The Alexandrian–nay, Gaussian–Solution

Carl Friedrich Gauss

A year ago, I wrote about “the Alexandrian solution” to the Gordian Knot. I saw this as a metaphor for all instances in which genius lies in espying the simplicity hiding in a complex situation.

It just occurred to me that Carl Friedrich Gauss was, at the age of 10, just such an Alexander the Great. (Alexander was young, too, of course. In espying simplicity, it seems to help to be young — ie, intellectually daring, unspoiled by the complexity of life, et cetera.)

In about 1787, the young Carl Friedrich sat in class when the teacher told the kids to find the sum of the numbers 1 through 100. In other words:

1 + 2 + 3 … + 100 = ?

Think of this as the Gordian Knot. The teacher assumed that the kids would be busy for a long time, practicing their addition skills. Gauss reacted just as Alexander would have (I take poetic license):

This is too f***ing boring. There must be a simpler way.

Did Gauss get nervous as the other kids pulled ahead adding numbers, while he was still at 1, searching for simplicity? I don’t know. But he found it:

He realized that the numbers came in pairs:

1 + 100 = 101
2 + 99 = 101
3 + 98 = 101

(and so on until:)

50 + 51 = 101

So the sum of the numbers is simply (simply!)

50 x 101, or 5,050

You might, if you’re a regular on The Hannibal Blog, be guessing that I’m much less interested in sums of numbers than in, shall we say, Gordian Knots and Alexandrian Solutions in general — meaning in other, preferably surprising, walks of life.

If you can think of any instances in which daring simplicity blasted through mind-numbing complexity, drop me a line.

47 thoughts on “The Alexandrian–nay, Gaussian–Solution

  1. One wonders how he arrived at the simple solution. Did he dutifully add the pairs of numbers until he realized (and the point of “51+50=101” and intuitively realized that the next 50 would be the opposite positioning? Or did he glance at the first few and see the pattern? Or did he just sit and stare at the blank paper until an epiphany hit him?

    • That should read “(at the point of….” not “(and the point of…”

      I am blaming this keyboard. It keeps doing what I tell it and not what I want it to do.

    • I suspect that it was intuition. But this, you see, is exactly what fascinates me about all “Alexandrian solutions”: How exactly DO they come about?

      What is the neuroscience of it? Do we relax into it, thus allowing some coincidental synapses to join hitherto unconnected neuron patterns? Or do we scan through possible patterns in a process of elimination? Etc etc.

    • I seem to recall an article in Psychology Today back in the late 70’s that talked about problem solving. As I dimly recall, there were two primary methods with all others being a blending of the two in various ways. The two were linear and intuitive. Intuitive problem solving was the epiphany type with a touch of trial and error. Linear involved following a logical progression. I may be misremembering the article, it has been over 35 years.

      As I learned over my years as a troubleshooter, we tend to not stick to one or the other but constantly varied the blending formula of the two. I would often walk away from a sticky problem that seemed unresolvable and work on something else for awhile. When I would return to the tough problem, a solution would seem to reveal itself (epiphany) and that solution was often the right one. It was, I think, the distraction of doing something else that allowed that to happen. The type of distraction was less important than the fact that there was a distraction.

  2. In the version of the story I heard which gives it a nice Hollywood twist, Gauss was being a pain in class by demonstrating that he knew more than the teacher. The teacher gave him this problem to shut him up and (so the teacher thought) keep him busy for a while.

    I can’t think of an answer to your question off the top of my head but will give it some thought. It would be very interesting to look at history and find examples of when he Gauss du jure came up with a solution of daring simplicity but it was beaten down by the establishment.

    • Yes, I heard that one, too. I wonder which makes the better movie script.

      “.. a solution of daring simplicity that was beaten down by the establishment…”

      You mean this, of course:

    • Yes, exactly what I had in mind. Plus there are probably lots of examples in politics–like George Ball’s opposition to escalation in Vietnam.

      I haven’t come up with an additional example in response to your challenge to find another Alexander or Gauss but I’m working on it. Would Samuel Johnson’s refutation of the concept of immaterialism (that matter didn’t really exist) by kicking a rock and saying “I refute it thus” qualify as “instances in which daring simplicity blasted through mind-numbing complexity”?

  3. Rumor has it that one day, Frank Sinatra was playing cards in his movie trailer when a production assistant entered to remind Sinatra and his buddies that shooting was running behind schedule and that he was expected on the set. Whereupon Sinatra allegedly reached for the script, ripped out a few pages, said “Now we’re back on schedule,” and turned back to his cards game.

    Although I don’t believe this tale for a second, it does demonstrate mind-numbing simplicity.

  4. This true story of Gauss has all the romance of hero-worship:

    CF Gauss 1777 – 1855

    Napoleon 1769 – 1821
    Nelson 1758 – 1805
    Beethoven 1770 – 1827
    Mozart 1756 – 1791
    JMW Turner 1775 – 1851
    Jane Austen 1775 – 1817
    Edgar Allan Poe 1809 – 1849
    Schubert 1797 – 1828
    James Watt 1736 – 1819
    Faraday 1791 – 1867
    Jefferson 1743-1826

    Where does one stop?

    • Those are great men, one great woman, and great lives. Some may have solved Gordian Knots with Alexandrian solutions. Which do you have in mind?

    • I missed out Darwin 1809-1882, who brought an ordered understanding of life itself.

      Each named person solved what appear to be different types of problem, but they all resolve in the end to the exercise of intuition, as someone has just said. The list is not intended to be complete and there are glaring omissions – the worst omission is the mathematician Ada Lovelace, 1815- 1852, daughter of Lord Byron and the pioneer of computing. The omission exposes my chauvinism.

      Austerlitz, Trafalgar, 1805, the year of the fifth symphony – Beethoven prefigures the solution to his personal problems not found until the late quartets disclosing the profoundest insights on the way – Mozart, who worked out his compositions in detail before committing them to paper, Turner the pioneer of impressionism within the representative, Austen who again found order in social chaos … there is so much to say and a myriad examples …

      Poe – The Murders in the Rue Morgue – Faraday who conceived the idea of a field –

      But the supreme problem-solver is Gauss. There are so many examples, but perhaps the one we take for granted is the theory of congruences, the arithmetic of remainders. In the field of Astronomy, there is his rediscovery of Ceres by calculating its orbit with amazing accuracy.

      The mathematician takes his intuition a step further and is not satisfied until his answers are confirmed with the utmost rigour. Even so, Godel showed that absolute rigour was never possible – using rigour itself! He is not on my list because he was not contemporary with the others and so not a pioneer of the romantic movement. So Mathematics is an invention drawn from the close observation of Nature.

      The drama of the lives of so many mathematicians is a reflection of the drama they unfold in their work. That drama is the drama of existence. Intuition may be swift and fertile, but we do not always know the pain and the struggles which prepare the ground.

    • That’s a very good brainstorm.

      Perhaps one day — not here, necessarily — you’ll spell out the rest.

      I’m especially intrigued by the non-science examples (the science examples tend to be “easier” in this context). For instance, Mozart and Beethoven: what makes their music beautiful is its complexity, no? But then, if Mozart really conceived it all in his head, he must have seen the simple thread running through it.

      Austen: “order in social chaos”… ORDER hiding in CHAOS. Like it. Intrigued.

      Trafalgar: must review the battle. Did he do some astonishingly simple maneuver?

    • Nelson cut through the enemy lines at a right angle. Extraordinarily daring, but it meant the enemy was shot through from stem to stern.

    • A subsidiary question is the timing of the instantaneous insight. Is it before, when or after it is reduced to language?

      I’d opt for before based not only on analogy with observations which have been held to indicate that the brain prepares itself for action before the decision is consciously made but also on that experience of thinking without language – meditation.

      This post is such a potent one, Andreas.

      Is all this a vindication of platonic forms? I suppose you have to decide whether truth is floating around in some unsensed paradise or is present for us all to see. The only way the two can be reconciled is to view the brain as a sense organ for ideas. Ideas would then need to be deemed external to us. But then, what is the difference between external and internal? Round in a circle we go!

      I use thinking here in its general meaning, not as defined by Jung.

  5. This article has a nice example of an elegant and simple solution to a geometry problem:

    I did spend some time staring at it, trying to come up with the intuitive solution, but I could only think of the ‘logical’ way to solve it, using the formulae I was taught at school. The ‘inductive chain’ method the article talks about does seem like a good way to ‘teach’ intuitive thinking…

    • Very interesting article. Piqued my curiosity… as it should. While reading it, I found myself thinking about why I disappointed my teachers more than I impressed them. And about why I was so bored as I went through school. And why I enjoyed my career as a technician with a huge telecommunications company. I will have to investigate Project Euler further. Even at my advancing age, I can still (and want to) learn.

      One thing that popped into my mind while reading the article was that children are very much like computers, we begin “programming” them immediately after birth. Some have suggested we can even do a bit of this while the child is in the womb.

    • This is a fine article, Susan, but relates less to the Gaussian reverse-the-mean insight and more to his rediscovery of Ceres.

      The most ground-breaking revelations are initially without algorithms. Learning to code in high-level language depends upon algorithms devised by others and so involves a requirement of conformity – a loss of freedom. There is something more to the human brain.

  6. I’m in DC, at a conference preoccupied with proposals for federal mortgage finance reform. Securitization, primary and secondary markets, QRMs and QMs (whatever they might be).

    The presenters seem agitated; the attendees look confused.

    Someone said: “It’s complicated, but it’s also very simple: Let’s remember that banks exist to serve the people, all the people.”

    At some other gathering of equally well-intentioned people, no doubt, someone is saying, “It’s all very simple: Banks exist to make money.”

    Makes me yearn for a math problem.

    • What a conference to be at. What did you do to deserve this?

      Actually, I already envision some Alexandrian Solutions for the Gordian Knot of mortgage reform, but that would be simultaneously boring and controversial (bad combo) to put here.

      My advice: Run. Just keep going, till you hit a poetry book…

    • Securitisation – swaps, vehicles, bonds and whatnot – must have its fascinating side, jenny, and it is after all a key player in the global economic rescue whether we like it or not. Tell me, isn’t it just a new-fangled way of raising capital and minimising risk going by a fancy name?

    • Richard, I suppose that securitization fascinates many people, though Andreas-the-Mortgage-Reform-Tease is right that I (at least mentally) ran fast to poetry when the topic was introduced. I don’t have a gripe with the idea of it. At least, I don’t think I do.

      I worry that specific reforms will shut people (the ones without a lot of cash) out of the market, or force them to borrow under very disadvantageous terms.

    • I was just teasing. Mortgages are as worthy of elegant thought as anything else.

      I worry more about the opposite of your worry: that too many people are included (when they really “should” be shut out), as happened in the run up to the recent crisis. Ie, people take on debt when they can’t service debt.

      As a separate point, when we favor owners (with the tax deduction, say) we necessarily punish renters. And why should we punish anybody? So out must go that silly tax deduction. (Admittedly, the deduction is not what’s being discussed at your conference.)

      Securitization is here to stay, but it was a bad innovation in my view: It introduced an “agency” problem into lending that used not to exist. In the old days a banker (picture Jimmy Stewart) lent to individuals and THEN HAD TO HOPE TO BE PAID BACK. Now a “banker” flogs a loan, takes a commission, and passes it on to a securitization entity that treats its assets (borrowers) and liabilities (lenders) anonymously. the banker doesn’t give a hoot about the lender’s credit. I think that’s the problem.

      So, in a very round-about way, you could say even in this context that complexity is to blame. In the old (simple) world, borrowers and lenders knew one another. in the new (complex) world, they do not, and that MUST be unstable.

    • Securitisation has a bad name because of its banking/ mortgage connection, but it is still a smart way of raising capital, not least by ailing EU economies from the ECB. 🙂

    • Andreas,

      Was the problem that too many people were included? Yeah, probably, in many cases. But how about the other story? It goes something like this:

      Banks steered unsuspecting borrowers toward bad loan products, even when those borrowers qualified for something better. In other words, these folks could service the debt, they just couldn’t service the terms of the loan they got. I follow these lawsuits against the anti-Jimmy Stewart:

      The tax deduction can go, for all I care.

      Traditionally (who knows, maybe those days are over), the most common way to amass wealth in America was through home ownership. Do we want folks who can’t come up with 20% down to be left out of that opportunity? 20% down appears to be the kind of loan that the reforms will encourage.

    • You’ve heard The Pool of Money on This American Life about all this, haven’t you? If not, highly worth it.

      The problem was lots of things, of course, and bad borrowing and bad lending cut like blades of a pair of scissors.

      But since you ask, we need to let people amass their wealth however they will, whether in houses or something else. And where there is nothing to be amassed, nothing will be amassed. Changing the rules to fake amassing would only hurt.

      Also, the word “tradition” refers to a time in history arbitrarily chosen. 19th century? 20th? Recent bubble?

    • OK, I’ll think about that, Andreas.

      Traditionally (here referring to that time in history that begins, arbitrarily, with my birth), I understand bupkis about economics.

  7. @Jenny – Regarding the topic of money, Robert Reich (remember him?) has reportedly said that if, in America, the rich were taxed at the same rates today as they were fifty years ago (the Age of Eisenhower) there would be a Federal budget surplus.

    This bespeaks a simple solution, made apparently insoluble because voters have been brainwashed, and also don’t know much about history.

    • Was he saying the age of deficits began because JFK cut taxes for the rich?

      Robert Reich is not my favorite economist. He forgets that there was no Medicare then, there was a draft, wages were tiny compared to today, that government was much smaller. For instance, the following departments did not exist during Eisenhower’s term:

      Housing and Urban Development
      Veterans Affairs
      Environmental Protection Agency
      Homeland Security

      The existence of deficits is caused by a very simple thing: Government spending more than it takes in. In fact, spending more than what is projected to be received.

      Simplest solution has nothing to do with tax rates, it has to do with spending within revenue.

  8. @Douglas – After tracking down the article that mentioned what Robert Reich said about the US budget deficit, I see that I misremembered it slightly. The piece, by [1] Michael Tomasky, then of the Guardian (UK), said in part:“…If wealthy Americans were paying taxes at the rate they did 50 years ago, says former Clinton labour secretary Robert Reich, the government would be taking in $350bn more a year: budget woes over….”

    Given that the US government has run budget deficits almost uninterruptedly [2] since at least the beginning of the twentieth century, I should have suspected that I’d quoted Michael Tomasky incorrectly.

    Looking at [3] Robert Reich’s piece, I see that he said the following: “…If the rich were taxed at the same rates they were half a century ago, they’d be paying in over $350 billion more this year alone, which translates into trillions over the next decade. That’s enough to accomplish everything the nation needs while also reducing future deficits…”.

    Lest you think I’m pointing fingers at the US unfairly, you should know that here in Canada the income tax rates on the very rich, at around 46%, are not that much higher than the 35% rate on the very rich in the US. And corporations – the playgrounds of the rich, after all – are taxed on profits at around 25% to 30%, which may be slightly lower than US corporate tax rates.

    In the UK, one of its biggest Banks, Barclays, last year made a profit of $11 billion, but paid only $100 million in corporate taxes – less than 1% of profits. So there we go. Compare all this to 30 and 40 years ago, when corporations the world over were taxed at around 50% of profits.

    You said: “…..The existence of deficits is caused by a very simple thing: Government spending more than it takes in. In fact, spending more than what is projected to be received……”

    I agree absolutely. Taxing the rich at the rates of 50 years ago would increase the amounts taken in, thereby making the US deficit manageable, as Robert Reich implies.

    You said of the times of fifty years ago: “….wages were tiny compared to today….”

    Given that, after eliminating inflation, the size of the economy then was less than half of that of today – perhaps even a third – wages then, for 90% of workers, certainly would have been smaller. But, in relation to the size of the economy, most workers then, would have earned perhaps the equivalent of twice as much as most workers today.

    On the other hand, people in the top 10% income bracket earn today, in relation to the size of the economy, many times more than people in the top 10% income bracket earned then. Income inequality today in the US is purportedly as big as it was in 1924 (or was it 1928?)




    • A most interesting analysis. It didn’t mention, though, that large unforeseen healthcare costs are a leading cause of bankruptcy in the US – a cause of bankruptcy presumably absent in the other countries on the list.

    • @Philippe, Here is something simple which Reich (et al) have overlooked in that premise. For all those deficit years, the government spent more than it took in. Reich fails to acknowledge the cause of deficits in this manner. He feels that government is lacking sufficient revenue, not that it is practicing fiscal irresponsibility.

      Take his premise to an individual’s situation. Joe Smith makes $50,000 per year. He nets say, $40,000 after income and payroll taxes. He and his family save no money but spend $45,000 on essentials and luxuries. Putting Joe in a $5,000 hole each year.

      How does Joe get out of that hole? By going to his boss and demanding he receive a $6,000 per year pay raise? Or by cutting his non-essential spending AFTER reviewing what is essential and what is not?

      Raising taxes treats the symptom, not the disease.

  9. I’m stuck on the daring and blasting, rather than the simplicity. Something that involves a duel with pistols would be perfect.

    Nonetheless, the first thing I thought of was the principle of least time. Most people would say that the shortest distance between two points is a straight line. Instead, according to Pierre Fermat, it would be the path of least time. For example, if you were a ray of light headed for the swimming pool, you would change direction slightly when you hit the water in order to get to the bottom faster.

    You can see the principle of least time (also see principle of least action) at Wikipedia. There has to be a story there in the history section. In objection to Fermat’s principle (from Wikipedia) ,
    Clerselier states:
    … Fermat’s principle can not be the cause, for otherwise we would be attributing knowledge to nature: and here, by nature, we understand only that order and lawfulness in the world, such as it is, which acts without foreknowledge, without choice, but by a necessary determination.
    I know this is a translation, but such an argument is suspect for being so confusing.

    • Clerselier (sorry, I haven’t read the article) confuses cause and observation. We observe consistency concerning light rays and geodesics, but that tells us nothing about the source of that consistency.

    • The principle of least time. I like that, Mr C.

      (You also got the right idea re blasting and pistols: We’re looking for explosive stuff here.)

      Let me invest some time so that I actually understand the theorem.

    • These seering, instantaneous realisations are often so obvious, though, in retrospect. That is what simple means in this context.

      Unwilling to accept intuition at face value, these great minds then work backwards, seeking consistency with the overall body of knowledge or exposing error or incompleteness in what has gone before.

      The example of Gauss and Ceres is, true enough, rather different because it uses established techniques, but it is the same mental process that sees through to the answer first and then justifies it.

      Trafalgar is not so far from this process either, because the manoeuvre was entirely novel, yet had to be worked out in detail. A sub-plot here is how Nelson allowed his commanders full discretion in the execution of the plan. What of the sniper who did for Nelson? No, he was an opportunist – opportunism is only one element of the “Alexandrian Solution”.

  10. @Douglas – In the matter of Joe Smith, if he’s your average Joe, he most likely does spend more each year than he earns, to the tune of *26% more* than what he earns. This is as it should be, for if all the average Joes stopped spending this way, the economy would go into a recession heaps worse than the one now.

    And the beauty of Joe’s debt situation is that he never has to repay all of what he owes to the credit card companies and banks. As he repays, he simply borrows more to cover his repayment. As it is for Joe, so it is for governments.

    Thinking that Joe, and governments, mustn’t spend more than they take in is so……….twentieth century.

    • I am hoping that there is a sarcasm dripping from your words. Though there is much truth there, too. Sadly.

      I never spent more than I owned except in my early years. Now I purchase outright on most things and pay off credit cards each month. My sweet (now departed) mother used to say “I never worry about hitting my limit on my credit cards. When I get close, they just raise it for me.”

  11. Cartesian geometry allows the easy solution of a wide range of problems of motion that were mind-numbingly difficult with a compass and straightedge, the geometry of the Greeks.

    Germ theory provides a sound theoretical basis for innoculation, vaccination, antiseptics, sterilization, and antibiotics, thereby helping extend human life, while pasteurization helps extend the life of milk and beer. 

    The Golden Rule encapsulates much of moral law.

    • Examples with potential, Jim. M. But Cartesian geometry and germ theory are not really Alexandrian Solutions, are they, but rather bodies of insight, in themselves containing complexity and Gordian Knots.

      the Golden Rule, however, is a bit of a knot-cutter indeed….

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