CTF Tutorial: counting

Decimal

Decimal (Greek “deca-”, meaning “ten”), or base ten, is the counting system most people use regularly, consisting of ten digits:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

To count past nine, one must use the “tens place”, starting by placing a 1 in the “tens place”:

10, 11, 12, 13, 14, 15, …

The presence of a 1 in the “tens place” means “add ten one time”. 2 in the “tens place” would mean “add ten two times", and so on. The decimal number:

372

Reads “three times one hundred, plus seven times ten, plus two”.

A “hundreds place” and “thousands place” can exist as well, and so on for any nth digit representing the place for ten times itself n times, or 10ⁿ.

Octal

Octal (Greek “okta-” meaning “eight”), or base eight, is a method of counting with only eight digits. In the interest of making it obvious when a number is octal, all octal numbers on this page will be preceded by the number “0”. The digits are thus:

00, 01, 02, 03, 04, 05, 06, 07

Octal features an “eights place”, a “sixty-fours place”, a “five hundred twelves place”, and so on. Counting from zero to ten in octal would therefore go like so:

00, 01, 02, 03, 04, 05, 06, 07, 010, 011, 012

Where “010” means “one times eight” (eight), “011” means “one times eight plus one” (nine), and “012” means “one times eight plus two” (ten).

The octal number:

0372

Reads “three times sixty-four, plus seven times eight, plus two”, or two hundred thirty four.

In general, the nth digit of an octal number represents the place for eight times itself n times, or 8ⁿ.

Hexadecimal

Hexadecimal (Greek “hexa-”, meaning “six”; plus “deca-”, meaning ten), or base sixteen, is a method of counting with sixteen digits (preceded by “0x” on this page):

0x0, 0x1, 0x2, 0x3, 0x4, 0x5, 0x6, 0x7,
0x8, 0x9, 0xA, 0xB, 0xC, 0xD, 0xE, 0xF

Hexadecimal (or “hex”) features a “sixteens place”, a “two hundred fifty sixes place”, and so on. Counting from zero to twenty in hex therefore would go like so:

0x0, 0x1, 0x2, 0x3, 0x4, 0x5, 0x6, 0x7,
0x8, 0x9, 0xA, 0xB, 0xC, 0xD, 0xE, 0xF,
0x10, 0x11, 0x12, 0x13, 0x14

Where “0xA” means “ten”, “0xB” means “eleven”, “0xF” means “fifteen”, “0x10” means “one times sixteen”, “0x11” means “one times sixteen plus one”, et cetera.

The hexadecimal number:

0x372

Reads “three times two hundred fifty six, plus seven times sixteen, plus two”, or eight hundred eighty-two.

Question

The key for this page is the decimal representation of the following sum:

12 + 072 + 0x5D

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