A mathematic insights to Calendar. Last Christmas, a kid in my neighborhood dressed in a pirate suit and knocked my door for "trick or treat". He explained to me that Dec 25, which is 25 written in base 10, is equal to 31 in base 8, which is Oct 31. So ingeniously, Halloween coincides with Christmas. 
I really wonder which other two dates would possibly coincides with each other.
Can you help me find them? And explain what kind the two dates are. Your birthday is also appreciated.
 A: The point in the original "date identity" is that the abbreviations  Dec and Oct for the month names are also interpreted as abbreviations for the base, i.e., "decimal 25 = octal 31" becomes "Dec 25 = Oct 31". There is no month name abbreviated as Bin or Hex, for example, so binary and hexadecimal can be excluded. Indeed, at best we might be somewhat willing to accept Sep for septimal (base 7) and Nov for novemal(?) (base 9). However, these are far from common enough to be understood from such abbreviations (and I am not even sure about the correct adjective forms). 
Nevertheless, in September, we have dates Sep 01, ..., Sep 06, Sep 10, ..., Sep 30 which can be seen as standing for the numbers one up to twenty-one.
In October, we have accordingly the numbers one up to twenty-five, in November the numbers one up to twenty-seven, and in December of course the numbers one up to thirty-one. Now you can pick any number that is common to at least two months and form such date equalities accordingly, e.g.,

Sep 01 = Oct 01 = Nov 01 = Dec 01

or

Sep 10 = Oct 07 = Nov 07 = Dec 07

or 

Sep 30 = Oct 25 = Nov 23 = Dec 21

or 

Oct 26 = Nov 24 = Dec 22

or

Nov 30 = Dec 27

and all cases in-between.
A: which other two dates would possibly coincide with each other 

There are too many to list out. 

Maybe you could find them using converting decimal number to octal number. Here is the rules you can follow to find. My birthday is 01 jan which is same in binary, octal and hexadecimal system . 
