Re: ISBN & undecimal counting

In article <slrndmhttu.m5m.tilford@xxxxxxxxxxxxxxxxxxx>,
Mark J. Tilford <tilford@xxxxxxxxxxxxxxxx> wrote:
>On Wed, 02 Nov 2005 10:01:03 -0000, Mark Brader <msb@xxxxxxx> wrote:
>>
>>
>> "Ramkumar" writes:
>>> A modern though little realised example of undecimal counting (i.e
>>> based on 11) is seen in the ISBN of published books. Any ISBN comprises
>>> ten digits. If you multiply the first by ten, the second by nine, the
>>> third by eight, and so on, summing the results as you go along, the
>>> result will always be divisible by eleven.
>>>
>>> Any special reasons why ISBN follows this?
>>
>> It allows the detection of any single-digit error and any transposition
>> of two adjacent digits. This is better than the Luhn checksum, which
>> does not detect the transposition of an adjacent 0 and 9.
>
>I'm pretty sure that the ISBN method also detects any transposition of any
>two nonadjacent digits.

Correct.

If digits number 11-i and 11-j in the original ISBN are a and b, the contribution of
these digits to the checksum is i a + j b. After transposition, it's j a + i b.
The change can be detected unless i a + j b - (j a + i b) = (i-j)(a-b) is divisible
by 11. Since |i-j| and |a-b| are at most 9, that can't happen unless i=j or b=a
(which both say there's no actual transposition).

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

Relevant Pages

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  • Re: ISBN & undecimal counting
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    (rec.puzzles)
  • Re: ISBN & undecimal counting
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