Yes it does, there are 4 versions of cards I know, 2 of which have white strips, but you will have to double check by downloading the Round1 app, using the kiosk at Round 1 or asking someone that works there next time you visit.
Depends on how the game card looks nowadays because every time I go to Round 1 Bowling with a group of people my age we always go bowling over playing arcade games. So, to answer your question I'm not quite sure if this location in Lakewood, California still accepts game cards with the white stripe on it.
Show me the card
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