The rules for an ability check are found on pg26 of the PHB. It basically states that a situation such as breaking down a door would apply your Strength Modifier + 1/2 level + any other applicable modifier.
My friends enjoy putting the most extreme scenario's in order to illustrate themselves...but it works. So based on this logic, they indicate that Stephen Hawking (high level character based on his experiences in life) would easily defeat a typical well built adult male in arm wrestling simply because he would add his half level bonus to the check.
Is the door stuck? Who needs a level 1 Minotaur when that level 20 dext based halfling has a far greater ability score check.
Can anyone help justify why they get a half level bonus to ability scores? In my mind, 1/2 level bonuses should be removed from all ability score checks. If this was done, would it unbalance or destroy any game mechanics?
The following is a bit of an aside where I get these questions from that people may enjoy:
I play with two D&D groups, my 4e group that I DM, and a 3.5 group who has not played 4th edition and hates the sound of everything <IE; Monsters utilize different rules from PC's, Saving throws are static and don't change based on level, you have to roll for Fort/Ref/Will, no BAB means a wizard can progress in the same manor as a fighter, multiclassing takes feats, classes feel to restricted with choices of A or B rather than anything you want, etc> . As such, I tend to try and defend some of the points they don't like about 4E. The above issue was brought up and I had no defense for it.
For the specific example of arm wrestling: Can you live with a Str8 Halfing beating a Str20 Orc in arm-wrestling? Because that's what you get if you allow rolling at all. If you don't like this, don't roll. Just compare the ability scores, highest score wins, ties result in a roll (and in that case, why shouldn't experience help?). That's a problem already known from 3E.
Likewise, if a 3E Bard uses Inspire Competence, for some reason a character can get stronger and break down a door he normally would have been absolutely unable to break down (regardless of the roll). Can you accept that? Or does it get a free pass because it's a supernatural effect?
It is not described what that half level bonus actually represents in the game world. For a Wizard, it can represent training and experience (like for his Arcana or Intelligence checks) - or it represents subtle magic (or heck, maybe even unsubtle) magic he uses to aid a task. If he breaks down a door, he looks at it closely, identifies the weak spots and speaks a small incantation, that will weaken it for a moment, and then he just has to apply the little natural strength given to him to break it down.
So, the Wizard gets the "free pass" from his supernatural (magic) abilities.
A Fighter making an Int check? He has seen so much, and experienced so much, of course this experience will count for something. Sure, he might not be the brightest man around, but he has learned a few tricks that surprise a less-experienced person, and he might be able to out-plan a Stephen Hawking.
Speaking of Stephen Hawking - this guy is a genius in our world. But he will not fight against Dragons or Ninjaes (unlike Einstein).
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So, he's an NPC that's not even a monster. He can have any modifier in the Science skill I want/need him to have. (Particularly since "Science" is not a skill in the core rules anyway)
If he was a D&D character, he would probably be a wizard. In that case, see above.
I think this explains it well.
I like to use the example of an 8 strength wizard swimming. The wizard doesn't swim well naturally, but he knows the exact body movements to conserve his strength, he mutters a quick magic incantation to boost his swimming knowledge, and oh he's epic....the god of oceans owes him a favor, etc.
In fact, skills and ability checks to me are the hidden power of 4e. Its not as flashy and overt as 9th level spells in 3rd, but the reality of 4e is that an epic level character is just plain awesome in nigh everything he does.
The only thing I don't particularly like about ability checks is my ability to "take 20" on most of them. For example, my current 6th level fighter can destroy any normal ADAMANTINE door if I try enough. I don't mind a character being able to take 10, but taking 20 can lead to even low level characters defeating obstacles that are supposed to be tougher.
Take 20 is the shorthand for "I try until I roll high enough to succeed". Take 10 is actually far more artificial, because it can guarantee a success even in situations where you could not retry anything (and it also guarantees a failure even if you could succeed at a task when rolling).