If you say that someone has the wit to do something, you mean that they have the intelligence and understanding to make the right decision or take the right action in a particular situation. The information is there and waiting to be accessed by anyone with the wit to use it. = sense
If you pit your wits against someone, you compete against them in a test of knowledge or intelligence. He has to pit his wits against an adversary who is cool, clever and cunning
a message whose ingenuity or verbal skill or incongruity has the power to evoke laughter
You can refer to your ability to think quickly and cleverly in a difficult situation as your wits. She has used her wits to progress to the position she holds today
A mental faculty, or power of the mind; used in this sense chiefly in the plural, and in certain phrases; as, to lose one's wits; at one's wits' end, and the like
A person of eminent sense or knowledge; a man of genius, fancy, or humor; one distinguished for bright or amusing sayings, for repartee, and the like
If you describe someone as a wit, you mean that they have the ability to use words or ideas in an amusing, clever, and imaginative way. Holmes was gregarious, a great wit, a man of wide interests