Special Agent Benford has been stymied by a third unsolved case of miscreant creative accounting and will voluntarily retire from the Fraudulent Numbers Task Force. Benford’s ability to detect falsified ledgers by merely counting the first digits of numerical entries is so renown that a law bears the name of our crestfallen hero.
Benford’s Law: On an honest accounting sheet, the first digit of almost 1⁄3 of all numerical entries should be ‘1’, much more often than ‘2’ or any other numeral. The count of each numeral’s being a first digit fits a decreasing pattern where ‘9’ is the first digit of the fewest numbers.
This methodology served Benford well through decades of service as unscrupulous accountants mostly juggled decimal numbers. Over the years, however, two cases remained uncracked because they involved other numbering systems. All positive whole numbers are suspect.
Unsolved Case One. Recognizing the oldest known numbering system, Benford knew that its first digits would not lead to conviction.
Unsolved Case Two. Benford recognized this numbering system as one employed by virtually all modern computers and again had to admit that its first digits were free of clues.
At last, alas, came the latest case.
- Unsolved Case Three. Benford had never before encountered this numbering system but upon seeing the entries realized that they too would defy “The Law.” Before giving up, Benford did learn that the same deviant numbering system had been seen among scholarly circles for centuries, though rarely, and has actually been the basis of some cleverly efficient digital computers.
What are the numbering systems of Cases One, Two and Three and why are they so Lawless?
Bounty challenge: In which of these cases can Benford’s replacement, Special Agent Successor, be more successful by counting entries’ second digits? What would be their expected numeral frequencies?(No foul wordplay is afoot.)