But how do you roleplay problem solving, without any system/dice assistance?
Lets say I introduce a puzzle that requires knowing how to manipulate an advanced mathematics formula (I wouldn't, for various reasons but let's say I do). The player of the "smart" character in the group (the character has the requisite knowledge skills) has an English background and avoids advanced math like the plague, how is he going to roleplay solving it?
Your example is not exactly general "problem solving" when your talking about doing a task like a math puzzle. If you would do this in a game, then sure your just going to do a "roll a check DC 10" type thing.
Though if I was doing a problem to solve, I would make it a problem that the player(s) had to solve for real. I don't use math ones often, I like more logic puzzles.
What about the shy player who has no "real life" charisma or social acumen wanting to play the dashing and socially gifted bard?
I explain to this player that they can't just casually with no effort role play "act" a character. If they want to role play "act" like something they are not, it will take practice and a lot of effort.
Though, a player is free to play a character however they wish on a whim; and when something comes up in the game that a socially gifted bard might do, they can make the mechanical rule roll to 'role play' "playing the game type". They can then be happy they 'role played' a socially gifted bard, by making the roll.
Does that player have any less right to want to (and be able to) play that character than the player who can't even bench the bar without any weights on it gets to play the STR 20 barbarian who can kick down obstacles with ease?
There is a huge difference between Role Playing "acting" and Role Playing "playing an RPG".
If your fine as counting "I rolled a d20 a couple times" as "role playing a character" , then you can do that in the game and it's all good.
Again, as above, to Role Play "acting" takes practice and effort
By extension then, you wouldn't let someone who can't pick a lock or scale a wall play a rogue?
Someone who can't properly swing a longsword play a fighter?
And, of course, if the player can't properly perform the gesture for magic missile - wizard is right out!
When your talking about a simple direct task, and like in your examples physical acts, your talking about very small amounts of time. A character does the action for seconds, or maybe a whole minute. But the player then is still 'playing the character' for all the hours of the gameplay.
And this is the big point. When a mechanical event comes up you can 'play' your character as charismatic for that short time as you roll the dice. But for the whole rest of the gameplay: the hours and hours of gameplay, the player is on their own to role play 'act' as whatever they want their character to be. And this is the bulk of the game time. This is where a player would really 'role play acting', not the combined couple minutes of mechanical rolls.