The humor lies in the fact that Halloween occurs on October 31 and Christmas occurs on December 25, thus equating "oct" in October and octal, and "dec" in December and decimal.
A mathematical joke is a form of humor which relies on aspects of mathematics or a stereotype of mathematicians to derive humor. The humor may come from a pun, or from a double meaning of a mathematical term. It may also come from a lay person's misunderstanding of a mathematical concept. These jokes are frequently inaccessible to those without a mathematical bent.
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive digits into groups of three (starting from the right).
Octal 31 is obviously equal to Decimal 25 but this is beyond my mathematic ability:
To convert integer decimals to octal, divide the original number by the largest possible factor of 8 and successively divide the remainders by successively smaller factors of 8 until the factor is 0. The octal representation is formed by the quotients, written in the order generated by the algorithm.
Thus 31 in octal is 3x8 + 1, or 25 in decimal (2x10 + 5).
As an aside, in binary place values are 8 4 2 1. Hence the quip, "There are 10 sorts of people - those who understand binary, and those who don't." (1x2 + 0 = 2)
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