KöMaL 2025 C1876

Winnie the Pooh tries to decide whether the following statement is true: 'If 30 days of a given year is chosen, there will not necessarily be at least five of them falling on the same day of the week.' He proposes the following argument: 'The statement is false. The day's of the week are the pigeon holes, and let's fill them with pigeons corresponding to the chosen days. We can distribute 28 (identical) pigeons such that every pigeon hole contains exactly 4 pigeons, however, the $29^{\text{th}}$ pigeon will surely be placed in a pigeon hole containing four pigeons, therefore there must be a day of the week containing at least five of the chosen days.' Pooh's friend, Christopher Robin points out the missing part of the argument: 'Pooh, you've only considered seven cases: when each pigeon hole contains four pigeons, except for one containing five'. How many cases were omitted by Pooh?