28 days, 29 days, 30 days, 31 days,
30 days 23 hours (to daylight saving time in March)
29 days 23 hours (to daylight saving time in April)
31 days 1 hour (from daylight saving time in October)
30 days 1 hour (from daylight saving time in September or November)
30 days 23 hours 59 minutes 59 seconds (minus coordination second in December)
31 days 1 second (plus coordination second in December)
29 days 23 hours 59 minutes 59 seconds (minus coordination second in June)
30 days 1 second (plus coordination second in June)

According to http://en.wikipedia.org/wiki/Leap_second , the coordination second is “typically scheduled either at the end of June 30 or December 31 (though leap seconds can be applied at the end of any month)”, so (assuming that that account is correct), technically the second could be subtracted or added in February as well, which would add four more possibilities (plus or minus, in or out of a leap year). This would stretch the question’s assumption of “simplicity” however!

A one-hour shift is customary, but Australia’s Lord Howe Island uses a half-hour shift.[26] Twenty-minute and two-hour shifts have been used in the past.

One thing that was not made explicit in the first response is that we are rounding to the nearest second for each month. If we round to any other measurement (or don’t round at all), our number of answers may change.

The question than arises why is it most appropriate to round to the nearest second?

The answer of four is still quite valid: 28, 29, 30 or 31 days.

Even a day that is only 23 hours or one that is 24 hours and 1 second is still just a single day.

A second is a precise and defined measurement. A “day” could even be a precise measurement, but a “Gregorian day” is built to flex to account for necessity of differing intervals. Just as a Gregorian month flexes to its amount of Gregorian days.

For example, let’s suppose the question was “how many different year lengths are possible?” The answer could be 1, 12 months. The answer could also be 2, 365 days or 366 days. The answer could also be as numerous as the number of times the earth has revolved around the sun, as it is likely none have been identical down to Planck time.

How many different answers are possible for the question given?

Leo: Leap seconds are inserted at newyear or mid-year UTC. Assuming you’re interested in local time (sure you are since you consider DST changes), this might be January and July in your timezone. This doesn’t change the number of cases, though.

## Leo:

28 days, 29 days, 30 days, 31 days,

30 days 23 hours (to daylight saving time in March)

29 days 23 hours (to daylight saving time in April)

31 days 1 hour (from daylight saving time in October)

30 days 1 hour (from daylight saving time in September or November)

30 days 23 hours 59 minutes 59 seconds (minus coordination second in December)

31 days 1 second (plus coordination second in December)

29 days 23 hours 59 minutes 59 seconds (minus coordination second in June)

30 days 1 second (plus coordination second in June)

Have I missed anything?

7 April 2010, 1:11 pm## Dan:

According to http://en.wikipedia.org/wiki/Leap_second , the coordination second is “typically scheduled either at the end of June 30 or December 31 (though leap seconds can be applied at the end of any month)”, so (assuming that that account is correct), technically the second could be subtracted or added in February as well, which would add four more possibilities (plus or minus, in or out of a leap year). This would stretch the question’s assumption of “simplicity” however!

7 April 2010, 1:45 pm## Gregory Marton:

Pretty good, Leo!

From http://en.wikipedia.org/wiki/Daylight_saving_time

A one-hour shift is customary, but Australia’s Lord Howe Island uses a half-hour shift.[26] Twenty-minute and two-hour shifts have been used in the past.

—–

Tanya, what made you ask?

7 April 2010, 2:14 pm## Tanya Khovanova:

Grem,

Just my students answer 3 or 4. I am happy that I got much more than that in the first reply.

7 April 2010, 9:55 pm## colorblind:

Good question.

One thing that was not made explicit in the first response is that we are rounding to the nearest second for each month. If we round to any other measurement (or don’t round at all), our number of answers may change.

The question than arises why is it most appropriate to round to the nearest second?

8 April 2010, 7:27 pm## Chris:

The answer of four is still quite valid: 28, 29, 30 or 31 days.

Even a day that is only 23 hours or one that is 24 hours and 1 second is still just a single day.

A second is a precise and defined measurement. A “day” could even be a precise measurement, but a “Gregorian day” is built to flex to account for necessity of differing intervals. Just as a Gregorian month flexes to its amount of Gregorian days.

For example, let’s suppose the question was “how many different year lengths are possible?” The answer could be 1, 12 months. The answer could also be 2, 365 days or 366 days. The answer could also be as numerous as the number of times the earth has revolved around the sun, as it is likely none have been identical down to Planck time.

How many different answers are possible for the question given?

15 April 2010, 10:17 am## egmont:

Leo: Leap seconds are inserted at newyear or mid-year UTC. Assuming you’re interested in local time (sure you are since you consider DST changes), this might be January and July in your timezone. This doesn’t change the number of cases, though.

22 November 2010, 6:58 pm