## No Magic In A Knight's Tour 278

Posted
by
Hemos

from the brute-force-it-baby dept.

from the brute-force-it-baby dept.

morgothan writes

*"As reported in an article on Math World the solution, or rather lack of solution has been found to the over one hundred fifty year old math problem of how many numbers of magic tours a knight can make on a standard 8x8 chessboard. It turn out that there exist one hundred forty distinct semimagic tours, but no magic tour. The solution came after 61.40 CPU-days, corresponding to 138.25 days of computation at 1 GHz, the project was completed on August 5, 2003 in which every possible enumeration was tried out. The author of the software that finally solved the problem has also put up a webpage in which he further explains the problem and his method of solving it."*Thanks to Mig for pointing out a great background page on Chessbase.com.
## Google Cache of the Web Page (Score:3, Informative)

## Magic Tours... (Score:3, Funny)

No wonder he didnt only got a Semimagic tour, damn you Travelocity! Damn you to hell!

## Re:Magic Tours... (Score:3, Funny)

maybe I'm on a "magic tour" right now!!

## Re:Magic Tours... (Score:3, Funny)

-B

## Re:Magic Tours... (Score:2)

## Re:Magic Tours... (Score:2, Funny)

Maybe if the Knight REALLY wanted to take a Magic Tour he'd consult a Wizard instead of a Computer...Did the Knight beat the Bishop on such a long, lonely journey, I wonder?

## Tired out (Score:3, Funny)

The solution came after 61.40 CPU-days, corresponding to 138.25 days of computation at 1 GHzI bet the horse was tired after hopping around so much.

## Clip clop (Score:5, Funny)

I bet the horse was tired after hopping around so much.Nope, but they got through about six coconuts.

## For our next experiment... (Score:5, Funny)

Now we're going to examine how many routes there are through all the bars in Amsterdam, and see if there are any "magic" routes that will let us complete the circuit without falling drunk in a bed of tulips.

## Re:For our next experiment... (Score:3, Funny)

Purely for medicinal purposes, you understand...

## That's nice, but not impressive (Score:5, Interesting)

Whatever happened to the classic style of problem-solving, whereby an actual

proofwas deduced, rather than simply employing brute-force to "count them all?" I mean, don't get me wrong, I'm glad that we finally have an answer to this pressing question (which I'm sure 95% of us had never heard of prior to reading this article), but does anyone else feel that these "brute-force" solutions are kind of cheating?## Re:That's nice, but not impressive (Score:5, Insightful)

150 yearslooking for a proof, or<150 daysof brute force, I would have to admit that the brute force approach was better.....## Re:That's nice, but not impressive (Score:4, Insightful)

## Re:That's nice, but not impressive (Score:2, Informative)

## Re:That's nice, but not impressive (Score:2, Funny)

Given: an 8x8 chess board, a black knight piece, a Cray supercomputer

1. Execute brutechess.exe

2. Therefore, there are no magic knight's tours.

## Re:That's nice, but not impressive (Score:2)

And you are correct about there being more pressing questions, such as prime numbers! [mersenne.org]

## Re:That's nice, but not impressive (Score:4, Insightful)

## 8x8 a special case. (Score:4, Informative)

Brute forcing to get a proof is better than no proof at all, but brute forcing a special case is even more acceptable.

## Re:That's nice, but not impressive (Score:5, Informative)

Not necessarily. As the article points out, there

isa proof for certain sizes of chessboard, but it doesn't extend down to 8x8. Not all mathematical proofs are quite as neat as we might like them to be in terms of how they apply to other variants on the same problem.## Re:That's nice, but not impressive (Score:2)

It also points out that everybody who actually cared one bit about the problem had fairly well already decided that there was no solve for boardsize == 8. So, effectively, someone blew 50+ days of CPU time proving something that nobody really cared about, and determined an answer that everybody already knew.

It's kind of like Microsoft Windows: The answer to a problem that no one really had.

## Re:That's nice, but not impressive (Score:2)

## Re:That's nice, but not impressive (Score:5, Insightful)

I mean this way we have one of the two, all that remains now is to turn the algorithm used into a formula for mathematical verification and there you have it.

In a way, the algorithm used is ALREADY a mathematical proof, inasfar as an algorithm can be proven correct using math...

## Re:That's nice, but not impressive (Score:2)

## Re:That's nice, but not impressive (Score:2, Interesting)

In Computer Science courses (university level, maybe before), you learn to take (some) algorithms and express them as mathematical formulas.

Sums using sigma notation and whatnot, if you are familiar.

So doesn't that mean that if you have the algorithm to solve the problem, you have the PROOF?

Admittedly, it might be very difficult to translate the algorithm into a standard mathematical formula, but it should be possible, right?

I am arguing that an alg

## Re:That's nice, but not impressive (Score:4, Interesting)

http://www.math.gatech.edu/~thomas/FC/fourcolor.h

By the way, I couldn't help but to notice the quote at the bottom of the page (generated by slashdot)

"Don't think; let the machine do it for you!" -- E. C. Berkeley

## Re:That's nice, but not impressive (Score:5, Interesting)

That said, there are arguments in favour of a classical proof as well. First, of course, is the matter of elegance; an elegant symbolic proof is a lot more pleasing than a brute-force approach (though an inelegant symbolic proof is as bad - or worse - than a method like this).

Second, a theoretical proof is sometimes more interesting for the secondary results it produces and the new avenues of progress in other areas, rather than in the proof itself. This is generally lacking for brute-force methods.

But in reality, these methods complement each other quite nicely. Knowing what the result should be, making an elegant classical proof of it becomes so much easier than before. And of course, you tend to need to know quite a lot about a problem (culled from classical methods) before you can formulate a prq'acticable brute-force approach.

## Re:That's nice, but not impressive (Score:5, Funny)

I spy an Nvidia engineer!

## Re:That's nice, but not impressive (Score:3, Interesting)

## Re:That's nice, but not impressive (Score:2, Insightful)

Build a combinatorial structure and ask some random question about it, and the odds are pretty decent you'll need to resort to brute force.

## Re:That's nice, but not impressive (Score:2)

isthe shortest proof.## Re:That's nice, but not impressive (Score:2)

## Re:That's nice, but not impressive (Score:2)

## Re:That's nice, but not impressive (Score:2)

## Re:That's nice, but not impressive (Score:2)

This work merely demonstrates the result; it does not do so in a way that's faster to verify than it was to generate. In general, the proofs of open questions tend to involve work which extends related theories; proving Fermat's Last Theorem, for example, involved developing and relat

## Re:That's nice, but not impressive (Score:2)

That said, any problem involving an infinite series of numbers pretty much require a logical proof, but a problem constrained to a chessboard is more analagous to "prove the quadratic theory is correct when a, b, and c are integers from 1 to 10."

## Source Code snippets (Score:2, Interesting)

Check this out...

#define true 7361

#define false 28456

## Article on Chessbase (Score:5, Informative)

## Oops. my mistake.. Too hungry (Score:2)

## on FARK.com this would have the following header (Score:4, Insightful)

Without trying to be too much of a Troll, can someone explain to me as a mathematical lay man as to why this problem has any significance? Given that a chess board is an arbitrary 8x8 set of constraints, is there anything that can be learned and applied to the real world (or even theoretical mathematics?) through solving this problem?

Also, I was under the impression that the objective of mathematical puzzles like this would be to find a simple, elegant proof. Does a brute-force calculation approach carry as much weight?

## Re:on FARK.com this would have the following heade (Score:5, Insightful)

Here's what I have to say about mathematicians (since I have a bachelors in mathematics): Most people would invent paint to cover a wall they have that's bare. Mathematicians would invent paint with no intended purpose, then later discover it could be used on a wall.

Mathematicians frequently create for the sake of creation, rather than for any specific purpose.

--RJ

## Re:on FARK.com this would have the following heade (Score:2, Interesting)

I am trying not to be too negative here, but why should everything have to have a significance to the real world? The world doesn't benefit from me solving a crossword puzzle. It's just me, wanting to know the solution to the puzzle.

In the same way, people have been puzzled by the magical knight's tour problem for 150 years. And someone finally has solved the problem. It might not provide a cure for cancer, but at least it entertained a few people for some time.

Maybe something as insignificant as that s

## Re:on FARK.com this would have the following heade (Score:5, Insightful)

However, try and think about what this problem really is mathematically, rather than in terms of chess or magic squares. A knight's tour requires the knight to visit every square on the board exactly once. Replace "squares" with "nodes," and "board" with "graph," and what we have here is the problem of finding a Hamiltonian path with some interesting constraints.

And that's where this problem gets interesting- finding a Hamiltonian path on a graph is known to be NP-complete, a designation which carries with it all sorts of baggage, including some of the million-dollar sort. [claymath.org] The fact that the question of whether magic knight's tours exist on an standard 8x8 board required 60+ CPU-days to answer speaks volumes about the complexity of the problem, and demands answers as to how this complexity comes about, and why there is no solution with the given constraints. Far from being just a silly mathematical curiosity, this is a problem that presents a lot of potential applications with regards to algorithms, complexity, and computability. Also, if you can find an algorithm that finds Hamiltonian paths in polynomial time, you get the aforementioned free money. [claymath.org]

So yes, the problem of finding a magic tour seems worthless if you only consider the problem literally in terms of a chessboard and magic squares, instead of as a model for other complex problems in mathematics, science, and engineering. But by the same token, how interesting and useful would the Traveling Salesman Problem [wolfram.com] be if it were only applied to traveling salesmen?

## A pedantic geek says . . . (Score:5, Funny)

The solution came after 61.40 CPU-days, corresponding to 138.25 days of computation at 1 GHzYeah, and I came to work in 12.34 horsepower-hours, corresponding to 666.13 hours of driving at 5K RPM. I mean, damn, I understand when my mom utters movie-level technobabble, but this?

## Re:A pedantic geek says . . . (Score:2)

I don't see what's so hard about that figure for you to grasp.Of course you don't; you're an AC. The point being made is that the figures have no real importance. Computing for 1 day at 1GHz is meaningless. It says nothing about what is actually happening in the code. I could spin on a NOP for 23.99999999 hours and kick out an actual work unit at the end and use the same figure as someone who was actually doing something in that time. It also doesn't give any indication on what processor is being

## 1GHz WHAT? (Score:4, Insightful)

Guys, please, this is SlashDot. A 1GHz G5 is not the same as a 1GHz Duron is not the same as a 1GHz P3. What sort of 1GHz chip is being referred to here?

## Re:1GHz WHAT? (Score:5, Funny)

Good grief. It's just an estimate. It's not the exact compute time that's interesting. It still tells me the interesting bits-- that it was a complexity that an ordinary PC could do in a reasonable time frame, not the sort of thing a gigantic cluster chewed on for 100 years.

## Re:1GHz WHAT? (Score:3, Funny)

## Re:1GHz WHAT? (Score:2)

OR

1 billion cycles per second for 138.25 days = 86,400,000,000,000 cycles (8.64 x 10^11 cycles, right?)

Which on my computer would take, hmm... 3 1/2 years if I turn kpp off.

## on the value of horse problems (Score:3, Insightful)

who also said

"never buy of the horse that is overridden as it will fetch less at the clackers"

## Re:on the value of horse problems (Score:2)

I like the horses; have done for some time but have never heard of a horse being sold at the clackers. The Knacker's yard maybe...

Where or what is the clackers?

## Re:on the value of horse problems (Score:2)

## Is math dying? (Score:4, Interesting)

Now before you object, I am aware of really nifty, clever mathematical breakthroughs in recent memory. The recent polynomial-time primity test comes to mind. But what is the point of "research" and "inquiry" if we only care about the answers to the problem, and not the reason why? What is the point of finding the next Optimal Golomb Ruler" [distributed.net] if we can't find a pattern relating one to the next? What would we really

dowith a magic knight's tour anyway, besides reading a clever and insightful proof of it's (non)existance?Even a computerized proof would be interesting, if a method for proof-solving on computers were possible. And yes, it is possible to write proof-solving software that is at least as good as a human mathematician without upsetting Mr. Godel. But this brute-force type proofing is just plain dull!

## Re:Is math dying? (Score:3, Insightful)

I would have thought that the slashdot crowd would be the first group to realise that computers are an

excellentaid to mathematics. Not every maths problem can be solved by hand, and there is often quite a bit of inginuity involved in these computer soloutions.I see comments like this as people being afraid of technology - The computer can potentiall be one of the mathematicican'

## Re:Is math dying? (Score:3, Interesting)

My problem is certainly not with computers, which I agree are a huge aid to mathematics. Suites like Mathematica etc. are tremendous tools, visualization method

## Re:Is math dying? (Score:2)

The brute-force proofs, whether you do them by hand or computer, are like saying you know how to drive because you can call a taxi.I'd say it's more like driving to the airport and taking a private jet. Sure, it doesn't involve much of driving skill, but you can get to places that are impractical or impossible to drive to. And even if you don't know how to drive, you (hopefully) know how to fly.

## Re:Is math dying? (Score:2, Interesting)

## Re:Is math dying? (Score:2)

infinite) set S having a property P?".Of course if such an x doesn't exist, you cannot solve this by applying brute force directly. You have to put some minimum of creative thinking into transforming such a problem into a finite case (as it was done in case of the 4-coloring problem).

## Re:Is math dying? (Score:2)

## Cheers (Score:2, Funny)

## Ahh the memories... (Score:2, Interesting)

Many years ago I saw some guy doing a kinght's tour blindfolded on tv. I thought that was amazing, he must have calculated every move after he was given the starting square. But

## For those of you who can't be arsed with the links (Score:5, Informative)

knights tourinvolves moving the knight onto every square of the chessboard without repeating a square (being a knight, it is of course allowed to jump over squares it has previously landed on).A

magic squareis a square grid, with each grid position numbered, such that the horizontals, verticals, and diagonals all add to the same number (you can also ask for all thebroken diagonalsto add to this number, but this doesn't have to hold in a basic magic square).If we number the knight's initial position 1, and increment with every jump, then a knights tour gives us a numbered n by n square (8 by 8 in the case of a normal chessboard).

The question: are there any knights tours which give us a magic square after we perform this numbering operation?

The answer: no, although there are many tours which give us a 'semi-magic' square (where the horizonals and verticals, but not the diagonals, give us the same sum).

The point: although magic squares have a variety of surprising uses, there seems to be no 'point' to the magic knights tour question other than another line in the book of 'solved minor problems' -- but if it's enough to write a paper on, then you can bet you'll be able to find a graduate student to do it

## A chess article without Go mentioned? (Score:3, Funny)

## Sure he can... (Score:5, Funny)

## Brute force--Bring it on! (Score:5, Interesting)

In my job, I do fine structural analysis and computational fluid dynamics for aerospace design. Aerospace engineering is rapidly reaching the point where simplified, elegant models are inadequate to the real-world structures they are meant to mimic. For instance, the Navier-Stokes equations, a set of second-order partial differential equations which describe fluid flow, are not susceptible to analytical solution in any but the most simplified cases, and as things like airfoils, turbine blades, and computers grow more refined, their properties require much hairier math to describe. Turbulent flow around a wing or turbine blade has no elegant solution, but a brute-force computational model can yield a solution that allows us to design a lighter wing, and thus a more fuel-efficient plane, or a stronger turbine fan, and a more efficient generator. Or, to cite an example near and dear to many slashdotter hearts, as we can better model the heat transfer inside a microprocessor, we can better devise ways to cool it, and thus build a faster, more stable computer. (And then, we can solve more iterations on it, and build an

even fastercomputer...and then we can solvemoreiterations... ^_~)There is an intense intellectual satisfaction to a 20 or 100 line proof (I still remember the triumph of my high school proof that a triangle's interior angles sum to 180), but for a lot of things, there simply exist no such proofs. As we tackle more and more mathematical problems, those with relatively simple proofs will quickly be solved and set aside, and we will move on to messier things. For instance, the proof of Fermat's Theorem runs to several hundred pages of dense elliptic curve math. It was an aesthetic disappointment for those hoping for a simpler solution, but it was a triumph for the field of mathematics as a whole.

There is a whole 'nother world of proofs involved in brute-force solutions. Estimated time to solution, order of error--elegant mathematical tools are still necessary, but they are increasingly used to delineate regions of uncertainty as the complete picture becomes messier. The closer mathematical models get to the real world, the more complex and beautiful, but the less elegant, they become. I do not think that the field of mathematics will stop producing elegant proofs, but I do think they will have less and less to do with the world we live in and more and more to do with hypothetical mathematical constructs, whose usefulness will then filter down in the form of special cases to more prosaic disciplines, like science and engineering. This diminishes neither science nor engineering, and adds to the knowledge available in all fields.

-Carolyn

## Topical Fiction (Score:3, Informative)

## Re:A note to the anti-MS zealots (Score:3, Funny)

>

The webpage mentions that the program is windows based and doesn't save state. That means that all of those CPU hours came in a row (at idle priority even). Bash MS all you want, but Windows isn't as unstable and problematic as all of your anti-MS zealots would like to believe.61.40 CPU-days spread between 10 people's computers, and you think that indicates enough uptime to brag about? Puh-leeze.

## Re:A note to the anti-MS zealots (Score:3, Funny)

## Re:A note to the anti-MS zealots (Score:2)

61.5 days uptime isn't something to brag about, it's a medical condition [emedicine.com].

## Re:A note to the anti-MS zealots (Score:2)

I'm just a lowly database programmer, so I have no clue about high end programs like this. Is it a common practice for time intensive processes to not have any sort of save file?

Shadow

## Re:A note to the anti-MS zealots (Score:2, Insightful)

Checkpointing is your friend.

## Re:A note to the anti-MS zealots (Score:2)

The operating system should be fault tolerant and not fail when a foreign(read: third party) application does some unexpected behavior.For the most part, modern windows does not crash and burn when a third party userland application does stuff. There are certainly the occasional bug and wierd corner cases, but those are to be expected in any software system that was not designed to be proven correct.

However, if a driver has bugs there really isn't much the OS can do. A driver is sitting there in kernel s

## Re:Interesting problem... (Score:3, Interesting)

magical tourhave been going around for quite a few centuries, with some of the brightest minds trying to work on it. So even if it doesn't have practical applications it's still important.## Re:Interesting problem... (Score:2, Interesting)

For me it is sort of like saying if I hurt my hand with a hammer it damn well hurts rather than investing the underlying reason.

## Re:Interesting problem... (Score:2, Insightful)

As discussed in Simon Singh's excellend Fermat's Enigma [amazon.com], the research done by many other people may be built upon the assumption that a particular mathematical statement is either true or false. Until a proof is presented -- either by brute force or more elegant means -- it us unknown whether the "ediface" built on the assumption will stand or fall.

Since pure mathematics underlies a great deal of applied research, having mathematical statemen

## Re:Interesting problem... (Score:2, Funny)

## Re:Interesting problem... (Score:2)

I don't know about you, but I get tired of hearing the plain old news. Most of it is kinda depressing, and hearing stories like this makes you realize how many interesting people there are out there. Sorry, but if I could moderate, your post would be troll. And as for the "nerd mountain", I think you are in the wrong place then.

## Re:Another interesting math problem (Score:3, Informative)

Post your code if you still think switching makes no difference.

## Re:Another interesting math problem (Score:5, Insightful)

\ 1 2 3

A y n n

B n y n

C n n y

The top row is the door number, the letters are the three cases, y means 'yes' (location of prize), n means 'no'. Suppose you chose door #1. In case A he'll show you door 2 or 3, in case B he'll show you #3, in case C he'll show #2. Only in Case A will you win by sticking with it. Case B and C you'll win by switching. That's 2/3 chance of winning by switching. Same thing regardless of what choice of door you made first.

## Re:Another interesting math problem (Score:2)

Eg.

Host says pick between door 1 and door 2.

Host says 'Are you sure? Do you want to switch?'

The way I see it, the third door is irrelevant. You are always making the following choices:

1) Pick from a winning or non-winning door.

2) Choose to keep the same choice as above or choose

## The Monty Hall Problem (Score:2)

There are two cases where you win by sticking. Where you are shown #2, and where you are shown #3.Yes, but the table you draw lists those cases equally with your initial choice, but that is false. Monly's revelation is

a seperate event. You're giving what is simply a subchoice the same weight as the main choices.Let's assume, for the same of simplicity, that you always pick door number one. Since the first choice is completely random, this doesn't affect the final odds -- and it reduces the number o

## Re:Another interesting math problem (Score:2)

Maybe it's me but I'm looking at this from a completely different perspective.

You have two doors. One has the prize. One doesn't. You've already picked door A. What's the chances of you winning the prize if you change and pick door B. All the host of the show has done is remove one door from your list of choices.

Either this problem is very very easy or I'm very very stupid.

## Re:Another interesting math problem (Score:2)

Although there are nine outcomes,obviously they are not equally probable.Monty cannot open the door that you picked, nor can he open a door with a prize behind it. Therefore, most of these have a probability of zero.True BUT your logic is flawed in that they're all either 0 or equally probable. S is just as likely to happen as T or W or Y.

Let's show all possible routes given a choice of door 1:\ 1 2 3 - shown

R y n n - 1

S y n n - 2

T y n n - 3

U n y n - 1

V n y n

## Re:Another interesting math problem (Score:2, Informative)

Suppose there are doors A, B and C. Let's say you pick door A originally, and the host then shows door B to be empty. Given this information, there is a one in three chance that the prize lies behind door A, and a two in three chance the prize lies behind door C.

Google for 'Monty Hall' for proofs of this.

## Re:Another interesting math problem (Score:4, Insightful)

## Re:Another interesting math problem (Score:5, Informative)

...Assuming the 'host' always opens one of the remaining doors and the remaining door never has a prize behind it...A note to all the people writing code to solve this, the key to this problem is that

Monty intentionally chooses a door that has no prize behind it. No doubt, some of you are interpreting this problem as, "Monty chooses another door at random, and we're only examining the cases where he chooses the door with no prize". If that is your interpretation, then switching doesn't matter, but the traditional Monty Hall Problem assumes Monty knows what's behind the doors and chooses accordingly.## Re:Another interesting math problem (Score:2)

...the logic is incredibly flawed.For someone who writes things like "I would have thought" and "That's what seems to make sense" I don't think you are in a position to posture on logic. In what way, exactly, is that poster's logic flawed?

## Re:Another interesting math problem (Score:4, Informative)

It really works like this:

Assuming you picked door A, then there are three possible situations:

-The prize is behind door A

-The prize is behind door B

-The prize is behind door C

So if you stick to the door you just picked, then you have a 1/3 chance of winning.

Now obviously there's a 2/3 chance that the prize is behind one of the two other doors. And the host is basically giving you the information

"IF the prize is behind one of the two remaining doors, then it is behind THIS one" by opening an empty door.

So switch, it will give you a 2/3 chance of winning.

And yes, I had to write a program, too, before I actually believed it.

## Like the paradox of check out queues (Score:2)

You pick queue 1. Queue 2 closes. Switching to Queue 3 is not going to help you. Unless the checkout operator in Queue 1 is an L-Plate operator.

So lemme see (I was always crap at probablility).

First you have a choice of three doors. So your chance of winning is 1/3 on door number 1.

Door number two is eliminated, leaving door number three with 2/3 probability against it.

But imagine, now you are only offered two doors to start with. Fifty fifty chance of winning. 1/2 chance.

## Re:Another interesting math problem (Score:2)

False. Chance, in this case, is controlled by the game show host. Hedoesknow there are three doors, and which one has the prize.That said, the 1/3-2/3 thing is only applicable if two things are true: That the host must open a door, and that the door must be empty. If those are not true (which they weren't in the original show) then the whole analysis flys out

## Re:Another interesting math problem (Score:3, Insightful)

Since it ALWAYS happens, the host-picked door (which is always empty) actually doesn't have any bearing on the problem. You can safely eliminate it from consideration.WRONG. Your choice had a 1/3 chance of being right because there were 3 doors to begin with, among which you chose randomly. Therefore all 3 of those doors are important to the problem, because they specify the conditions in which your first choice is made. Just because the host removes doors doesn't suddenly change the probability of your

## Re:Another interesting math problem (Score:4, Informative)

Now imagine that you always switch. If you picked the right door to start with, you're screwed as you will always lose by switching. However, if you pick one of the two wrong doors then the host must to open the other wrong door, leaving just the right door remaining. So if you picked a wrong door in the first place, you are guaranteed to win by switching. Your chance of selecting a wrong door initially is 2 in 3 and therefore your chance of winning by always switching is 2 in 3.

1/3 + 2/3 = 1, hence all possibilites are accounted for and it is proved that by switching you twice as much chance of winning than by sticking.

## Re:Another interesting math problem (Score:2)

I don't buy it. When Monty Hall opens the door, he changes the problem. A simple brute-force technique will show you that your odds of winning are 50-50:

P = Prize behind this door; N = No prize behind this door;

Bold= The door you initially picked;Italics= The door Monty Hall reveals; K = Keep your original door; S = Switch doors; W = You win with this strategy; L = You lose with this strategy.1 2 3 K S

PNN W LPNNW LP

NNL WP

NNL WNPNL WNPN W LN

PNW LNPNL WNNP L WNNP L WNN## Re:Another interesting math problem (Score:2)

Prior Restraint (179698)said:The error that you've made in this analysis can be summed up in the first two lines of your table. In the case where the player chooses the correct door, it is irrelevant which door Monty choses, so

## Re:Another interesting math problem (Score:2)

You have 3 doors

* X X

A B C

The prize is on door A (the *), you pick door A and the host opens the door C:

* X

A B

You already picked door A, and regardless of the probabilities in the past step, this is a new problem, and you are reduced to a 1/2 probability. The pro

## Think of words ending in 'gry.' (Score:4, Funny)

## Re:Think of words ending in 'gry.' (Score:2)

## Re:Another interesting math problem (Score:4, Informative)

The point is that the host knows where the prize is, and that he is giving away a lot of info by opening one door. It would be useless to switch if he opened the door at random and could possible find the prize himself.

so you should switch. As simple as that

And, your demo prg must be wrong. Remember that the host should use his knowledge and always open and empty door, never the correct one

## Everybody gets this wrong [especially "experts"] (Score:2)

This is wrong since it is never stated that the host HAS to open a door and give you a chance to switch. What if he only opens a [wrong] door when he feels like it ? In this situation, you should NEVER switch [assuming the host does not want

## Re:Everybody gets this wrong [especially "experts" (Score:2)

Then you have never seen a correct statement of the problem, and the people you have heard it from are indeed idiots trying to look smart. A true Math geek would understand the importance of accurately stating the problem:

There are three doors. Behind one is a million dollars. Behind the other two there are goats. You pick one door. Monty opens one of the other doors to reveal a goat, and offers you the option of switching your choice to the remaining door. Assuming the location of the million dollar

## Found in new age bin: Magic Knight (Score:2)

Found in some remainder bin somewhere: their debut album: "Enchanted Castle", and maybe even a copy of "Mystic Joust".