If you like word puzzles (and who among us doesn’t, really?), you’ll love the Word Games forum over on Podcacher.com.
Yes, I know the past few posts have been a bit heavy on the plugging of Podcacher.com, but if you listened to the show and realized how aweome Sonny and Sandy are, you’d completely understand.
FYI – Puzzlehead.org along with the PS101 series and the Lamp Skirt Micros 101 Final Exam were mentioned on the May 17, 2009, episode of Podcacher’s podcast (starting at about 12:40 into the podcast). Whoo-hoo! Thanks, Sonny and Sandy!
Podcacher is a great service, one to which all puzzleheads and geocachers should subscribe.
It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.
-Pierre de Fermat, 1637
Pierre de Fermat was a French lawyer and mathematician who lived from 1601 to 1665, and his work has formed the foundation of many advanced areas of mathematics. His work on maxima, minima, and tangents to curves formed the basis of differentiation. His work on reducing general power functions to the sums of geometric series was influential to Newton and Liebnitz in their development of the fundamental theorem of calculus. Fermat’s work with Blaise Pascal formed the basis of the theory of probability. His principle of least time lead to the principle of least action and the development of classical physics. Fermat was instrumental in establishing the basis upon which modern scientific and mathematical understanding is derived.
However, he was not always as rigorous as one might like. Although he dabbled in mathematics, he always considered himself an amateur. He communicated most of his work to his friends in letters, often with little or no proof of his assertions. Many mathematicians doubted several of his claims, especially given the difficulty of some of the problems he attempted to solve and the tools available to him with which to solve them.
His most famous assertion was made in a note he scribbled in the margin of the book Arithmetica by the Greek philosopher and mathematician Diophantus. The assertion concerns the equation a^n + b^n = c^n (where “^” means “to the power of”). While solutions for the case of n=2 were well known since the time of the ancient Greeks, Fermat asserted that for any other integer greater than 2 there were no numbers a, b, and c that could make this equation true. This assertion has come to be known as Fermat’s Last Theorem, as it was the last of his unproven assertions to remain unproven.
The most tantalizing part of his assertion was the quote at the end: I have discovered a truly marvelous proof of this, which this margin is too narrow to contain. For over 350 years, mathematicians searched in vain for this elusive proof. Fermat challenged his peers to prove the cases of n=3 and n=4, but he himself offered no proof of those cases. Euler proved the case for n=3 in 1770, n=5 was proved by others around 1825, and n=7 was proved in 1839.
Additional proofs were developed in the 1800s for 6 ,10, and 14. Sophie Germain proved that Fermat’s Last Theorem was true for any prime number p such that the expression 2p+1 was also prime (such values of p are called “Sophie Germain primes”). By 1993, computer-based proofs had been used to demonstrate the truth of Fermat’s Last Theorem for all values up to 4 million. But the proof of the general case of n>2 remained unproven.
Mathemetician Andrew Wiles was one of many to become enamored at a young age with the search for Fermat’s marvelous little proof. He described the problem this way: “Here was a problem, that I, a ten year old, could understand and I knew from that moment that I would never let it go. I had to solve it.”. Wiles worked on the proof off and on for years, with no success. Eventually he abandoned work on proving the theorem and went on to other areas of research, specifically into the advanced mathematical concept of modular elliptic curves. (Don’t ask me what they are – I’ve read about them a bunch and still don’t understand them.)
In 1985, mathematician Jean-Pierre Serre asserted that if a special case of the theorem concerning elliptic curves called the Taniyama-Shimura Conjecture was true, then Fermat’s Last Theorem would have to be true. In 1986, Ken Ribet proved this assertion which transformed the proof of Fermat’s Last Theorem into the proof of the Taniyama-Shimura Conjecture. This development meant that Andrew Wiles’ dream of proving Fermat’s Theorem was now a matter of proving one of the most important conjectures in his particular area of mathematics specialization.
Wiles’ search for a proof was renewed in earnest. Wiles worked for years on the proof in secret, starting in the summer of 1986. He dedicated all of his research time to proving Taniyama-Shimura. He compared the proof to walking into a giant mansion where all of the lights were turned out – you wander around in one room, feeling your way around the walls and the furniture, until eventually you find the light switch. Once you turn on the switch, the room is illuminated, and you can make your way to the next room. Repeat this process long enough, and eventually the entire mansion will be lit.
In a series of lectures presented at the Isaac Newton Institute for Mathematical Sciences over three days from June 21-23, 1993, Wiles presented his proof to the world. As he began lecturing on day one, word spread quickly through the mathematics community that his work might be the long-awaited proof of Fermat’s Last Theorem. On the third day, after finishing his proof and stating that his proof implied the correctness of Fermat’s Last Theorem, he concluded by saying, “I think I’ll stop here.” For his proof, delivered 354 years after the original theorem was posited, he not only received a standing ovation, but a host of other awards and accolades.
His proof is not at all the “marvelous proof” that Fermat had envisioned – it uses mathematical techniques that were not developed until the 20th century and beyond. At the time the proof was presented, there were only a handful of people in the world that had any potential of understanding it. Most likely, Fermat believed that he had found a proof using only 17th century methods, and the potential existence of such a proof is what has driven mathematicians for years to seek its solution.
Wiles journey has been documented in many books and television specials. His own description of his journey contains many words of advice not only for mathematicians, but for puzzle-solvers everywhere, including:
Wikipedia has a marvelous description of the proof – the proof itself is over 200 pages long. It also has a number of references to other articles, books, and TV shows to learn more about the proof.
Of course, the best pop culture reference to the proof is in Tom Lehrer’s timeless classic That’s Mathematics (YouTube).
In 1912, the Collegio Romano (now known as the Pontifical Gregorian University) faced a financial crisis. Short on cash, the school decided to raise funds through a discreet sale of a portion of the holdings in its library. Polish-American book dealer Wilfrid M. Voynich acquired 30 of the texts, which included what has become one of the most studied and least understood books in history – a book now known as the Voynich Manuscript.
This 272-page hand-written book (of which only 240 pages remain) is filled with writing in an unknown language as well as beautiful illustrations. So far, no one has managed to decipher the text or ascertain its meaning.
The language in which the text is written is perhaps the most mysterious part of the entire book. There are over 170,000 unique glyphs (or “letters”) in the text, and an alphabet of approximately 20-30 glyphs would account for nearly the entire text. The glyphs are arranged into approximately 35,000 words of varying length that seem to follow some basic rules of spelling similar to other known languages – certain glyphs must appear in each word (as vowels do in English), some glyphs can never follow others, and some symbols may be doubled while others may not. Some words are quite common, while others appear sporadically or only once. But the letter frequency, word frequency, and word relationships are unlike those in any other known language – it is far more complicated than a simple substitution cipher.
The book is organized into 17 groups of 16 pages each, divided into six major sections of different style and subjects (as indicated by the illustrations) on matters that appear to be herbal, astronomical, biological, cosmological, pharmaceutical, and instructive. The relationship between the illustrations and the text is uncertain.
The book’s history is also somewhat uncertain. By the style of dress of people depicted in the illustrations, most historians believe the book was written between 1450 and 1520. The earliest reference to the book was in a letter written in 1639 by Georg Baresch, an obscure alchemist living in Prague, to Jesuit scholar Anathasius Kircher at Collegio Romano asking for assistance in deciphering “this Sphynx that had been taking up space uselessly in [his] library for many years.” Kircher acquired the book in 1666 after Baresch’s death.
There is no mention of the book for the next 200 years, although it was likely kept at the library of Collegio Romano. When the forces of King Victor Emmanuel II of Italy captured Rome in 1870, the college moved much of its library collection to the Italian countryside for protection.
After Voynich’s death in 1930, the book made its way through the hands of various book collectors and dealers who were unable to find a buyer for it. In 1969, the book was donated to Yale University.
Fortunately, Yale has made high-res scans of the entire Voynich Manuscript avaialble online. Click here to take your own tour of the book’s secrets.
Many theories abound about the book’s authorship, content, purpose, and language. My favorite one was postulated by the folks over at XKCD.
I just upgraded the site to WordPress 2.8, which was released yesterday. WordPress describes the 2.8 release as a “fit and finish” release, meaning they fixed a bunch of bugs and cleaned up some annoyances in the user interface.
You shouldn’t notice anything different on the site other than it should generate pages somewhat faster. The Dashboard looks a bit different here and there, but it also looks mostly the same. (Most of the changes are visible only to administrators.)
If you notice anything that’s behaving too oddly (other than the site administrator), please let me know.
Cryptic crosswords are a fun variation on the basic crossword puzzle concept. The most important difference between a traditional crossword and a cryptic one is in the clues: in a cryptic, every single clue is a lateral thinking puzzle unto itself.
In essence, a cryptic clue leads to its answer as long as you read it in the right way. What the clue appears to say when read normally (the surface reading) is a distraction and usually has nothing to do with the clue answer. The challenge is to find the way of reading the clue that leads to the solution.
A typical clue gives you two ways of getting to the answer, either of which can come first. One part of the clue is a definition, which must exactly match the part of speech and tense of the answer. The other part (the subsidiary indication, or wordplay) gives you an alternative route to the answer. One of the tasks of the solver is to find the boundary between definition and wordplay and insert a mental pause there when reading the clue cryptically.
Either the definition or the wordplay can come first, and they never overlap. As a further hint, the clue is followed by a number in parentheses that indicates the number of letters in the answer.
There are a number of different but reasonably standard techniques employed in the wordplay, including anagrams, charades (breaking down larger words into smaller components), containers (inserting one word within another), reversals (spelling words backwards), homophones (different words that sounds alike), deletions (removing letters from the beginning, middle, or end of a word), double definitions (two different interpretations of the clue give the same answer), and more.
This can sound terribly daunting, but it’s a lot easier than you might think. For example, take the following clue:
Returned beer fit for a king (5)
The phrase fit for a king is the definition and returned beer is the wordplay. Beer is LAGER, returned implies a reversal which gives REGAL, and the definition of REGAL is fit for a king.
Here’s another example:
Power plant lacks a spiritual leader (6)
A power plant is a REACTOR, and a REACTOR that lacks “a” is a RECTOR.
How the heck were you supposed to know all of that? Trial and error, experience, and lucky guesses … that’s what makes cryptic crosswords fun! Here’s a few guides to help you solve cryptic clues:
Ready to try your hand at an entire puzzle? Although not as common in America, newspapers throughout the British Commonwealth regularly run them. Here are a few online sources for good puzzles that serve them to you absolutely free:
As a final note here, try solving this one:
Obscure shout before headless lens (7)
[spoiler]CRYPTIC. A lens is an OPTIC, headless implies a deletion of the first letter, leaving PTIC, and a shout is a CRY. Okay, so it’s not a great clue … it’s my first attempt at writing one though, so your forgiveness is greatly appreciated.[/spoiler]
I saw an interesting query in the statistics gizmo used to run this web site. Someone entered the following search string: “how to solve music puzzles”.
Well, here’s your answer. This article is somewhat spoilerific. While it doesn’t tell you how to solve any specific puzzle, the information it contains is derived from puzzles I’ve solved in the past. I’ve tried to supply enough hints to get moving in the right direction without totally spoiling any particular puzzle.
Solving a music puzzle requires some understanding of music theory, which is the study of the language and the notation of music. Musical notation is any system which uses written symbols to represent aurally perceived music. Many types of notation systems have been created throughout history, but most written music you are likely to encounter will use only modern musical symbols.
The topic of music theory is vast – far too big to include in a single article here. But the links presented so far will take you to a great set of resources to understand how music works so that you can get started in cracking puzzles that use music.
Here are just a few of the many possible ways in which music could be used to conceal information (such as a secret message or the coordinates of a geocache):
Notes have letter names, from A to G. A puzzle constructor might begin with a word that uses only those letters, such as BAG, ACE, BADGE, or CABBAGE, then replace each letter in the word with a corresponding note.
An interval is the difference in pitch between two notes, played either at the same time or played successively. An interval of a single half step is called a minor second, two half steps is a major second, three half steps is minor third, and so forth. A puzzle constructor might encode a series of numbers as a series of intervals.
A beat is a pulse that constitutes the fundamental unit of time in a piece of music. A measure is a segment of time, and the number and note value of beats in a measure is called the time signature. For instance, a measure of four beats in which a quarter note gets the beat is said to be in 4/4 time. A measure of six beats in which an eight note gets the beat is said to be in 6/8 time. Patterns of beats can be used to encode just about any kind of information, including letters, numbers, symbols, and more.
That should be enough to get you going. Good luck!
The concept behind GoogleEarthing.com is very simple: imagine a scavenger hunt akin to Where’s Waldo whose search space consists of the entire visible surface of the Earth. Seriously.
The rules of the game are very simple:
The first person with the correct coordinates, name, or otherwise completely specific description of the location will will a valueless prize chosen by the site operator.
Just for grins, here’s an idea of the sort of image you are tasked to find:
The above image is actually puzzle #92, posted November 12, 2006. As of this blog entry, this puzzle has not been solved … will you be the first to crack it?
