Special Birthdays for Nerds (like me)

I called my bank today to try to straighten out something.  In the process, they asked my birthday for verification purposes.  I said, “Twelve, Five, Sixty.  My birthday is special.  It multiplies.”

After a longish pause, the guy on the other end said, “Huh, it does multiply.  Cool, I wonder how common that is.”  I replied, “Good question.  I have known that my birthday multiplied since I was seven, but I never thought of that question until you asked it, and I must say that I don’t know.”

Well, now I do know.  Part of the question here stems from two digit date fields – mm/dd/yy, in the American system. 12 X 5 = 60.  How common is that?

Over a 400 year period, there are 146,097 days.  (365.25 * 4) – 3 days.  No leap years in 1700, 1800, and 1900, but yes, leap year in 2000.

MonthPossible Days
131
228
331
424
519
616
714
812
911
109
119
128

In this situation, over 100 years, there are 212 days where the month and day multiply to be the year.   That’s 848 over a 400 year period.  February 29th would work in XX58, but that is never a leap year.

Then I said, why limit it to multiplication, why not consider addition, subtraction and division?  With addition, any combination of month and year will produce a year, so over a 100 year period, there should be 365 additive special birthdays.  February 29th would work in XX31, but that is never a leap year.  So, 1460 over 400 years.

Subtraction is more scarce for special birthdays.  The potential number of special birthdays in any month is the month number itself.  All of the special days would pack into the beginning of a century, for us ending at 12/1/2011.  78 per century, or 312 over 400 years.

Finally there is division, which is the rarest.  The number of days in a given month is equal to the number of factors for a month.  In order that would be 1, 2, 2, 3, 3, 2, 4, 2, 4, 3, 4, 2, 6.  That makes 35 over a century, or 140 over 400 years.

So how many mathematically special birthdays are there in total?  Wait, we have to net out double counting.  Without boring you with the math, double counting happens with multiplication and division when the day is one. 02/01/2002 is special for division and multiplication.  Addition and multiplication match for 02/02/2004.  Over 400 years, there are 52 days of overlap in total.

This brings us to the final table:

AdditionMultiplicationSubtractionDivisionTotal
Denominator146,097146,097146,097146,097146,097
Special days1,4608483121402,708
Percentage1.00%0.58%0.21%0.10%1.85%
One in1001724681,04454

So, by my definition, one in 54 have mathematically special birthdays.  I’m sure there are other ways to view this, and I look forward to comments from those that will broaden my mind, and/or correct my errors.