a hot wire coming down from a light-fan or such, to power up the two switches. We still designate the cable as "m," even though its use is reminiscent of 2-wire h-cable.
16- Take a look at switch-16. Something different is going on... As #6 in Fig.1, it was acting as the leg-end of a 3-way system. Now it is the hot-end. Cable-t still carries red and black as travelers and the white as neutral, but this time the neutral is coming from our box, and the leg cable (not the L shown here!) for this 3-way system is going to the light from another switch box than ours. Cable-L above switch-16 no longer belongs to 16...
17- ...It is being lit up by switch-17, which is no longer the hot-end of its 3-way system; it is the leg end. Our box could be where that cable-L would get its neutral, via the wirenut that is also feeding neutral to cable-t. But L's white wire is not in that nut! Did I goof up? I've been known to, but in this case this "White to Nowhere" is just x-tra. As author of this scenario, I know that the light which cable-L is going to, will have a neutral from another source. If all whites that were not hot, switched, or needed as neutrals were always hooked to neutrals, we would get some situations of "parallel conductor" paths for current to run on. Though not generally dangerous, it makes for confusion -- maybe more confusion than seeing a white wire cut off by itself here and there! Then why is that tempting wirenut even there near cable-L? Couldn't cable-t's neutral have gone directly to the big wirenut in the lower left of our box? Yes, but let's say there wasn't enough wire to reach. It is OK to extend wires in a box or panel this way.
18- Switch-18 now relates to a 2-wire cable in addition to its original 3-wire cable. Still true to n-ness, cable-n's white is Not Neutral. So what is going on? The two-wire cable-h is bringing a hot from a light box (or switched receptacle) and "returning" the switched result (whatever the 3-way switches decide) on its other wire. Which wire is the hot and which is the return leg? If it is connected according to Code at the light, the black of two-wire cable-h should be the return leg for the light; therefore our switch-18 is the leg-end of the system and its partner elsewhere is the hot-end; therefore the non-traveler wire in cable-n (that is, the black) is that hot, provided via the white of cable-h from the light.
19- Switch-19 (a 4-way switch) is the middleman of its 3-way system, but now using 2-wire cables instead of the 3-wire cables of switch-9. No hots, legs, or neutrals are needed here; they are being taken care of in other boxes and cables of the 3-way switch system. Life is simple after all. Travelers in, travelers out; which pair is incoming and which outgoing could be determined but is not usually important, even if instructions from the switch manufacturer seem to say so. If you want more challenge, how would a 4-way switch be hooked up when the two cables were n-n? How when they were t-L-n or n-n-h?
A good review now would be to go back and account for what became of six black wires that used to be in the big wirenut in FIG.1.