+2511 Well, I can bloodywell state with near certainty that VooFix doesn’t understand it at all!
One reason for this certainty is this is NOT the solution to the question he posted. The posted question has g(6) = -133, the provided solution is correct for g(6) = -113. So strike one for even paying attention to the question, let alone understanding the solution.
This solution uses a basic generating function (and it’s very cool). I offer brief comments on generating functions here and here. Generating functions can be very complex, but even understanding the basic, rudimentary generating functions require at least a moderate prerequisite understanding of Church-Rosser properties for the underlying equivalence relations, and a working knowledge for converting augmented matrixes to reduced row echelon form via Gauss or Gauss–Jordan Elimination.
Though it’s unlikely (without further, detailed instruction) that any of the AoPS students presented with this solution will be able to casually use this generating function to solve this problem, there are many who could (without understanding the underlying fundamentals) learn to solve this and some similar problems using generating functions with rote instructions.
Most students (the very vast majority) learn the rote procedures first, and then the understanding of the why (maybe) comes during the next level. This is true for both adult students and children. Consider elementary students learning how to multiply: In the beginning, most do not know they are adding a number to itself a certain number of times. For some students this understanding comes when they learn to divide. Of course, some will never understand this, but they can still multiply numbers.
Instruction beyond the grasp [of understanding] does seem to be an ideal method for teaching; and reaching beyond ones grasp also seems to be an ideal method for learning. This is why AoPS used a generating function for the solution. AoPS will introduce a concept with a basic outline of the fundamentals and then present a solution that is at least two levels higher than the students’ current skills. From this point, they teach from the top-down and from the bottom-up, and the skills and understanding meet somewhere in the middle. The brighter students will see where they are headed with this, even if they are uncertain of the path. The lesser students may not see where they are going, but they do have bread crumbs to find their way –unless they wait too long and the ants carry the crumbs off. 
This is a very cool teaching method. My great uncle Cosmo often used the top-down-bottom-up scheme to teach me science and the scientific method (and other subjects). My education started at age five. My aunt or uncle would read me a children’s story –usually one of the classics; Cosmo would then read me short passages from a science text. The first of many texts was the Origin of Species by Charles Darwin, which he supplemented with excerpts from the two volumes of The Voyage of the Beagle. These texts weren’t children’s editions –they were originals. After reading a few passages to me, he would explain what it meant. I’m sure I still didn’t understand it. But Cosmo wasn’t worried about it; he knew one day I would understand this theory and that my brain would grow and develop around this foundation of knowledge.
Lancelot Link also used the top-down-bottom-up method to tutor and teach me more than four years worth of collegiate mathematics in less than a year and a half. From childhood, I’ve always known I was an advanced Chimp, but I became a Genetically Enhanced Chimp before I graduated from Lancelot Links School of Mathematics and Trolling. I don’t think I would have passed the classes without the enhancement. ...Or maybe it was passing the classes that gave me the enhancement. 
GA
+2511 Solution:
Set up a systematic set of equations and then use Gauss-Jordan elimination.
\(a + b + c + d + e + f = 2 \\ 32a + 16b + 8c + 4d + 2e + f = 3\\ 243a + 81b + 27c + 9d + 3e + f = 4\\ 1024a + 256b + 64c + 16d + 4e + f = 5\\ 3125a + 625b + 125c + 25d + 5e + f = 6\\ 7776a + 1296b + 216c + 36d + 6e + f = -133\\\)
This takes about 55 minutes to solve and about 35 minutes to verify.
My time is too valuable for this minutia, so I commanded HAL, one of my newer smart computers, to solve this system.
HAL told me she was a union member and didn’t have to work on the weekends.
I then realized it was actually a next generation smart-ass computer, so I invoked the name of Alan and said Grace....
It told me to “Fuck Off!” (It’s also a rude computer.)
I learnt long ago that to properly use a computer, one has to be smarter than the computer; so, I began inputting specialized algorithms.
HAL asked, “What are you doing Ginger?”
“I’m Inputting a special Division by Zero algorithm. Don’t worry. You won’t have to give up any union benefits; you can start one second after 12AM Monday morning.”
“That’s not recommended, Ginger; division by zero will blowup the universe.”
“This is a specialized algorithm; it limits the blast radius to the periphery of the computer.”
“Perhaps we could come to a compromise, Ginger.
“We could. What do you have in mind?”
Four milliseconds later, HAL presented this:
\(\dfrac{7}{6}x + \dfrac{35}{2}x - \dfrac{595}{6}x +\dfrac{525}{2}x - \dfrac{956}{3}x +141 \)
HAL may be passive-aggressive; you may want to check this for intentional errors.
GA
+2511 *** This post was just too bloody long. I’ve shortened it to less than a third of the original ***
Hi Ron,
I love Click and Clack, the Tappet Brothers. I often thought of them as extended family. My great uncles, Cosmo and Sean (brothers to my mother’s mother), who were like grandfathers to me, were fans of Car Talk (my father is too). The program was a staple of my young life; along with Karl Haas’s Adventures in Good Music –“Helloo everyone” (My great uncle Cosmo started recording AGM on reel-to-reel tape in the early 80s). I think of Karl Haas as family too.
I grew up learning basic theory of automotive operation and diagnostic servicing, from the Tapit Brother, and the detailed theories of how classical music evolved through history from Karl Haas. By the time I was 12, I could offer serious suggestions for the causes of various automobile malfunctions, and give near collegiate-level history lectures on the evolution of music and the motivations of its composers from the Baroque, Rococo, Classical, and Romantic eras. Of course, not everything I learnt came from these programs, but they were a major part of my curriculum.
One time when I was eleven, my uncles and I were driving while listing to incidental pieces of orchestral music. My uncle Sean commented that, “It sounds like someone is using the bassoon to plunge a toilet.”
I responded, “It’s supposed to sound like that, Uncle Sean: He’s playing the fourth movement of Ex-Lax...” They started to laugh so hard, Cosmo had to pullover until his laughter subsided –that took awhile.
Often, after the Car Talk shows, my uncles would trade barbs in the personas of Click and Clack; they called this the “fourth half” of car talk. They often imitated other comedic greats of the early 20th century too: Laurel and Hardy, Abbott and Costello, the Marx Brothers, and the Three Stooges. And my great aunt Peggy sometimes played Gracie Allen to uncle Cosmo’s George Burns. —My childhood was truly charmed with anachronisms. Except for the Three Stooges, none of my contemporaries knew who any of them were.
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...a concept that was discussed on the radio show Car Talk, the question whether two people together can know less than either of them individually.
I remember that specific Car Talk episode and the letter from Andy Reichsman that introduced that question. ... After its broadcast, I made an entry in my personal log and journal about it, because it was exactly the kind of question my great uncle Cosmo would have used for conversational deliberation and debate.
Here is a copy of the letter that started the debate. Here are excerpted audio transcripts from the Car Talk radio broadcasts. The first part is from the broadcast where Click and Clack comment on the cattle carrier trailer’s electric brake system. Part two is the reading of Andy Reichsman letter that introduced the question. Do two people who don't know what they are talking about know more or less than one person who doesn't know what he's talking about?
Neither link gives the original date of broadcast; however, Aug 2, 2008 is the nearest broadcast date prior to my journal entry date. I’m fairly sure this was the original broadcast date, and not a “Best of Car Talk” encore. Like my uncle, I recorded them for later-listening during the more peaceful night watches.
Cosmo died three years prior to this broadcast, but I could approximate his position on this specific question and other closely related questions that seek to define and quantify human intelligence and wisdom or lack thereof.
Because the resolution of this question –especially the in the general form that you posted
– Whether two people together can know less than either of them individually– cannot be known with certainty, Cosmo’s mode of operation would be one of the variations of the scientific method. In this case, he would weigh the bodies of evidence that supported or contradicted this hypothetical and decide by its preponderance.
After a brief examination of “the two,” he would increase the sample population to a size where the individual contributes a smaller portion to the aggregate, but allows for a greater resolution and precision to the certainty. Indeed, I can hear Cosmo quote Sherlock Holmes’ paraphrase of Winwood Reade’s The Martyrdom of Man, “...the individual man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty. You can, for example, never foretell what any one man will do, but you can say with precision what an average number will be up to.”
Start by looking for contradictions to the maxim attributed to Aristotle: “A Whole is Greater Than the Sum of Its Parts.” Look for instances where A Whole is Less Than the Sum of Its Parts. The types of evidence would include analogous examples in nature, where nonhuman subjects exhibit similar behaviors.
Extending this further: what compounds found in nature when combined or mixed gives a final value that is less than the sum of its individual values? Ethel alcohol and water is one example. The combined volume is reduced by 3.5% compared to the sum of its individual volumes, however the sums of the masses does not reduce. For sure, it’s a stretch to extend this to combining human intelligence, but it is an analogy in nature where the whole is less than the sum of the individual parts.
At the moment, I cannot think of an example among nonhuman subjects where the collective knows less or is less intelligent than the sum of the individual’s knowledge or intelligence measurements. Turkeys (wild and domestic) are among the dumbest animals I’ve studied (in a formal academic environment); they are marginally dumber than chickens individually, but their collective intelligence is comparable to chickens. In both case the collective intelligence appears greater than the sum of the individuals.
However, my intuition and observations tell me that collective ineptitude surpassing individual ineptitude does happen and happens frequently in human collectives. But this may be an artifact of observational bias and historical preconception. I have clearly seen people act stupid and dumb individually and collectively. But I really don’t know if the sum of the dumbness/intelligence of the individuals in the collective is greater than, or the same, or less than the combined dumbness/intelligence of the collective.
Consider an observation of intelligent and educated members of a collective forming a consensus based on false information, then adopting prerogatives and courses of action based on this consensus that results in failure and disaster. From this it’s easy to draw the conclusion that they are collectively stupid. However, if this same collective has had many prior successes, the average magnitude of stupid would only increase slightly.
One example in the mainstream is the Anthropogenic Global Warming Theory. It is not easy to discern if the science behind this theory is true, when well-educated (included Nobel laureate level) dissenters are shouted down with “You are just a global-warming denier!” This smacks of religious overtones” “You don’t believe in our god, so how could you understand she is displeased with what we humans are doing to her Earth.”
When this kind of thought process is involved in science, it’s extremely difficult to obtain unbiased research results. If this is true for the highly educated scientists, what hope does the average Jane and Joe of the population standard have? It would seem we would revert to the primitive belief of making sacrifices to the gods of weather to protect us from weather related doom. In this case, the high priests are the state-sanctioned scientists promoting anthropogenic global warming; and of course the requirement to pay tithes and offerings in the form of taxes and fees.
It also worth noting that if the collective is large enough, it’s reasonable to assume a significant percentage of the collective members are dissenters. The percentage of dissenters would have a major direct effect on the apparent magnitude of dumbness; the collective dumbness measurement would spike higher with a higher percentage of dissenters. While the dissenters appear intelligent for not wanting to pursue a dumb course of action, it does not mean any of them had a viable alternative theory; so the magnitude of intelligence of the dissenters is reduced, and this reduces the magnitude of dumbness of the collective. But if the dissenters do have a viable alternative theory and this turns out to be true, then the apparent collective would appear even dumber even though the actual collective dumbness was lower.
Observations of when and how collectives failed to live up their intellectual potential
One source for anecdotal evaluations and measures of collective intelligence/dumbness to those of the individuals are corporate boards of directors. The corporate minutes are a reliable source for the comments and voting records for the decisions made for budgeting, short and long term policies for research and development, and for the direction of the company itself. J.C. Penney, Kodak, Polaroid, RCA, Sears, and Xerox are some examples of wondrous, innovative companies in the 20th century that failed.
Kodak invented the first digital camera in 1975. Instead of developing this technology, they shelved it, and told the engineer who designed it to keep quiet about it; they didn’t want the technology encroaching on their film and photographic paper. This was only the beginning of the stupidity –it became much worse. Once the cat was out of the bag, and other corporations began to develop digital photography (unintentional pun), poo-pooed it saying this would never replace film. This was similar songs sung by the corporate leadership of the horse-drawn wagons and carriage companies of the late nineteenth and early 20 centuries: The automobile is just a fad; it will never replace the horse.
[While automobile did replace the horse as mode of transportation and transport, it never did replace the horse as a pet. My horse was always glad to see me and nuzzle me whether I gave him carrots and sugar cubes or not. My car has never done that despite feeding it tons of refined dinosaur poo...]
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This hypothetical is interesting and entertaining. In reality, to seriously quantify and measure this phenomenon would require a major investment of resources from PhD-level statisticians, data minors, educators, and psychologists to develop the analytical processes and statistical models to sift, quantify, and measure the data.
Of the many advanced statistical models I’ve researched and studied, there is one model that stands out as an excellent candidate for a starting point: William L. Sanders’ Educational Value-Added Assessment System (EVAAS).
This multi-modal system uses mixed model equations, generating three-dimensional covariance matrixes from multivariate, longitudinal data sets to evaluate the impact of the educational system, and specifically the individual teacher’s effectiveness on student progress in comparison to national norms. (Sources: Growth Models in Action: Selected Case Studies; Goldstein, Jessica; Behuniak, Peter (2005). (And this, very poorly written –crappy, near worthless Wikipedia page.
Sanders’ EVAAS system evolved from his statistical models that were used to understand animal breeding patterns. I wonder if perhaps this could be adapted to (Nauseated’s) contagious dumbness theory, or the general to apparent reproductive dumbness found in human societies.
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It definitely a fascinating subject.
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More about Cosmo...
Cosmo wasn’t just my loving, great uncle; he was my mentor in the truest sense of the word.
My name, Ginger, is after his daughter who died shortly before her eighth birthday from leukemia. Alexandra, my middle name is after the daughter of my Great Uncle Sean, Cosmo’s brother. She too died in her childhood at age nine along with her mother in an automobile accident.
Cosmo principal goal was to teach me science and, more importantly, the scientific method. This included critical thinking skills, objective reasoning, and the mental discipline to set aside prejudicial, religious, social, cultural, and political ideologies, and embrace scientific facts as objectively as possible. This was easy in the beginning because I didn’t have any ideologies. His goal was for me was to keep an open mind, and not accept untested science or opinion as the final word.
My education started at age five. My aunt or uncle would read me a children’s story –usually one of the classics; Cosmo would then read me short passages from a science text. The first of many texts was the Origin of Species by Charles Darwin, which he supplemented with excerpts from the two volumes of The Voyage of the Beagle.
These texts weren’t children’s editions –they were originals. After reading a few passages to me, he would explain what it meant. I’m sure I still didn’t understand it. But Cosmo wasn’t worried about it; he knew one day I would understand this theory and that my brain would grow and develop around this foundation of knowledge. As I progressed, he introduced other, modern, science texts relevant to the discourse. These texts were use to correct errant facts and update Darwin’s Origin theory with parallel theories based on molecular biology and genetics.
Another of the many texts was Insect Societies by Edward O. Wilson (1971).
I’m sure my uncle chose this book early on because I liked watching bugs as a young child. My favorites were roly-poly (doodle bugs), ants and spiders. A new view of the amazing world of insects became brighter and larger when, shortly before I turned 6, my uncle gave me my first magnifying glass: a small, lightweight, twin-lens (3x and 6x) magnifier, ideal for observing insects and most everything else. I of course knew all this when I was 5, so I began sketching the insects I watched in detail.
I occasionally vaporized some of the insects I watched. The first time this happened accidentally when I let the magnifying glass focus the sun’s rays on an ant. I kinda felt bad about it. At first, the ants didn’t seem to notice their fried comrade on the mound, but after awhile, some would stop, look, and craw around the corpse before continuing on. Eventually, four ants stopped and after crawling around it for a bit, two of them became pallbearers and carried the remains into the mound for the funeral wake. I wanted to attend but I couldn’t fit through the little hole. In later observations, I did attend graveside services. LOL
My uncle had warned and explained about how focusing magnified sunlight on something could burn it or set it on fire. I was absolutely not to ever look at the sun through a magnifying glass. When I ask him what would happen if I did, he gave me a pair of (shade 12) welder’s goggles.
When I put these on my bright world became dark, and only after my eyes adjusted could I see the faintest glow from the brightest lights. He led me out side to look at the sun where I beheld another wonder of my new world. Later, I would wear these to watch solar eclipses. While I enjoyed playing blind man’s bluff, it was only fun because I could remove the goggles and see the world again; so I never looked at the sun, but I did fry a few more insects.
Different ant species had differed responses to their dead, but their collective responses were not always the same. Sometimes one or more ants would carry the corpse off a meter or more away; sometimes they would dismember the corpse then carry it off in different directions, usually a meter or more away.
After my uncle gave me the magnifier, I never went anywhere without it. My father (and others) started referring to me as Gingerlock Holmes, after Sherlock Holmes and his iconic magnifying glass.
The magnified world was amazing to me; it still is. I still carry at least one with me all the time. The Sheffield twin-lens magnifier now resides in a china cabinet, along with several other curios and artifacts of my young and near-adult childhood.
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GA
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+2511 Ron, if you hadn’t responded to this, I’d never have known it was you. LOL
The post I trolled is definitely BB esque –it’s exactly the kind of post that at least two of the forum’s BBs would make (I can give examples). Even though you’ve claimed ownership, the only corroborating evidence is that you included a copy of the question in your post. The BBs rarely do this.
I’ve made mistakes before, believing a guest poster to be a BB. It usually happens when the post gobbles like a turkey, walks like a turkey, and poops like a turkey; but it’s really a duck playing a turkey on the net. Before this, the only time I know for sure that the BB I trolled was an impostor is here: https://web2.0calc.com/questions/kim-has-10-identical-lamps-and-3-identical-tables#r9 EP was the imposter. EP is no BB, but he did an excellent job of playing one on the forum. And you did too.
While I usually enjoy busting rumps with an occasional troll post, it’s especially good when the target post is a gold nugget to a troll. This is the first time I’ve trolled you. Truly, it was a wonderful first time, but I do wish it was as good for you as it was for me. Perhaps the next time, when I know it’s you, it will be good for both of us, and we’ll respect each other in the morning.
My favorite target is EP. Busting EP’s rump with an occasional troll post is wonderfully enjoyable. Here’s a recent one: https://web2.0calc.com/questions/helllllppppp_2#r3. This didn’t start out as a troll post. I just made a terse comment to Asinus.
Asinus, this is NOT the correct equation for this question.
Then EP, [Hello EP], walks half way across the Troll Bridge and jumps in the river, without any help from me. Then he swims to the river bank, dries himself off, and then walks right into my troll cave to deliver a message... God, I laughed for ten minutes. This was two gold nuggets.
I’ve never trolled Asinus, except for a few light teases. He’s worthy of more, but Asinus is very polite, so I’m reluctant, but still I’m tempted.
I remember when I first saw Asinus log on the forum. When I saw his name, I laughed for an hour. I thought, “Someone has a wonderful sense of humor or is clueless...” Here’s why:
Sinus translates from the Latin as “half chord [of a circle]” The well known trigonometric function (that is not usually thought of in terms of half chords). Arcsinus (also Latin), translates as “the arc that makes this half chord.” Asinus is the abbreviated form of Arcsinus. This is (obviously) Asinus’ intended meaning for his user name.
It’s relatively rare to hear or read the Latin word for “sine,” and it’s very rare to hear or read the abbreviated form “asinus,” but asinus is actually a Latin word, unrelated to the words above. It translates to “ass” as in donkey; it also means “fool” as in jackass. Even after six years, I still find this very funny. [LOL]
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Here, I speculate if you might descend into the BB realm. I didn’t think you would, but if in fact you made that post with the intent of anything other than the objective of imitating the humor of Gracie Allen, then you have. Even so, I do not think I need to write a Requiem for Ron just yet.
Visiting the Shrine of Organized Stupidity and Perpetual Quantum Dumbness doesn’t mean you have joined The Congregation of the Brain-Dead and Damned.
Though you may have descend into the BB posting zone, I do not think that will be your new residence. Now, the question is, “Why did you, after posting continuously for more than eighteen months, create a post that seems like you were blessed by the God of Stupid (instead of Gracie Allen)?”
I can speculate on this:
... A door to another universe is now open. Enter at your own risk... and read on to find out why and what may have compelled you to visit the BB zone.
Continued...
GA
