There is an old English rhyme that goes like this:
As I was going to St Ives
I met a man with seven wives
Each wife had seven sacks
Each sack had seven cats
Each cat had seven kits
Kits, cats, sacks, wives
How many were going to St Ives?
According to Wikipedia, the earliest known published version of this rhyme was published in a manuscript dated to around 1730 (but it differs in referring to 'nine' rather than 'seven' wives). The modern form of the rhyme was first printed around 1825. Both of those mathematical puzzles, however, are late comers. "A similar problem appears in the Rhind Mathematical Papyrus (Problem 79), dated to around 1650 BC." The answer to the riddle is not as straight forward as it first might seem. To learn why the solution could be either 2,753 or 2,802, click on the Wikipedia link and read the explanation. The Wikipedia article also provides the solution to the puzzle found in the Rhind Mathematical Papyrus. That papyrus "introduces ... roughly 85 problems" to be solved ["Math Puzzles’ Oldest Ancestors Took Form on Egyptian Papyrus," by Pam Belluck, New York Times, 6 December 2010]. Belluck, like Wikipedia, notes that "the Rhind Mathematical Papyrus, contains a puzzle of sevens that bears an uncanny likeness to the St. Ives riddle." She continues:
"It has mice and barley, not wives and sacks, but the gist is similar. Seven houses have seven cats that each eat seven mice that each eat seven grains of barley. Each barley grain would have produced seven hekat of grain. (A hekat was a unit of volume, roughly 1.3 gallons.)
The goal: to determine how many things are described. The answer: 19,607.The Rhind papyrus, which dates to 1650 B.C., is one of several precocious papyri and other artifacts displaying Egyptian mathematical ingenuity. There is the Moscow Mathematical Papyrus (held at the Pushkin State Museum of Fine Arts in Moscow), the Egyptian Mathematical Leather Roll (which along with the Rhind papyrus is housed at the British Museum) and the Akhmim Wooden Tablets (at the Museum of Egyptian Antiquities in Cairo). They include methods of measuring a ship’s mast and rudder, calculating the volume of cylinders and truncated pyramids, dividing grain quantities into fractions and verifying how much bread to exchange for beer. They even compute a circle's area using an early approximation of pi. (They use 256/81, about 3.16, instead of pi's value of 3.14159....) It all goes to show that making puzzles is 'the most ancient of all instincts,' said Marcel Danesi, a puzzle expert and anthropology professor at the University of Toronto, who calls documents like the Rhind papyrus 'the first puzzle books in history.'"
Based on that, I guess it shouldn't surprise us to see so many travelers working word or mathematical puzzles to help pass the time. Belluck continues:
"Dr. Danesi says people of all eras and cultures gravitate toward puzzles because puzzles have solutions. 'Other philosophical puzzles of life do not,' he continued. 'When you do get it you go, "Aha, there it is, damn it," and it gives you some relief.'"
You must admit that you feel some sense of satisfaction when you successfully fill in a crossword puzzle or complete a Sudoku grid. Belluck reports, however, that the ancients were looking for more than personal gratification when they generated puzzles. She continues:
"Egyptian puzzles were not just recreational diversions seeking the comforting illusion of competence. They were serious about their mission. In the Rhind papyrus, its scribe, known as Ahmes, introduces the roughly 85 problems by saying that he is presenting the 'correct method of reckoning, for grasping the meaning of things and knowing everything that is, obscurities and all secrets.' And the documents were practical guides to navigating a maturing civilization and an expanding economy."
Divvying up diverse goods in a fair way so that trade could be conducted was obviously serious business. Puzzles helped equip those early traders with the tools they needed to compete and profit in what was still a barter economy. Belluck continues:
"So the Akhmim tablets, nearly 4,000 years old, contain lists of servants’ names, along with a series of computations concerning how a hekat of grain can be divided by 3, 7, 10, 11 and 13. The Egyptian Mathematical Leather Roll, also from about 1650 B.C., is generally considered a kind of practice test for students to learn how to convert fractions into sums of other fractions. The Rhind papyrus contains geometry problems that compute the slopes of pyramids and the volume of various-shaped granaries. And the Moscow papyrus, from about 1850 B.C., has about 25 problems, including ways to measure ships’ parts and find the surface area of a hemisphere and the area of triangles. Especially interesting are problems that calculate how efficient a laborer was by how many logs he carried or how many sandals he could make and decorate. Or the problems that involve a pefsu, a unit measuring the strength or weakness of beer or bread based on how much grain is used to make it. One problem calculates whether it's right to exchange 100 loaves of 20-pefsu bread for 10 jugs of 4-pefsu malt-date beer. After a series of steps, the papyrus proclaims, according to one translation: 'Behold! The beer quantity is found to be correct.'"
That may be the world's first tailgate moment or Octoberfest! In a companion article, Benedict Carey asserts that puzzle solving is one of the innate qualities of modern humans that allowed us not just to survive but thrive ["Tracing the Spark of Creative Problem-Solving, New York Times, 6 December 2010]. He writes:
"The puzzles look easy, and mostly they are. Given three words — 'trip,' 'house' and 'goal,' for example — find a fourth that will complete a compound word with each. A minute or so of mental trolling (housekeeper, goalkeeper, trip?) is all it usually takes. But who wants to troll? Let lightning strike. Let the clues suddenly coalesce in the brain — 'field!' — as they do so often for young children solving a riddle. As they must have done, for that matter, in the minds of those early humans who outfoxed nature well before the advent of deduction, abstraction or SAT prep courses. Puzzle-solving is such an ancient, universal practice, scholars say, precisely because it depends on creative insight, on the primitive spark that ignited the first campfires."
Carey reports that "neuroscientists are beginning to tap [the] source" of creativity that permits us to solve problems -- or, at least, discover ways to increase the chances of tapping into that creativity. He reports, for example, that "researchers at Northwestern University found that people were more likely to solve word puzzles with sudden insight when they were amused, having just seen a short comedy routine." You might call these Steve Allen or Steve Martin moments. Both Steves are noted for having remarkably eclectic tastes and a broad range of talents. Comedians, in general, appear to be a perceptive group that sees connections or paradoxes in life that most of us simply ignore -- and they make us laugh when they bring them to our attention.
"'What we think is happening,' said Mark Beeman, a neuroscientist who conducted the study with Karuna Subramaniam, a graduate student, 'is that the humor, this positive mood, is lowering the brain's threshold for detecting weaker or more remote connections' to solve puzzles. This and other recent research suggest that the appeal of puzzles goes far deeper than the dopamine-reward rush of finding a solution. The very idea of doing a crossword or a Sudoku puzzle typically shifts the brain into an open, playful state that is itself a pleasing escape."
Carey agrees with experts interviewed by Belluck that one of the attractions of puzzles is that they "are reassuringly soluble; but like any serious problem, they require more than mere intellect to crack." He continues:
"'It's imagination, it's inference, it's guessing; and much of it is happening subconsciously,' said Marcel Danesi, a professor of anthropology at the University of Toronto and the author of 'The Puzzle Instinct: The Meaning of Puzzles in Human Life.' ... For almost a century scientists have used puzzles to study what they call insight thinking, the leaps of understanding that seem to come out of the blue, without the incremental drudgery of analysis. In one classic experiment, the German psychologist Karl Duncker presented people with a candle, a box of thumbtacks and the assignment of attaching the candle to a wall. About a quarter of the subjects in some studies thought to tack the box to the wall as a support — some immediately, and others after a few failed efforts to tack wax to drywall."
Carey notes that "the creative leap may well be informed by subconscious cues." I've noted before that most creative people are extremely observant. They see things that appear odd, out of place, or perform less effectively than they could and start asking questions. They generally don't need help solving problems. The less observant often need a nudge. Carey explains:
"In another well-known experiment, psychologists challenged people to tie together two cords; the cords hung from the ceiling of a large room, too far apart to be grabbed at the same time. A small percentage of people solved it without any help, by tying something like a pair of pliers to one cord and swinging it like a pendulum so that it could be caught while they held the other cord. In some experiments researchers gave hints to those who were stumped — for instance, by bumping into one of the strings so that it swung. Many of those who then solved the problem said they had no recollection of the hint, though it very likely registered subconsciously."
For you really old timers, you might remember an old television show called "Beat the Clock" hosted by Bud Collyer. The show ran on CBS from 1950–1958 and ABC from 1958-1961. One of the challenges that participants had to solve was to get two wigs hanging from strings into a lidless top hat seated on their heads. The only way they could perform the stunt successfully was to get one wig in the hat then get the other wig swinging like in the problem described above. Some participants saw that solution immediately while others never caught on in the allotted time. Perhaps the most interesting study cited by Carey was one that reviewed available research results and "concluded that the abilities most strongly correlated with insight problem-solving 'were not significantly correlated' with solving analytical problems." In other words, people who are very analytical may not have many "Eureka" moments and people who repeatedly come up with interesting insights may not be very good at crunching numbers. Carey concludes: "Either way, creative problem-solving usually requires both analysis and sudden out-of-the-box insight." According to Adam Anderson, a psychologist at the University of Toronto who was interviewed by Carey, a single person can be both analytical and intuitive; but, he thinks, "they are truly different brain states." The science seems to be proving him correct.
"That is what brain-imaging studies are beginning to show. At first, such studies did little more than confirm that the process was happening as expected: brain areas that register reward spiked in activity when people came up with a solution, for instance. Yet the 'Aha!' moment of seeing a solution is only one step along a pathway. In a series of recent studies, Dr. Beeman at Northwestern and John Kounios, a psychologist at Drexel University, have imaged people's brains as they prepare to tackle a puzzle but before they've seen it. Those whose brains show a particular signature of preparatory activity, one that is strongly correlated with positive moods, turn out to be more likely to solve the puzzles with sudden insight than with trial and error (the clues can be solved either way)."
Carey concludes: "The punch line is that a good joke can move the brain toward just this kind of state." In a recent two-part series on happiness [In the Pursuit of Happiness -- Part 1: National Happiness and In the Pursuit of Happiness -- Part 2: Personal Happiness], the author of one of the articles I cited asserted that being happy is a handicap because most driven and productive people are miserable. These studies on problem solving seem to indicate that being happy (or at least in a good mood) is critical to the problem-solving process. My suspicion is that both misery and happiness put people in states that nudge creativity to the front of mental processes and motivate people to achieve something -- there have been a number of unhappy artists and playwrights. Good moods, however, appear to do this better than other mental states when it comes to problem solving. Carey concludes:
"This diffuse brain state is not only an intellectual one, open to looser connections between words and concepts. In a study published [in 2009], researchers at the University of Toronto found that the visual areas in people in positive moods picked up more background detail, even when they were instructed to block out distracting information during a computer task. The findings fit with dozens of experiments linking positive moods to better creative problem-solving. 'The implication is that positive mood engages this broad, diffuse attentional state that is both perceptual and visual,' said Dr. Anderson. 'You're not only thinking more broadly, you're literally seeing more. The two systems are working in parallel.' The idea that a distracted brain can be a more insightful one is still a work in progress. So, for that matter, is the notion that puzzle-solving helps the brain in any way to navigate the labyrinth of soured relationships, uncertain career options or hard choices that so often define the world outside. But at the very least, acing the Saturday crossword or some mind-bending Sudoku suggests that some of the tools for the job are intact. And as any puzzle-head can attest, that buoyant, open state of mind isn't a bad one to try on for size once in a while. Whether you're working a puzzle or not."
Before simply blindly accepting the notion that people in good moods are better problem solvers, I'd like to bring a couple of articles by John Tierney into the discussion. In the first article ["Discovering the Virtues of a Wandering Mind," New York Times, 28 June 2010], Tierney agrees with Carey "that a distracted brain can be a more insightful one." He writes:
"At long last, the doodling daydreamer is getting some respect. ... researchers have been analyzing those stray thoughts, they’ve found daydreaming to be remarkably common — and often quite useful. A wandering mind can protect you from immediate perils and keep you on course toward long-term goals. Sometimes daydreaming is counterproductive, but sometimes it fosters creativity and helps you solve problems."
On the other, Tierney notes that daydreaming is probably a symptom of unhappiness (like the character in James Thurber's short story, "The Secret Life of Walter Mitty." ["When the Mind Wanders, Happiness Also Strays," by John Tierney, New York Times, 15 November 2010]. All of this seems to imply that if you want to solve a problem avoid daydreaming until you are in a good mood.
