A store wants to celebrate its anniversary and will give a $200 shopping certicate to the first customer to enter the store whose birthday is the same as that of two other previously admitted customers.
Also, there are extra prizes of $5,000 that will be for the first two customers that have consecutive birthday dates.
(a) What is the greatest possible number of customers that will enter the store before the $200 winner is announced?
(b) What is the greatest possible number of customers that will enter before the winners for the $5000 is announced?
For part a, I got 732 customers, and I applied the Pigeonhole principle. But for part b, I am thinking that it will be infinite because I don't know how many customers will enter the store.
(a) It may indeed happen that there are $2\times 366=732$ customers before the $733$rd enters and wins.
(b) is a bit ambiguous: Are 1 March and 28 February consecutive? Are 31 December and 1 January consecutive? If we arrange the $366$ dates on a circle, we can pick at most $183$ of them with gaps between, so the answer should be $183$ (and the $184$th wins together with some other customer).