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So in this question Can I create +1 stat boosting item? we see it suggested the game shouldn't have items that give odd numbered boosts. What's with items like Circlet of Persuasion which do exactly that? Aren't they breaking that same rule/advice?"

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Nope. The question you linked specifically speaks of odd-numbered boosts to attributes (strength, dexterity etc). These bonuses will have a varying benefit for the player depending on their unmodified attribute: because of how attribute modifiers are calculated, a character with an odd dexterity score would benefit a lot from a +1 dexterity item, but a character with an even dexterity score wouldn't as their modifier would remain unchanged. Therefore you'd have a magic item that only works for some people and not others, with no real in-universe justification.

The circlet, on the other hand, doesn't affect one's attributes, it just gives a direct bonus to rolls. It works equally well for anyone using it without weird mechanical kinks.

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