The same artifacts reappearing in Temple Run 2

There is the "Coupon Collectors Lemma" which estimates how long on average it will take to accumulate all of a set of n items, assuming each is equally likely, which gives about n log n steps. In the case of collecting 10 things, this works about to about 30. So generally by the time you get all 10 things, you'll have on average 3 of each; with some more, some less, of course. The variance is significant (resulting in a standard deviation of about 12, meaning there is a good chance it could be anywhere in the 15 to 50 range instead of the expected 30.) For 5 items (relics, holiday artifacts,...), the numbers are about 11.5 for the expected mean and about 6 for the standard deviation.

Note that when there are multiple objects (rings, masks, critters, etc.) this analysis applies just to that category. That is, on average, by the time you have all 10 rings, you will have collected 30 rings.

Note that one of the updates (Dec 2014) included a comment about adjusting the artifact probabilities to make it more likely to get ones that you don't already have. The analysis above applies to the case where all items are equally likely (already collected and uncollected alike) and from the update note, it sounds like they increased the likelihood for the uncollected ones. So depending upon how much they adjusted things, the averages could be a little or a lot lower. My speculation is that they did this in response to people complaining that it could take quite a long time to get that last item.

There is absolutely no way to trick the game into obtaining a specific artefact rather than those you already have. Carry on, you will end up getting them all !