
GRU |

Actually, the horse covers four squares (2x2 5' squares), as it is a Large creature.
Mounted characters take up the exact same space the creature they're riding does. In the case of a character riding a horse, the character occupies the same 2x2 square the horse currently occupies.
Thanks Cartigan & Zurai!
GRU
Quandary |

2 wide, 2 long, 2 high.
4 squares on the battle mat.
8 cubes in 3D space. This is why you get +1 for attacking from higher ground.
The rider is assumed to occupy all 8.
I play it as the rider of a horse occupies the upper 4 cubes of the horse´s 8 cubes, unless they´re doing a fancy maneuver by hanging below the horse or to one side (for cover) or whatnot (in which case it is whatever appropriate hemisphere).
This means that creatures hiding in trenches (below ground level) can´t be hit if the rider is wielding a longsword (and vice versa) unless lunge is in play. Basically it just matches up better to how I visualize different combat scenarios, i.e. opponents in trench or trap targetting ground-level cube or 0-reach opponents on foot/on ground CAN ONLY target the horse, while normal 5`reach opponents on foot/on ground can target either rider or mount as normal... And the mounted rider would need a reach weapon to target those sub-ground level opponents (like when riding over a low bridge or a causeway).
But as far as RAW is concerned, the rider should be considered occupying each of every 8 cubes.

ohako |
here's a follow-up question, along the same lines, but more annoying :)
Let's say I take Child of Winter and the Vermin Companion feats from the 3.5 Eberron setting. I pick as my animal companion a Giant Praying Mantis.
And now let's say I ride around on my giant mantis like a pony, and have it wear chitin armor barding (slightly larger mantis bits for armor, etc).
So, I'm in the same squares as it. Now let's say it hits something with its claws and starts a grapple. The grappled (really grabbed) bad guy is in the same squares as the mantis, as am I. Can the enemy make grapple checks against me, the mantis rider?

Quandary |

Well, since this is a thread in the PRPG Rules Question Forum, no you aren´t in the same square and neither is your mount nor it´s grabbed opponent, because that´s no longer how Grapple works. But since you are inhabiting the same squares/cubes as your Mantis Mount, as long as the opponent can target your Mount with an attack, it can target you. So... Yes

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here's a follow-up question, along the same lines, but more annoying :)
Let's say I take Child of Winter and the Vermin Companion feats from the 3.5 Eberron setting. I pick as my animal companion a Giant Praying Mantis.
And now let's say I ride around on my giant mantis like a pony, and have it wear chitin armor barding (slightly larger mantis bits for armor, etc).
So, I'm in the same squares as it. Now let's say it hits something with its claws and starts a grapple. The grappled (really grabbed) bad guy is in the same squares as the mantis, as am I. Can the enemy make grapple checks against me, the mantis rider?
I don't think the rider would be grappled. A grappled creature can only make attacks, including grapple checks, against opponents not in the grapple with one free hand, regardless of which square they are in. So the grappled enemy could try to grab the rider (initiating a grapple against him) but would take a -4 penalty (p. 200-201).