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If I use a reroll, say from my character folio, can I take 10 on the reroll (assuming that it would have been legal to take 10 on the original roll, of course)?
Here is the situation that I am wondering about. My character has Climb -1. She is faced with needing to make a DC 10 Climb check, without distractions, so taking 10--while perfectly legal--won't get her up the wall she needs to climb. She rolls a 2, and I use my reroll. Since I have 1 star my reroll gets a +1 bonus, so her effective Climb skill is now +0. Taking 10, if allowed, would prove sufficient for her to bypass the troublesome wall.

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Interesting idea.
You're asking for two things.
(1) To take 10 on the second roll
(2) To use your GM bonus with the take 10
In the absence of an official rule, personally, I'd allow you take 10, but wouldn't allow the GM bonus on taking 10. My understanding is the bonus should be added to the random roll, not the total. Since you're taking a flat value of 10 for the roll (that's the idea of how taking 10 works), it doesn't apply.

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Interesting idea.
You're asking for two things.
(1) To take 10 on the second roll
(2) To use your GM bonus with the take 10In the absence of an official rule, personally, I'd allow you take 10, but wouldn't allow the GM bonus on taking 10. My understanding is the bonus should be added to the random roll, not the total. Since you're taking a flat value of 10 for the roll (that's the idea of how taking 10 works), it doesn't apply.
Hmm, that's an interesting take. Would you then say also that if the natural roll was altered that the following are true:
1) The GM bonus would never raise the result higher than what could be rolled on a natural 20 (so for instance if I reroll to a 17 on my Day Job with 5 stars, I would not get the results of 22 + my character's bonuses but instead 20 + my character's bonuses).
2) Since the actual roll was altered, this also increases the chance of a critical threat (so if I roll a 10 with a +5 star reroll, I threaten a critical with a keen scimitar).
I would take both 1 and 2 to be true with your interpretation. I don't think there's anything wrong with that, but if it's true I'd like to know since I do it differently.

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The Fox, this is how I run that at my table:
The magic shirt / folio re-roll does exactly that, re-rolls the dice. It doesn't re-set the entire action. If you used some sort of bonus before the d20 was rolled (like the current GM boon for running at conventions, or the Investigator's Inspiration points) then you have still spent that resource, and still gain its results.
Taking 10 is a decision. I won't let you change your mind on that decision. So, if you decided to Take 10 originally, you need to do so after using your magic shirt, and if you'd originally chosen to roll the d20, you'd do that on the re-roll as well.
Until this season, with the bonus for GM stars, there was never any reason to "re-roll" a Take 10. Now there is. So, at my table, if you have two GM-stars, you can (a) Take 10, (b) immediately announce a magic folio re-roll, (c) show me your PFS card, and (d) Take 10 with the +2 bonus.

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That seems fair enough, Chris.
Also, even without the GM stars, I think there is reason to allow take 10 on a reroll: Joe needs to make an Acrobatics check to jump across a chasm 15 ft. wide. No problem though, he has a +10 to that skill. He gets a running start, rolls, and the die comes up "2". Joe decides to reroll instead of lose a character to a 300-ft. drop.
His friend Amy remarks, "Joe, why didn't you take 10 on the first roll?"
Joe replies, "I'm new, what's that? I didn't know that was allowed." Or even, "I had a late night last night cramming for my medical school exams, I guess I spaced it."
Of course, you COULD force him to roll the second time. Personally, I would let him take 10, and I would be kicking myself as GM for not suggesting it in the first place (to be fair, I try to always make that suggestion to players, but often players roll the dice before I get the chance).

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I don't think those really come into it, Mark.
To keep it simple - if you don't roll a dice and instead choose to take the flat total of taking 10 instead of rolling for a random number, you don't get the bonus that comes with "a second chance to try a little harder".
That means the total between the dice and the GM reroll can go over 20, and only the natural number on the dice is taken into account for the purposes of critical attacks.
Again, I may be wrong, this is just my perspective and I imagine others would quite possibly easily either disallow taking 10 on the reroll or just allowing the bonus on a reroll regardless of an actual reroll or taking 10.

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His friend Amy remarks, "Joe, why didn't you take 10 on the first roll?"
Joe replies, "I'm new, what's that? I didn't know that was allowed."
"I'm new" is an exception to anything as far as retcons or allowances go.
We're assuming the average play experience with players who understand the basic rules.

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The Fox wrote:His friend Amy remarks, "Joe, why didn't you take 10 on the first roll?"
Joe replies, "I'm new, what's that? I didn't know that was allowed."
"I'm new" is an exception to anything as far as retcons or allowances go.
We're assuming the average play experience with players who understand the basic rules.
You're right. I retract that as an example. There is still the possibility that Joe simply spaced it. I would feel pretty bad making a player roll again if the second roll failed also and everyone at the table knew that taking 10 would have succeeded.

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I don't think those really come into it, Mark.
To keep it simple - if you don't roll a dice and instead choose to take the flat total of taking 10 instead of rolling for a random number, you don't get the bonus that comes with "a second chance to try a little harder".
That means the total between the dice and the GM reroll can go over 20, and only the natural number on the dice is taken into account for the purposes of critical attacks.
Again, I may be wrong, this is just my perspective and I imagine others would quite possibly easily either disallow taking 10 on the reroll or just allowing the bonus on a reroll regardless of an actual reroll or taking 10.
I think your viewpoint as expressed in your first post is certainly a valid possibility, but it seems it would produce the opposite results for going over 20 + modifiers and critical threats.
I can also see the "if you rolled the first time, you weren't taking it easy in character and taking 10, so the reroll shouldn't be a T10" perspective too, but then what if the player wanted to T10 for both the original and the second roll?

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but then what if the player wanted to T10 for both the original and the second roll?
Then he can take 10 for the reroll, but as there's no roll, should there be a GM bonus which is applied to rolls?
Although, it may be moot if you apply that same logic to even allowing a reroll on a take 10 as there is no roll to roll again.

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I'm curious. It seems the GM star bonus is an untyped bonus. Those of you who would not allow it to apply to a take 10, I assume you allow other untyped bonuses to apply, no?
So far, I think Chris has the most balanced interpretation, and closest to, in my mind, the spirit of both the reroll and the star bonus. I'm still interested in other opinions, however (in particular, Mark has teased out Avatar's take on it pretty well, without providing his own).

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Since taking 20 takes more time to complete, I wouldn't think a reroll would be eligible for it. I have always thought of a reroll as a sort of luck, divine intervention, whatever alteration that happens in an instant, not a rewrite of decisions or a chain of events. To say you are going to quickly climb the rope, fail, and then say I am going to redo it with a slower more meticulous form, I don't see that as much different than someone saying "Instead of a touch spell, I am going with magic missile." It just feel like too many things have changed to me.
Taking 10 has a similar feel, but with increased carefulness, by the line that you can't do it under duress, though it doesn't have the time clause. I am still inclined to view it the same way and say it is a different type of action than the random roll, but could be convinced otherwise on the 10.