The game is Chinese Checkers (or Sternhalma).
Standard Chinese Checkers board:
Could be a variant board (some travel sets compress to 12 per star) or a different game like Star Checkers or a solitaire marble puzzle (Peg Solitaire has 33 holes, cross-shaped).
Search "Chinese Checkers board layout" — if that's it, the "12" is a misremembered 10. If not, describe the track shape (star vs. cross) and I'll identify it!