Problems with analyzing cases

Methodological Remark. One of the most important goal of the high-school education is to teach students consistent and clear thinking; that is, how to not confuse cause and consequence; how to analyze cases carefully, without skipping any; how to build chain of propositions and lemmas properly. The following problems in logic can help you in handling this.

Problem1. Peter's mom said: ``All champions are good at math.'' Peter says: ``I am good at math. Therefore I am a champion!''Is this implication right or wrong?
Problem2. Assume that the following statements are true:

• among people having TV sets there are some who are not mathematicians;
• non-mathematicians who swim in swimming pools every day do not have TV sets.

Can we claim that not all people having TV sets swim everyday?
Problem3. During a trial in Wonderland the March Hare claimed that the cookies were stolen bad the Mad Hatter. Then the Mad Hatter and the Dormouse gave testimonies which for some reason were not recorded. Later on in the trial it was found out that the cookies were stolen by only one of these three defendants, and, moreover, only the guilty one gave true testimony. Who stole the cookies?
Problem4. Three friends -- sculptor White, violinist Black, and artist Redhheat met in a cafeteria. ``It is remarkable that one of has white hair, another has black hair, and the third has red hair, though no one's name gives the color of their hair'' said the black haired person. ``You are right,''answer White. What color is the artist's hair?