There are 10 rings one need to find in order to complete a game's objective. Me and my daughter just finished finding them all. She was really happy to see all rings.
When do they pop out? There is no pattern as to when the treasure chest (artifact) will appear. It can show after a few kilometers or it can also be as if infinity. But one things for sure, in every game, there will always be one. I'm not sure however if you can get 2 artifacts in a game, we haven't tried it yet.