A Grain-Bank Ledger That Catches Its Own Mistakes: Check Digits, Cross-Casts, and the Errors That Still Get Through

A grain bank keeps its members' balances in a handwritten ledger, and a single copying error can take grain from one family and give it to another. This paper designs a checking scheme that a clerk can run with pencil and paper and computes what it catches. Each member receives a five-digit account number whose final digit is a modulo-11 check digit computed from the first four digits with weights 5, 4, 3, and 2; account payloads that would require a check value of 10 are not issued, which removes 909 of 10000 payloads (9.09 percent) and leaves 9091 issuable accounts. Exhaustive enumeration over every issued account finds that the check digit catches all 409095 possible single-digit substitutions (100.0 percent) and all 32728 possible adjacent transpositions (100.0 percent). Amounts are controlled separately by cross-casting each 25-entry page, comparing column totals against the page net computed from the rows. Under a stated clerk-error model totalling 27.0 errors per 1000 entries, the combined scheme leaves 1.56 undetected errors per 1000 entries, detects 94.22 percent of all errors, and improves on an unchecked book with member-spotting alone (21.6 undetected per 1000 entries) by a factor of 13.84. The residual is dominated by one class the scheme cannot see: a whole entry omitted from the book altogether contributes 0.0015 of the 0.00156 undetected probability per entry. The scheme costs 17.5 extra seconds per entry, or 291.7 minutes per 1000 entries, nearly half of which is time spent resolving pages that the checks flag, which happens for 47.49 percent of pages. All results are simulated, not measured.