Here is the full list of all 27 4X4 achievements.
Stage1 clear_Green
Unlock stage2
Stage2 clear_Green
Unlock stage3
Stage3 clear_Green
Unlock stage4
Stage4 clear_Green
Unlock stage5
Stage5 clear_Green
Unlock stage6
All stage clear_Green
unlock Infinite mode
Record 180 seconds in infinite mode
Record 540 seconds in infinite mode
Record 1080 seconds in infinite mode
Stage1 clear_Blue
Stage2 clear_Blue
Stage3 clear_Blue
Stage4 clear_Blue
Stage5 clear_Blue
Stage6 clear_Blue
Stage1 clear_Red
Stage2 clear_Red
Stage3 clear_Red
Stage4 clear_Red
Stage5 clear_Red
Stage6 clear_Red