@caps reports that an errata has fixed this issue, making reach weapons work just as they did in 3.5 (i.e. the solution originally suggested in this answer anyway). This answer is retained for historical purposes but is now out of date.
In 3.5, you’d get all of the
?s. If it weren’t for the exception made for reach weapons, and we calculated distances for them normally, you would miss out on the four
?s in the furthest corners:
The reason for this exception is pretty simple: the above figure does not make a continuous ring around the character
C. That means that someone could slip in through the corner, avoiding an attack of opportunity and defeating a lot of the purpose of having a reach weapon in the first place.
Pathfinder does not make this exception, and so this possibility of slipping through the corner becomes a problem. However, the developers at Paizo have ...for lack of a better word, we’ll call it a clarification, though it honestly just confused me more. From the FAQ, which Paizo likes to treat as official:
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 is, you don’t threaten 15 ft. away, so you don’t get the corner, but you do 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 is going to provoke.
E moving to the point marked
C with a reach weapon provokes an attack of opportunity (assuming this isn’t a 5 ft. step of course), because somewhere between
E’s square and
_, there is a point that is 10 ft. away from
E has to pass through.
Presumably, you would adjudicate this square as being the square
E starts in, before moving. In this sense, the end result is identical to the 3.5 version for movement towards you: creatures leaving that corner square to enter a square inside your reach provoke an attack of opportunity as if you threatened that square. You are not eligible to make an attack of opportunity if
E performs any other action that provokes from that square, including movement in other directions, because you do not actually threaten it.
This is a headache.
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.