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Don't quote me on this because I don't know the formula for it but it seems to be 0.20% chance per piece for an artifact to appear. Because let's think of it this way. If you put in 500 pieces all at once you are guaranteed to get an artifact. So by that logic, it's 0.20% chance per piece you put in added on to whatever progression you are up to.
For example: On a brand new statue, 0 pieces progress. You put in 100. That's 20% for an artifact. Now every single piece you put in, 101,102,103, and so on will be 20.2%, 20.4%, 20.6% to break.
Basically if you just want the item (and i'm warning you now, the item is most likely not very good but it's randomly generated, anything can happen), put in as little as possible and do the same process for each statue you come across.
The benefits of putting in more pieces is more stats on the item, but what's the say you're gonna get a crappy ass item with lots of stats, it's still crap.
So in the end I just do the portion donation method on every statue, get the item and hope for the best. Saving up to 500 is not worth it because by the end of the game you'll most likely just get/have something better. By that logic it's actually better to do it immediately to use an item now than be pissed at something you get later.
Pro-Tip: use the ALT key to select only some of your lutefisk, do not put all 500 into one shrine or you will waste them. There is an arbitrary random number of fish you must reach on each shrine and going over that number just wastes lutefisk. I usually put 5 fish into the shrine at a time, it seems to be a good balance between conservation and time.