In searching for answers to this, I've come across an interesting point on the Paizo forums. If you are in one of the far corners and you consider these not threatened, then you can diagonally move towards the character and never provoke an AOO (that is, per RAW)... and yet, logically, a threatened area should make an uninterrupted circle around the creature. This may explain why 3.5 made an exception out of this.
Yes, that is precisely why 3.5e made that exception, and it’s also why Paizo issued an official FAQ that changed Pathfinder’s rules to add the same exception, as @caps reports in this fine answer that you should go upvote.
Thus, a reach weapon can attack the following \$X\$’s from \$C\$:
\begin{array}{c|c}
\phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} \\ \hline
& X & X & X & X & X & \\ \hline
& X & & & & X & \\ \hline
& X & & C & & X & \\ \hline
& X & & & & X & \\ \hline
& X & X & X & X & X & \\ \hline
\\
\end{array}
Before the FAQ change
However, the FAQ entry that caps reports did not exist at the time this question was asked. At that time, you did not get the four corners, and so could only attack these \$X\$’s from \$C\$:
\begin{array}{c|c}
\phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} & \phantom{X} \\ \hline
& A & X & X & X & & \\ \hline
& X & B & & & X & \\ \hline
& X & & C & & X & \\ \hline
& X & & & & X & \\ \hline
& & X & X & X & & \\ \hline
\\
\end{array}
Here, it would appear that someone could step from \$A\$ to \$B\$ to avoid an attack of opportunity altogether. However, even before the FAQ just changed this to be like 3.5e, the developers at Paizo had ...for lack of a better word, we’ll call it a clarification, though it honestly just confused me more. From an earlier FAQ:
Can you or can you not attack diagonally at a distance of 2x squares (15'=10' exception) with a reach weapon?
James Jacobs: Nope. A reach weapon gives a specific extension to your reach. When you count out squares, since every other square is doubled when you count diagonally, that means that there’ll be corners where you can’t reach.
Sean K. Reynolds: It's an artifact of the grid. The closest the rules come to addressing this is in Large, Huge, Gargantuan, and Colossal Creatures, which says:
Unlike when someone uses a reach weapon, a creature with greater than normal natural reach (more than 5 feet) still threatens squares adjacent to it. A creature with greater than normal natural reach usually gets an attack of opportunity against you if you approach it, because you must enter and move within the range of its reach before you can attack it.
So just because the grid has a square for "15 feet away" and a square for "5 feet away," but no square for "10 feet away," using that corner path doesn't mean you're magically teleporting from 15 feet to 5 feet; you are passing through a 10-foot-radius band around the creature, and therefore you provoke an AOO.
Admittedly it's not clear, and obviously it doesn't have the diagram in the 3E book to provide a non-textual example, but it's supposed to work as I described above.
Basically, the idea was, under the rules at the time, you didn’t threaten 15 ft. away, so you don’t get the corner, but you did threaten 10 ft. away and there’s no way to move from 15 ft. away to 5 ft. away without passing through a point that is 10 ft. away. Thus, someone moving from 15 ft. away on the diagonal to 5 ft. away on the same diagonal was going to provoke even under these rules.
So the enemy at \$A\$ moving to the point marked \$B\$ towards \$C\$ with a reach weapon provoked an attack of opportunity (assuming this isn’t a 5 ft. step of course), because somewhere between \$A\$ and \$B\$, there is a point that is 10 ft. away from \$C\$ that the enemy has to pass through.
Presumably, you would have adjudicated the enemy’s position for the purposes of the attack of opportunity as being \$A\$, though this was never made clear. In this sense, the end result was identical to the 3.5e/post-FAQ version for movement towards you: creatures leaving that corner square to enter a square inside your reach provoked an attack of opportunity as if you threatened that square. You were not eligible to make an attack of opportunity if the enemy performs any other action that provokes from \$A\$, including movement in other directions, because you do not actually threaten it.
This was a headache. Even before the FAQ changed things to match 3.5e, that was precisely what I recommended:
Reach weapons are one of the few fairly-nice things that melee can get. There’s really no need to nerf them. I strongly suggest that you straight-up ignore this nonsense and use the 3.5 rule. The exception to the usual calculation of ranges in the case of reach weapons is weird, but clearly there was a good reason for it: without it, you wind up with this mess.
Best Answer
How do the Kobolds remember which parts are trapped?
Basically, this answer is about weaving the Kobold's own marking system into the narrative. It does assume you draw your own maps and don't use Dungeon Tiles or anything.
Obtain 6 or so pretty looking symbols (they don't need to have meaning, but if they look Draconic it's bonus awesome) Mark every square of the map with one of them. In the narrative, explain that the Kobolds did just that; they covered all the floors with all sorts of markings. For each room, assign 2 symbols to mean "trap" and the others to do nothing. Vary the symbols per room.
This will clearly signal to the players "this room is trapped", but then the word Kobold is basically a synonym for "traps" anyway, so that's okay. It will keep your players attention strongly on the traps, they will try to figure it out by watching Kobold movement (which is good! traps exist in the narrative to be interacted with) and will mostly stumble right into a trap in room 2 before they realise that the Kobolds vary which symbols mean trap in every room.
You could even tie a meaning to each symbol (in Draconic) and explain the words to players who speak it, and then have the chosen symbols make sense per room. Since Kobolds employ deadly traps in their home, they need a system to keep down their own losses, and that has to be simply enough to teach to their kids while still being confusing enough that confound and/or kill enemies.
To tie it even stronger into the narrative, include a room that the Kobolds use for training their young (probably one with few or no Kobolds in it, since they know it's less hurtful) where the traps just drop a bucket of water on their heads or launch sticky paper at them.
And maybe in the newest room, the Kobolds haven't had time to place traps yet, but they did place the markings. Your players will be properly trained by this point to move very carefully, figure out the system, and probably be amused when they realise the Kobolds have trained them like Pavlov's dogs to avoid certain squares, even if they don't do anything. (If they are trained so well they never trigger any, make sure they find a scrap of paper later that has instructions on which traps should go where that makes it clear none of them have been installed yet. Remember: if you don't tell them, it doesn't exist.)
(Regarding the last part, I once trapped my players in a room inside the Temple of the God of Theater, by giving them a huge grid full of letters whose only effect was that certain letters made the floating balls of electricity in the corners light up ominously. They were told that the room was "very dangerous and none had ever traversed it". It took them an hour to figure out they were being played and they loved it.)