Can/May (能 vs 可以)

能 (néng) and 可以 (kěyǐ) are both modal verbs meaning "can," but they differ in nuance. 能 focuses on ability, capacity, or possibility, while 可以 focuses on permission or acceptability. In many situations they overlap, but in certain contexts only one is correct.

能 is used for: (1) Physical or innate ability — "他能跑十公里" (He can run 10 kilometers). (2) Circumstantial possibility — "明天你能来吗？" (Can you come tomorrow? — Is it possible given your circumstances?). (3) Capacity after recovery — "我的手好了，能写字了" (My hand is better, I can write again).

可以 is used for: (1) Permission — "我可以进来吗？" (May I come in?). (2) Feasibility or suitability — "这里可以停车" (You can park here / Parking is allowed here). (3) Suggestion — "你可以试试这个" (You could try this).

For negation, there is a critical difference: "不能" means "cannot" (inability or prohibition), while "不可以" means "may not" or "not allowed." To deny permission, use 不可以 or 不能 (both work). But 不会 is used to say someone lacks a learned skill ("I can't swim" as in "I never learned" = 我不会游泳), which is distinct from both 能 and 可以.

At HSK 1, you learned 会 (huì) for learned abilities. Now the full picture is: 会 = learned skill, 能 = capacity/possibility, 可以 = permission/feasibility.