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I'm building an ice elemental based on the iron golem. I want to give it an ability that regenerates 1d10 hp at the beginning of each of its turns as long as it is in a cold environment. I'm trying to figure out what affect this would have on the CR if any.

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Increase the CR by 1

Regeneration is simply converted to more HP for CR calculation:

Increase the monster's effective hit points by 3 x the number of hit points the monster regenerates each round. - DMG 281

The defensive CR of the Iron Golem is 11.5 (10 from HP, 1.5 added for AC). With regeneration the base is 12 now, AC means the same modifier.

As final CR is the average of defensive and offensive CR, it increases by 1.

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