The Great Houdini is performing again! Houdini has built up lots of reputation by showing his great card guessing tricks in [predecessor puzzle-1] and in [predecessor puzzle-2]. Everybody wants to see the famous magician, and the magic show is sold for many weeks in advance. Today Houdini shows the following awesome trick:
- A girl from the audience carefully shuffles a deck of $52$ cards ($13$ spades, $13$ hearts, $13$ clubs and $13$ diamonds), and hands the deck over to Houdini's assistant.
- The assistant looks through the entire deck (but without changing the order of the cards and without revealing them to Houdini).
- The assistant then repeats the following step $52$ times: He takes the top card and puts it face-down on the table. Houdini guesses the suit. Then the assistant flips the card over and reveals it to everybody (including Houdini). Then the step is repeated.
Houdini's goal is of course to make as many correct guesses as possible. To reach this goal, Houdini and his assistant apply a sophisticated cheating strategy:
The backsides of the cards are not perfectly symmetric. The assistant has two different ways of putting down every card, and hence may communicate to Houdini one bit of information per card.
Question: Is there a cheating strategy for Houdini and his assistant that guarantees Houdini to make (a) at least $27$ correct guesses (b) at least $32$ correct guesses (even in the worst possible case)?