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Say I have 2 influence and want the shotgun which is 4 successes. If I roll 2 successes, can I spend 2 clue tokens to reroll my success to attempt to get the shotgun or does rerolling a die remove that success and in theory I wouldn't be able to get the shotgun with my default influence and without a loan?

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You only count successes once all rerolls have been made. This can be inferred from this clause in the rulebook:

Rerolling Dice
Immediately after rolling dice during a test, an investigator may spend a Clue token to reroll one die. He may perform this multiple times, as long as he spends a Clue token each time. He determines whether he passed or failed after resolving all rerolls.

(Emphasis mine)

Also, in the reference guide, you'll find:

When a die is rerolled, use the result of the reroll, even if the new result is worse.

So, clearly a reroll can remove a success if that dice is rerolled! This makes it entirely clear you are just manipulating the result of a roll of a fixed number of dice. You can never get more successes than the original number of dice in the pool.

This becomes even clearer when you remember that there are effects that allow an investigator to roll additional dice. If the intent was to allow rerolls to create additional successes, there wouldn't have been any need for the additional dice mechanic.

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    Thanks, that makes sense - this is more along the lines of what I was looking for in an answer. So just to be 100% clear - I would not be able to get the shotgun proposed in my example because rerolls would overwrite the success I have. – Howdy_McGee Jan 18 '16 at 16:00
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    @Howdy_McGee Yes, you do not get the shotgun. Of course, if you have two successes from your roll, you can just take on some debt and get the shotgun that way. – Kris Jan 18 '16 at 16:04
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Successes and failures are only counted after you finish rerolling.

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    Thanks for the answer! Do you have any references? Rerolls sound like sound like a reroll of the dice already in play but in this scenario it's acting an additional die. – Howdy_McGee Jan 18 '16 at 8:06
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    It would act like an additional die in the scenario you described. I have checked the reference and it's not clarified. I guess the creators never thought it needs to be clarified because rerolls work the same way in every single game that uses dice, of the many games I played. "Reroll", just as "roll" isn't a term specific to one game or another. It's just an english word, like "redo", "restart", "remake". Neither of these words mean "create another of the same object". It's "recreate an existing object". In reroll's case the "object" recreated is your die roll. – tsuma534 Jan 18 '16 at 8:17

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