04 Binary, octal and hexadecimal numbers

“There are 10 kinds of people. Those who understand binary code and those who don’t.”

Another, much better known, place value system is the binary system. A place value system with a base of two. There are only two states, zero and one, on or off, dot or dash, there or gone, true or false. Computers work this way because circuits work this way. Because electricity can only take on these two states. It is there or it is gone.

This gives an immense number of possibilities, especially when encoding messages, which are often binary (i.e. based on two states) but not necessarily written in the dual system (i.e. with zero and one). Very many things can have two states. Upper and lower case letters, for example. Dashes and dots (also completely “unbinary”, such as Morse). Something can be true or false, for example, hidden in text. Which makes the binary of coordinate hiding one of the most frequently, but also one of the most creatively used geocaching cipher types.

The dual system


The decipherment of dual numbers, i.e. binary code consisting of zero and one, is very simple. Dual numbers are written one after the other. The first digit on the far right counts 2^0, i.e. one (in our decimal system), as long as it has been filled with a 1 (current there). If there is a 0 there, it also counts as zero, i.e. nothing. The next digit, to the left of the first, has the value 2^1, i.e. two. If it is filled with a 0, it has the value zero, if it is filled with a 1, it has – decimally – the value two. It continues with the third digit (2^2), one, i.e. four, then comes the eight (2^3) , and so it goes on and on.

While such dual numbers can easily be transferred into our usual decimal system (at least up to a certain length, even in our heads), they have the disadvantage that they can become extremely long. A coordinate represented in the form, for example 52 45 123, simply written together as a large number, already has an impressive length when expressed in zeros and ones:


For this reason, the hexadecimal representation has been adopted in the field of data processing. Hexa = from the Greek for 6, decimal from the Latin for 10, i.e. a mixed place value system on the basis of 16, whereby the letters A to F are used in addition to the digits from 0 to 9. Here, one counts normally with the decimal digits up to 9 and then, if there is a jump to the next place value level (ten) in the decimal system, uses the letters instead of these. A (hexadecimal) is thus a 10 (decimal), F a 15. Only at 16 is there a jump to the second “dimension”, a second digit, only that this is not worth ten, as in our decimal system, but 16.



So far, so clear? But of course you don’t have to learn it by heart, there are various conversion tools on the internet and even the otherwise rather frowned-upon Windows calculator manages the conversion of hex/dual/decimal and even octal in the programming view (in
the calculator menu under View -> Programmer).

The octal representation just mentioned is another place value system, this time in base 8. It counts from 0 to 7, then follows a place value jump to 10, which is decimal eight!


Another possibility that comes from data processing is binary coded decimals – BCD. This means that the digits 0 – 9 are each encoded with a binary representation. Four binary characters are most common, but there can also be five, seven or ten.

Maybe you can find clues in the listing like

Summary :

  • Do you find 0+1 or some kind of “binary”, two-level representation in a puzzle? Perhaps binary code?
  • Codes containing digits from 0-9 and letters from A-F could be hexadecimal encoded.
  • Numbers containing only digits from 0-7 “smell” strongly of octal…