Round about the time Lewis Carroll wrote the "Two Clocks" problem he also wrote a neat demonstration of the need for a date line as a puzzle. I can't find a reference to it on line, but the basic idea is this.

You set off at noon, so the sun is directly overhead, and ask everyone you meet what day of the week it is. "Tuesday", they say.   You keep walking with the sun directly overhead at all times, and asking people. "Tuesday" they say.


After 24 hours, you are back where you started and ask the day of the week. "Wednesday" comes the response.

SO there must be some point on the journey where you moved but a short distance but people stopped saying "Tuesday" and started saying "Wednesday".