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Mr. Cole , his sister, his son, and his daughter are chess players.

I. The best player's twin and the worst player are of the opposite sex.

II. The best player and the worst player are of the same age.

Who is the best player?

Sounds like the best player is

the daughter

And that

Mr. Cole's sister is NOT his twin -- he is much older. She is the worst player, and the same age as her twin niece and nephew.

EDIT to argue against the other answer:

I disagree, because, the best player has a twin, and is also the same age as the worst player -- but the worst player and the best player's twin are of the opposite sex, therefore NOT the same person. Therefore there are three people of the same age.

EDIT again: As for it being unrealistic to have this age difference -- well, not at all. Happens in my own extended family.

There isn't enough information to solve this puzzle.

because

the two age groups are indistinguishable. If the best player could be Mr. Cole, it could also be his son. If it could be Mr. Cole's sister, it could also be his daughter. I don't even need to attempt the puzzle to know this.

But I will attempt it anyway, and

it can literally be anyone, because I realized there isn't really a difference between the genders in this case either.

EDIT: Yeah, I have a reason for why the other response is incorrect as well.

Think about the ages of Mr. Cole's parents. Assuming a minimum age of 20 or so to have kids (this is being a bit generous), Mr. Cole's parents are at least 40 when Mr. Cole's sister is born (even this is being generous), which is starting to become a little old. There's just too little information to determine whether this is true or not.