Vfrdtyky

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.

I disagree [with an older answer, now deleted], 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.

As for [the claim, now deleted, that a generation gap between siblings is unrealistic] -- well, not at all. Happens in my own extended family.

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.

For a little brevity in this long winded answer, we name Mr. Cole's son Adam, his daughter Betty, and his sister Dzsesszika. We can restate the premise and the facts in a more convenient way.

If we assume there is no trickery with reassigned genders, gender pronouns, relativity, cloning, complete family information, tied skills, non-binary genders, or incest - we must conclude

the daughter is the best chess player

1. male_opposite = female


2. female_opposite = male

From the pronouns / identifiers we have

3. A_gender = C_gender = male


4. B_gender = D_gender = female


5. A_sibling = B


6. B_sibling = A


7. C_sibling = D


8. D_sibling = C

Cole is older than his children

9. C_age != A_age
   


10. C_age != B_age
    

We have (I) which also implies the best player is a twin

11. Best_age = (Best_sibling)_age
    


12. ((Best_sibling)_gender)_opposite = Worst_gender
    

We have (II)

13. Best_age = Worst_age
    

From (12) and single genders we know that

14. Best_sibling != Worst

From (11,13,14) we know there are three distinct people where

15. Best_age = (Best_sibling)_age = Worst_age
    

From (9,10,15) and complete family information we know that Cole could only be the same age as one other person (his sister), and thus

16. C != Best


17. C != Best_sibling
    


18. C != Worst

So the sister is the same age as the children.

From complete family information and (7,8,16,17) we know the best and their sibling must be the children; from (18) we know the worst must be the sister

19. {A, B} = {Best, Best_sibling}


20. D = Worst

Applying (2,4,20) into (12) we know the best player's sibling is male

21. (Best_sibling)_gender = male

And applying (3,4,5,6,21) to (19) we can conclude it is the daughter

22. Best_sibling = A


23. Best = B

Commenting on the issue of apparent symmetry:

While facts (I) and (II) are symmetric across age, the crucial assumption from the premise that Mr. Cole is not the same age as his son or daughter allows us to reach a conclusion.

Here's a shorter proof-type answer. I'll be using Cireo's notations for the players, so A: son, B: daughter, C: Cole, D: Cole's sister.

First we note that every sibling-pair is a brother-sister pair. So we can deduce from (I) that the best player and the worst player are of the same sex, since the best player and their twin are necessarily of the opposite sex.
So, because the same player cannot be the best player and the worst player at the same time, this means that the (Best player, Worst player) pair is one of the following:
(A,C), (C,A), (B,D), (D,B) (*)

Now, because C is A and B's father, we know that:
age(C) $not =$ age(B) and age(C) $not =$ age(A) (#)

(I) implies that age(Best player)=age(Best player's sibling)

(II) says that age(Best player)=age(Worst player)

So we have age(Best player)=age(Best player's sibling)=age(Worst player).

So we have a chain equality involving three players. This chain equality cannot involve C because then it would also involve at least one of A or B which can't be true by (#).

So C is neither the worst player nor the best, and we can refine (*) as follows: the (Best player, Worst player) pair is one of (B,D), (D,B).

And the remaining allowed player, A, must be the best player's sibling, so the best player is B and (Best player, Worst player) is (B,D).