Editing Beale ciphers (section)

=== Statistical analysis ===
Another method to check the validity of the ciphers is to investigate some statistical aspects in different number bases. For example, one can investigate the frequency of the last digit in each number in the ciphers. These frequencies are not uniformly distributed – some digits are more common than others. This is true for all three ciphers.<ref>{{Cite book|last=Wase|first=Viktor|title=Proceedings of the 3rd International Conference on Historical Cryptology HistoCrypt 2020 |chapter=The Role of Base 10 in the Beale Papers |date=2020|volume=171 |pages=153–157 |chapter-url=https://ep.liu.se/ecp/171/019/ecp2020_171_019.pdf|doi=10.3384/ecp2020171019 |isbn=978-91-7929-827-2 |s2cid=219488310 }}</ref>

However, if one considers a base that is relatively prime to 10, then the last digits of the numbers in the unsolved ciphers turn uniform – each digit is equally common. The frequency of the solved cipher stays non-uniform. This indicates a complex behaviour in the solved cipher as one might expect from an encoded message, while the unsolved ciphers have a simpler behaviour.{{original research inline|date=April 2024}}

Humans have limited abilities when it comes to generating random numbers out of thin air.<ref>{{Cite journal|last=Wagenaar|first=W. A.|date=1972|title=Generation of random sequences by human subject: A critical survey of literature|url=http://wexler.free.fr/library/files/wagenaar%20(1972)%20generation%20of%20random%20sequences%20by%20human%20subjects.%20a%20critical%20survey%20of%20literature.pdf|journal=Psychological Bulletin|volume=77|pages=65–72|doi=10.1037/h0032060 }}</ref> One explanation for the difference between base 10 and the other ones is that the numbers were produced by a human in base 10, which would mean that the ciphers are fraudulent.
[[File:Beale last dig.png|center|frame|Statistical analysis of the last digits in the Beale Ciphers. The solved cipher (2) differs wildly from the uniform distribution in all bases, but this is only true for the unsolved ones in base 10. This indicates that the ciphers are fraudulent. The analysis is based on a permutation [[Kolmogorov-Smirnov test]].]]