A Deep Thought, Monty Hall, and Trisecting Angles

Brian Hayes, an American Scientist senior writer, wrote this month's Computing Science column titled Foolproof. It discusses the increasingly complex nature of mathematical proofs, and at one point he expresses this, at least to me, rather deep thought: But it's an interesting predicament. If something is beyond your ability to understand, how do you know it's correct? You can only take someone else's word for it, something we're reluctant to do here. But this is the boat I'm in with regard to population genetics and information science. I can follow the math up a to point, but beyond that point I have to just accept that the math supports the theory.Hayes follows his observation with several examples, one of which is the now infamous Monty Hall problem. Though he tells us the correct answer, he doesn't describe why it's the correct answer. When I first read about this problem I didn't really follow why it was correct, so this time I began writing a computer program to prove it's correct.I didn't have to code for even five minutes to discover the answer, and I didn't bother completing the program. The answer emerges immediately from the way you have to structure the conditionals. When you make your initial guess you have a 1/3 probability of being correct. Now Monty Hall opens one of the other doors showing the big prize is not there. This action does not change the probability at all. Whether you've chose correctly or not, there will always be a door Monty can open that does not contain the prize. In other words, Monty has not actually provided you any additional information. There is still a 1/3 probability that the prize is behind the door you've chosen. Which means, since there's only one remaining door, there's a 2/3 probability that it hides the big prize!!!To really understand this it helps to consider each case. Imagine you've chosen correctly, which happens 1/3 of the time. If you change you lose, so 1/3 of time is a loss, which means 2/3 of the time is a win. Imagining you've chosen incorrectly yields the same answer. You'll choose incorrectly 2/3 of the time. If you change you win, so 2/3 of the time is a win.Hayes goes on to explain why trisecting angles is impossible. I'd never seen it explained before, and he tells us that it is explained very infrequently. It turns out it's impossible because you can do square roots and 4th roots and 8th roots using geometry, but not cube roots. To trisect an angle you need to use geometry to compute a cube root, something known to be impossible.But, I would ask him, how does he know that taking a cube root is the only approach to trisecting an angle. True, one way of trisecting an angle requires taking a cube root, but is that the only way, and can it be proved?--Percy

Not to diminish Percy's post, but I felt that people who aren't as enthusiastic about mathematical puzzles might appreciate an up-front description of the Monty Hall problem:

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Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

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Monty Hall problem - Wikipedia

I find considering it in terms of ten boxes/cups/doors/whatever You pick one and Monty says you either get to open that door, or the other nine. Which do you choose?As for trisecting an angle, I couldn't help you. Geometry was so far in my past and has rarely been used since.

It's important to note that the usual argument relies on Monty knowingly choosing a door which does not have the prize behind it. If he chooses a door at random the probabilities balance again - it's a straight 50/50. Working it out is a bit more complicated but it's true.

You have email at your EvC address.--Percy

During plane geometry in the 8th grade, construction component, we learned how to bisect any angle and demonstrate the proof; the teacher told us it was impossible to trisect an angle with construction, and merely replied to my question that the math that proved this was beyond my comprehension.So I set about trisecting an angle with construction. Many weeks and reams of paper later, I came back with three methods: one for a 90 degree angle, one for acute angles, and one for obtuse angles. He pointed out that my solution required knowing which category any given angle belonged to and so was no solution at all. I still refused to accept the impossibility, though I stopped my mad nights with compass and protractor. I did learn a great deal about plane geometry in the process, though, and appreciated the difficulty as a matter of converting "odds" to "evens"...The difficulty of our increasingly specialized world goes beyond mathematics: we are encouraged to take part in medical decisions, but our knowledge can never equal that of the physicians asking us to make those decisions; we necessarily put our lives in the hands of the engineers who design our vehicles, ground and air, the workers who build them, and the mechanics who repair and maintain them. The list is as long as the roster of specialized roles in a technological society.The best we can do, I think, is to maintain a healthy skepticism and to examine past results and the opinions of the experts' peers: the aeronautical engineer who claims her radical half-wing design is perfectly safe has to be judged in light of the opinions of other engineers and the track record of her invention. The real danger lies in accepting the first "credentialed" opinion one encounters.I never completely accepted the impossibility of trisecting an angle, because it was never demonstrated to me, mostly because my six semesters of college math stopped after calculus and analytic geometry, though in fact I never asked about it again.I'm comfortable living with my uninformed skepticism for two reasons: the cost of informing that skepticism is high, and the relevance of the truth of the matter to my daily life is low. I suppose we all negotiate those parameters.Edited by Omnivorous, : topy

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Hi, Percy.

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But, I would ask him, how does he know that taking a cube root is the only approach to trisecting an angle. True, one way of trisecting an angle requires taking a cube root, but is that the only way, and can it be proved?

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I think you may have misread the argument. Hayes' argument is basically the same as the one I am looking at in Hungerford's Algebra, pp238-240. Here is how the argument goes:1. It is a fact, proved using elementary geometry, that 2. Therefore, 3. It is a fact from elementary trigonometry, proven using elementary geometry, that cos 60 = 1/2.4. Therefore, 5. Multiplying by 2 and moving all the terms to one side, we have 6. Hence, cos 20 is a root of the polynomial 7. This polynomial is irreducible, that is, it cannot be factored, that is, it has no rational solution.8. Therefore, cos 20 must be irrational.9. The polynomial is of degree 3, which is not a power of 2.10. If an irrational number is not a root of a irreducible polynomial of degree a power of 2, then it cannot be constructed solely with a straight-edge and compass.11. Therefore, a segment of length cos 20 cannot be so constructed.12. Now assume that any angle can be trisected. In particular, a 60 degree angle can be trisected.13. Therefore, a 20 degree angle can be constructed with compass and ruler.14. Therefore, a segment of length cos 20 can be constructed.15. (11) and (14) are contradictory.16. Therefore, not all angles can be trisected.Most of these points I understand well; I'm right now trying to understand (10).Anyway, the argument in the essay is not that you need to construct a cube root in order to trisect an angle. The argument is that being able to trisect a particular angle would imply something about the roots of a particular polynomial that we know is not true.Hope this helps.Added by edit:Oops. It's been too long since I've had algebra. I've used the word irreducible inappropriately here. But I didn't want to explain what it means to say "cos 20 must be algebraic of degree a power of 2 over the field of rational numbers."Added by another edit:Maybe my use of irreducible here is alright. Man, I have to review my basic (graduate level) algebra.Edited by Chiroptera, : typo in (12)Edited by Chiroptera, : No reason given.Edited by Chiroptera, : No reason given.

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Never believe anything in politics until it has been officially denied. -- Otto von Bismarck

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He pointed out that my solution required knowing which category any given angle belonged to and so was no solution at all.

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Your teacher was wrong: in principle it is possible to determine whether an angle is acute, obtuse, or right, and so a solution that requires this knowledge is not invalid for that reason.We know a right angle (90 degrees) can be trisected; 1/3 of 90 is 30, and a 30 degree angle can be constructed.But your solution for the acute and obtuse cases must have had a flaw which your teacher did not discover.

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Never believe anything in politics until it has been officially denied. -- Otto von Bismarck

I'll echo Chiro by saying that the problem of trisecting an angle is specifically that it is impossible to trisect an arbitrary angle using only a compass and a straight edge. As Chiro demonstrated with a concrete example, there is a property derived from Galois theorey called 'constructable' for which you can prove that certain geometric elements can or cannot be constructable.You can prove lots of neat things such as the constructability of certain n-gons.

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Of course, biblical creationists are committed to belief in God's written Word, the Bible, which forbids bearing false witness; --AIG (lest they forget)

Agreed, the solution may have looked right but it most certainly was not a solution in the QED sense.

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Of course, biblical creationists are committed to belief in God's written Word, the Bible, which forbids bearing false witness; --AIG (lest they forget)

He didn't try very hard. I'll take your word for the flaws .

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That's not quite the argument. The argument is that if you had any straightedge-and-compasses method of trisecting the angle, then you could use this method to find non-rational roots of certain cubic equations (the author of your article instances 8u³ - 6u = 1). But this is impossible to do using straightedge and compasses, so you can't trisect the angle. It's not: "To trisect an angle you need to use geometry to compute the root of a cubic equation" but: "If you could trisect an angle then you would be able to use geometry to compute the root of a cubic equation".Edited by Dr Adequate, : No reason given.

That's true. They did it this way, because that's what Monty Hall really did on the show. He wouldn't have opened the door with the prize behind it, because that gives everything away.You probably already know that. I'm only pointing this out, because your statement threw me, and I thought you were arguing that the problem was being given wrong.

I've always felt very dwarfed out that the 19-year old Carl Friedrich Gauss proved a straight edge & compass can only make a few regular polygons up the astounding 17-sided one. 19 years old!RAZD & I & our other brother were challenged by our dad once to prove the construction of a regular pentagon was correct & true. Years went by. Other things got in the way. Finally I had some time to address the problem and proved it, but only with a very inelegant procedure ultimately relying on the graph of a cubic and showing that of the 3 roots, only one was usable and it was the correct one, using the sqrt(5) as derived from the other direction. I had basically drilled a tunnel from both sides and met in the middle with this solution. I presented it to my dad & my brothers and could see in all their eyes the same dissatisfaction I had. Oh well.19 years old!
There are some who think Newton was the Genius of Geniuses, some might toss in a vote for Einstein. But Mr. Gauss has to be way up there in my mind.

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- xongsmith, 5.7d

I also worked at this, and first noted that the ratio of the diagonal to the side of a pentagon appeared to be the golden ratio, then proved it with algebra. (this was in the process of making a geodetic mobile using all 6 of the polyhedrons with equilateral sided faces plus ones made in the manner of Buckminster Fuller's domes).
PolyhedronsAnd like Omnivorous, I spent many a night after college trying to trisect an angle. I developed a methodology that got close enough for practical use (imho), but never an exact solution. Wish I'd had access to the internet and the formula provided by Chiroptera.Enjoy.

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