One of my players is playing a mounted character, and is claiming that since a lance is a two handed weapon, that it deals double Strength damage, even when wielded one handed while mounted. While this doesn't make sense, does it follow the rules as written?
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The lance does double damage if used from the back of a charging mount.
This has nothing to do with the separate rule that if you wield a one-handed weapon two-handed, or a two-handed weapon, it does 1 1/2 x STR damage (not 2x).
I will note that in Hero Lab it correctly simulates this behavior. My cavalier with a +4 STR mod, when mounted, can hold in one hand for d8+4 or two for d8+6. If I am charging, this becomes 2d8+8 one-handed or 2d8+12 two-handed.
Furthermore, this isn't some strange superheroic fighting style - before the development of the vamplate, holding on with both hands was really the only way to employ a spear or lance on horseback. Also note that lances aren't fragile or designed to break (jousting lances perhaps, but not ones made for war).
The Lance has this special quality:
Note that wielding the lance one-handed is optional. You can, not you must.
The Ride skill gives you the following benefit:
So yes, RAW you can wield a lance with two hands, while guiding your mount with your knees.
However to do so, you must still meet all the normal requirements of a two-handed weapon. So, no shield. You must be guiding your mount with your knees to have the weapon ready to strike. Etc.
Here's what the rules say about two-handed weapons:
Note that the higher strength bonus is a function of wielding a weapon two-handed, not of using a two-handed weapon. So wielding a two-handed weapon one-handed (as can be done with the lance) doesn't grant a higher strength bonus (if he wants to use a shield, for example).
(And to clarify: One-handed vs. two-handed changes the strength bonus from 1x to 1.5x. The double damage from using a lance on a charging mount is always present, whether wielded in one or both hands)