Why does the date sometimes show the 31st when it is actually the 1st?

As mentioned above, most mechanical calendar watches take a very simplistic approach to handling the date. This is because the modern Gregorian-based calendar follows somewhat complex rules. While the rules are simple for a digital watch to handle, they get very complex to execute in a tiny mechanical device. Most particularly complicated is that correct computation of the number of days in each month requires the watch to keep track of not only the current date, but also the month and year.

A conventional calendar watch is mechanically very simple--it takes the elementary approach that all months are 31 days. The watch simply rolls the date forward by one every other time the hands pass forward across 12:00. The date goes up to 31 then restarts at 1. They are not aware of any concepts of month, year, decade or century.

The only down side to this greatly simplified approach is that five times a year, you have to manually roll the date forward to correct for this. In addition to the normally short month of February, there are four months of the year that have only 30 days: April, June, September and November. So on the 1st day of the following months, you will have to advance the date by one so your watch will correctly display the 1st.

The "perpetual calendar" watches that do handle this are much more cumbersome to set. They have to understand not only the day but also the month and the year. Making a mechanism that can not only do that but can be changed and reset when needed is pretty intricate. Since it is far simpler to merely correct the date five times a year, few manufacturers bother to add the expense of such a feature to their watches.

Digital watches have no problem with the number of days in a month because they can easily programmed to track both the month and day.