Casting out nines
Casting out elevens
Improved checksum
Credit card

Checksum calculator simulations

This page shows some interactive animations of simple checksum calculators.
For more information on these and other machines, see my Calculating History site.

Select one of the tabs

Malý's device for casting out nines.

“Casting out nines” is a method to check simple calculations by deriving a checksum.
To check C = A + B, or C = AB, or C = A × B, or C = A / B:

Reumhelm Controlex

A device for casting out elevens.
To check C = A + B, or C = AB, or C = A × B, or C = A / B:

Improved check sum calculator

A general formula for checksums is
nΣi=1   wi ai = 0 mod p
where   ai = i th digit of the number a
wi = i th weight (integer)
p = an integer
For a specific application, wi and p can be choosen to best detect errors that are often occurring in that application.
Different wi's for each i and a prime number p are a good start.
Alfred Henry Fielding Richardson, from New Barnet, Hertfordshire, UK, proposed an algorithm for account numbers with a complicated weighting scheme, and p = 11
3a1 + 6a2 + 8a3 + 5a4 + 7a5 + 10a6 + 2a7 + 4a8 + 9a9 + a10 = 0 mod 11

a10 is shown in the small window at the top left.
Because p = 11, it is possible that a10 = 10, which is indicated by an 'A'.



Pure Oil Company credit card number calculator

Credit card numbers obey a checksum invented by Hans P. Luhn, of IBM.
For the digits ai of a number a, with a1 being the right-most digit.
12Σi=1 {   i is odd:    ai
  i  is even: 2ai mod 9
}    = 0 mod 10
IBM patented a device for computing the checksum.
A much simpler device was invented by Leonard C. Zitnik, for the Pure Oil Company. Oil companies were early credit card issuers, for easy payment in their gasoline stations.
Try 4408 0412 3456 7890, if the credit card number is correct, the procedure ends with the disk pointer at the red triangle.
Please wait

A calculator for Verhoeff's algorithm

Unlike the devices shown in the other tabs, the checksum algorithm derived by the Dutch mathematician Jacobus (Koos) Verhoeff is not based on weighted-adding-modulo-some-number.
Verhoeff's algorithm uses multiplications in point group D5, and a permutation.

Verhoeff's algorithm was used for numbering old German banknotes (Deutsche Mark). These numbers also contain letters, which are converted to digits by the table at the right.
Select the numbers by setting the red sliders, left to right. Note that when setting the slider for a number, the next column of numbers changes. The last column gives the rightmost digit of the banknote number.

Bank note GN4480100S8

Try GN4480100S8 !   More info