100 fare evaders want to take a train, consisting of 12 coaches, from the first to the 76th station. They know that at the first station two ticket inspectors will board two coaches. After the 4th station, in the time between each station, one of the ticket inspectors will cross to a neighbouring coach. The ticket inspectors take turns to do this. A fare evader can see a ticket inspector only if the ticket inspector is in the next coach or the next but one coach. At each station each fare evader has time to run along the platform the length of no more than three coaches – for example at a station a fare evader in the 7th coach can run to any coach between the 4th and 10th inclusive and board it. What is the largest number of fare evaders that can travel their entire journey without ever ending up in the same coach as one of the ticket inspectors, no matter how the ticket inspectors choose to move? The fare evaders have no information about the ticket inspectors beyond that which is given here, and they agree their strategy before boarding.
A pack of 36 cards was placed in front of a psychic face down. He calls the suit of the top card, after which the card is opened, shown to him and put aside. After this, the psychic calls out the suit of the next card, etc. The task of the psychic is to guess the suit as many times as possible. However, the card backs are in fact asymmetrical, and the psychic can see in which of the two positions the top card lies. The deck is prepared by a bribed employee. The clerk knows the order of the cards in the deck, and although he cannot change it, he can prompt the psychic by having the card backs arranged in a way according to a specific arrangement. Can the psychic, with the help of such a clue, ensure the guessing of the suit of
a) more than half of the cards;
b) no less than 20 cards?