Checksum Magic

[BrianG]

Turtle, over at http://hypography.com/forums/physics-and-mathematics/1343-katabatak-math-exploration-pure-number-theory.html just taught me an easier way to explain the Euro bill checksum. If you convert the letter at the start of the serial number to its number value and add all the digits together, over and over until you’re left with one digit, the Katabatak function, you always end up with 8, the checksum. I’ll presume to use Turtle’s words, “Now to the Katabatak function, K(n). Two principles, addition & repetition, make up the function. It is that operation of repeatedly adding an integer's digits until arriving at a single digit” and in a Euro serial number, you will always get the result, 8. Here are the two examples:

 

X27678791315

24+2+7+6+7+8+7+9+1+3+1+5=80=8+0=8

 

S63050315029

 

19+6+3+0+5+0+3+1+5+0+2+9=53=5+3=8

 

So, if any single digit is missing, all you do is take the Katabatak function of the numbers you do have, and find the difference from 8. Note that if a 9 and 0 are still ambiguous, if either is missing you’ll be left with 8, the Katabatak function of the converted letter and the 10 numbers you have yield the result, 8 and 8+0=0 and 8+9=17=1+7=8. If an 8 is missing, the Katabatak function will leave you with 9 giving you a negative one difference from the checksum, 8 and since we can’t go there, the missing digit is 8.

 

I hope this sheds some light on this little illusion.