Challenging calendar problem..

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Pick a time on the date Jan 31, 2004 and the same time nine years later (2013). If you figure out the midpoint between these two date/times, it will be the same time of day (plus one hour since DST would be in effect then) but the date would be Aug. 1, 2008.

The question is "Why isn't the middle date July 31, 2008 instead of Aug 1.?"

Because there is an odd number of leap-days (3) between the two boundary dates.

I know it's an even number of days (9 years @365 plus 3 leap-days), but why Aug 1 instead of Jul 31?

Maybe because February has less days?

Part of the answer. In non-leap years, the first quarter has 90 days. The other quarters have 91, 92, and 92 respectively every year. July 31 has four 31 day months and two 30 day months till next Jan 31 while going from Jan 31 there are only three 31 day months, two 30 day months, plus February.

So is that the answer?

Another way of looking at is between July 31 and Jan 31 there are four 31-day months and two 30-day months for 184 days. Between Jan 31 and July 31 there are months of 28,31,30,31,30, and 31 for 181 days (182 in leap years). For an equal number of days before and after Jan 31, the answer has to be later than July 31. Between 2004 and 2013, the answer is 7pm on Aug 1 (daylight savings time is in effect).

If you substitute some other odd numbered years like 2009, 2011, or 2017, the answer will be Aug 2 at 7am.