Here is the full list of all 21 PUZZLETIME: Lovely Girls achievements.
Complete Mei's first puzzle
Complete Mei's second puzzle
Complete Mei's third puzzle
Complete Mey's fourth puzzle
Complete Kimiko's first puzzle
Complete Kimiko's second puzzle
Complete Kimiko's third puzzle
Complete Kimiko's fourth puzzle
Complete Rika's first puzzle
Complete Rika's second puzzle
Complete Rika's third puzzle
Complete Rika's fourth puzzle
Complete Izumi's first puzzle
Complete Izumi's second puzzle
Complete Izumi's third puzzle
Complete Izumi's fourth puzzle
Complete Akemi's first puzzle
Complete Akemi's second puzzle
Complete Akemi's third puzzle
Complete Akemi's fourth puzzle
Complete the game