Page 21 : Beale Ciphers Analyses

Benford's Law - A clue to the Beale ciphers?

The contributor for this page is Richard 'Tony' Rogers. Tony is a database software specialist with the U.S. Government, focusing in particular on neural networks and pattern recognition. He is optimistic that his work on the Beale codes, based partly on the information below, will lead to a verifiable final solution later in 2005.

Benford's Law is relatively new in mathematical terms. Its first indications were in 1881. It was formally promulgated in 1938, and further developed in the 1990's.

As described in the link above, it is an astonishing mathematical theorem, a powerful and simple tool for pointing suspicion at frauds, embezzlers, tax evaders, sloppy accountants and even computer bugs. The income tax agencies of several nations and several states, including California, are using detection software based on Benford's Law.

Simply, it states that naturally occuring number series, including such wildly disparate and unrelated categories as the areas of rivers, baseball statistics, numbers in magazine articles, and street addresses, all follow the same first-digit probability pattern, which can be charted as follows:

The graph below shows fraudulent tax data. In general, fraudulent or concocted data appear to have far fewer numbers starting with 1 and many more starting with 6 than do true data. Note also the "random guess" data in this chart.

So how does this apply to the Beale codes? Here is the first digit frequency graph for the three ciphers:

For comparison, I generated three random number files with a computer program, with the same quantity of numbers and upper and lower limits as the three ciphers, and charted the results as follows:

The "Random 520" is different because C1 has limits of 1 to 2906, whereas the other two are 1 to 1005, and 1 to 975. The point is, as one would expect with random numbers, the frequencies of first digits are relatively equal, unlike the Benford's Law predictions, and the Beale ciphers.

What does this prove?

It is a strong argument that the Beale codes are not just random numbers as some analysts have concluded; and the opposite of random is a logical structure.

Note that C2, with a known solution, is the closest to the Benford prediction.

Note also, that C1 and C3 have very similar patterns: they essentially adhere to the prediction, but show identical discrepancies, with depressions at numbers 5 and 7, and subdued spikes at numbers 6, 8, and 9. Peaking at 6 is a sign of fraudulent data, but this one is weak. These two graphs remind me of the teeth of a house key... and these two keys would likely fit the same door.

Can these patterns be a clue to the encoding method?

• Not random
• Logical structure
• Identical structures for C1 and C3

If solution seekers need refreshment, this is it.

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