Beale Cipher

It’s the stuff of legends: A group of men comea across what would be today worth $65 million in gold and silver while on expedition in early-19th-century New Mexico territory. Then, they transport said treasure thousands of miles and bury it in Virginia. One of them, named Thomas Jefferson Beale, leaves three ciphertexts, simply strings of comma-separated numbers, with an innkeeper in Virginia, who forgets about it for more than 20 years.

One day, the innkeeper, realizing that Beale isn’t coming back, opens the box and tries to solve the riddle. Frustrated, he then tells the story and passes along the texts to a friend, J.B. Ward, who cracks one of the three ciphertexts, but not the one that actually gives the precise location of the treasure. More than a hundred years go by, and no one can solve the remaining two ciphers, not even with the benefit of modern computers, and the treasure, if it exists, may still be out there, waiting in the mountains of Virginia.

Picking up on this unsolved mystery, modern storyteller Andrew S. Allen created a short film The Thomas Beale Cipher, a refreshingly modern take on this century-old mystery. In Allen’s story, Professor White, a cryptographer who has recently run into some poor luck, has figured out a way to solve the Beale ciphers. But this knowledge is dangerous, and federal agents are hunting him down.

What does Professor White know? Why are the agents after him? Like all good mysteries, it’s as much about what isn’t told, as what is told.

What is told in The Thomas Beale Cipher has been gorgeously rendered by a combination of old-school, hand-drawn rotoscoping techniques with rich fabric and paper textures. The effect is vibrant and simple, but still haunting — a perfect match for the short, thrilling plot. The Thomas Beale Cipher has been applauded for both its design and story and won numerous film festival awards in 2010.

Fast Company recently praised the film’s rotoscoping techniques, while The Thomas Beale Cipher has been mentioned at BoingBoing and Gizmodo as well. In all, it’s a pithy, well-crafted piece, combining a one-of-a-kind aesthetic with a puzzling and sometimes humorous storyline that is sure to get the wheels turning.

But The Thomas Beale Cipher is much more than an interpretation or retelling of the story. Within the film itself, Allen has embedded ciphers of his own—there are 16 hidden messages in the film, at least two of them require a “genius mind” to crack.

According to Allen, “The root of the film started from a short four-line poem I’d written while riding the train. This was at a time when I was riding the train a lot.

From there I began weaving in various elements that had interested me for a long time like cryptography, Rube Goldberg contraptions, and the idea of an intellectual hero who doesn’t use his fists but his cleverness to thwart attacks. From there, fellow screenwriter Josh Froscheiser and co-producer Jason Sondhi did wonders in helping to pull together the story and dialogue.”

But, how does Allen’s short film connect to the “real” story of the Beale ciphers? Eveything now known about the Beale Cipher is based on a single pamphlet, published by J. B. Ward and submitted to the Library of Congress in 1885.

In this pamphlet, Ward purportedly transcribes what the Virginian innkeeper William Morriss recalls near the end of his life in 1862 and reprints the ciphertexts in full. The first ciphertext allegedly describes the location of the treasure. The second text, the only one deciphered, explains what the treasure is, and the third supposedly details the names of relatives that should get the treasure.

As for Thomas Jefferson Beale, he went off West to seek out a new destiny. He was never heard from again.

J.B. Ward claimed that he solved the second cipher “by accident.” The second text involved a modified book cipher in which the numbers are keyed to words in the Declaration of Independence and the first letter of that word makes up the plain text.

But, the real meat of the mystery is the first cipher of the Beale collection, which supposedly points to the exact location of the treasure. All three ciphertexts, while likely using the same encryption method, seem to involve different books as keys to the cipher.

Here is where the Beale ciphers have inspired people to not just seek the treasure but to seek the truth. Scholars, from historians to statisticians to cryptographers, have tried to unravel the Beale mystery, but to no avail. Many claim, quite compellingly, that the ciphers are a hoax, perpetrated by Ward to increase sales.

But still others claim to have solved the remaining ciphers and/or found Beale’s vault (minus the treasure, unfortunately). However, since none of them are willing to explain how they derived at their solutions, their veracity is questionable at best.

So, is the Beale treasure real or fiction? Is this whole Beale affair nothing more than a hoax and the supposed code-breakers hoaxers building upon a hoax? While the legend might seem implausible, the attractiveness of an unsolved mystery like this one is undeniable, and perhaps sometimes the truth just simply does not matter.

Rather than confronting the Beale ciphers head-on or engaging in the historical debates about whether the ciphers, and Thomas Beale, are real, Allen toys with the Beale controversy in The Thomas Beale Cipher. Adding to the mythos with his own cleverly embedded ciphers, Allen enhances the mystery, even while drawing upon it:

I stumbled upon [the Beale ciphers] one day while doing some research and found it to be such a fascinating story that no one had really told yet. I was interested in weaving in a real cipher to blur the boundaries between fact and fiction. The best stories, I think, are those that continue to live beyond the screen. I really believe there’s great potential to tell stories that engage with viewers on a level beyond the moments spent in the seat of a movie theater. This is part of the reason behind the idea of including hidden messages in the film itself. We also had a lot of story we wanted to tell that just wouldn’t fit in 10 minutes, so the ciphers became a way for me to reveal more about the characters without eating up screen time.

A fine addition to the mythos of the Beale ciphers, The Thomas Beale Cipher is on a limited run, with screenings coming at the Oxford Film Festival in Mississippi in February and at the Cinequest Film Festival in San Jose, California, in March. The film will also be screened as part of the Best of the Northwest Film Festival and Tour.

The entire film is available for viewing at The Thomas Beale Cipher website, which also contains a few hidden story artifacts.

A article i found interesting to share

The Congruent Number Problem

The question of determining whether a given rational number is a congruent number is called the congruent number problem. This problem has not (as of 2009) been brought to a successful resolution. Tunnell’s theorem provides an easily testable criterion for determining whether a number is congruent; but his result relies on the Birch and Swinnerton-Dyer conjecture, which is still unproven.

Fermat’s right triangle theorem, named after Pierre de Fermat, states that no square number can be a congruent number.

The question of whether a given number is congruent turns out to be equivalent to the condition that a certain elliptic curve has positive rank. An alternative approach to the idea is presented below (as can essentially also be found in the introduction to Tunnell’s paper).

Suppose a,b,c are numbers (not necessarily positive or rational) which satisfy the following two equations:

		a^2 + b^2 &=& c^2\\
		\frac{ab}{2} &=& n.

Then set x = n(a+c)/b and y = 2n2(a+c)/b2. A calculation shows

y^2 = x^3 -n^2x

and y is not 0 (if y = 0 then a = –c, so b =0, but (1/2)ab = n is nonzero, a contradiction).

Conversely, if x and y are numbers which satisfy the above equation and y is not 0, set a = (x2n2)/y, b = 2nx/y, and c = (x2 + n2)/y . A calculation shows these three numbers satisfy the two equations for a, b, and c above.

These two correspondences between (a,b,c) and (x,y) are inverses of each other, so we have a one-to-one correspondence between any solution of the two equations in a, b, and c and any solution of the equation in x and y with y nonzero. In particular, from the formulas in the two correspondences, for rational n we see that a, b, and c are rational if and only if the corresponding x and y are rational, and vice versa. (We also have that a, b, and c are all positive if and only if x and y are all positive; notice from the equation y2 = x3xn2 = x(x2n2) that if x and y are positive then x2n2 must be positive, so the formula for a above is positive.)

Thus a positive rational number n is congruent if and only if the equation y2 = x3n2x has a rational point with y not equal to 0. It can be shown (as a nice application of Dirichlet’s theorem on primes in arithmetic progression) that the only torsion points on this elliptic curve are those with y equal to 0, hence the existence of a rational point with y nonzero is equivalent to saying the elliptic curve has positive rank.


The following deal with the problem in a more rigorous way with some illustrations included:






A math joke

Found the following joke from Math OverFlow:

Hilbert had a student who one day presented him with a paper purporting to prove the Riemann Hypothesis. Hilbert studied the paper carefully and was really impressed by depth of the argument; but unfortunately he found an error in it which even he could not eliminate. The following year the student died. Hilbert asked the grieving parents if he might be permitted to make a funeral oration. While the student’s relatives and friends were weeping beside the grave in the rain, Hilbert came forward. He began by saying what a tragedy it was that such a gifted young man had died before he had had an opportunity to show what he could accomplish. But, he continued, in spite of the fact that this young man’s proof of the Riemann Hypothesis contained an error, it was still possible that some day a proof of the famous problem would be obtained along the lines which the deceased had indicated. “In fact,” he continued with enthusiasm, standing there in the rain by the dead student’s grave, “let us consider a function of a complex variable….”

This reminds me of another one form my friend, it goes on like this:

Sir comes into the class, “Today i am going to tell you a story”, then he turns to students and says, “ofcourse my stories begin a little differently, let f(x) be a polunomial…..”