The balloon point of view
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Leif Andersson - 24 Jan 2005 17:05 GMT The usual way of describing special realativity is to describe Michelsson-Morleys experiment, draw the conclusion that either is the Earth not moving with respect to the ether or there is no ether, and then exclude the first alternativ.
This is due to the fact that the physicists and mathematicians that discussed these matters did it on suburban-trains on the way to and from work. For a train-passenger it is natural to think of the Earth running through the ether like a train running through the air. But if they had traveled by balloon they had never even come up with the idea of using waves in the surrounding media to measure speed. No wind ever blows on board a ballon.
And after all: Is it not obvious that the Earth hovers like a ballon in the ether rather than it moves like a train through the ether?
So what does the world look like from a balloon-passenger point of view? Of course also a balloon-passenger would realize the need to change the Galilean transformation to the Lorentz-transformation but the interpretation would be different from the usual way of handling relativity.
More about these matters on my home-page
http://w1.877.telia.com/~u87701228
Leif Andersson
stephen@nomail.com - 24 Jan 2005 17:45 GMT : The usual way of describing special realativity is to describe : Michelsson-Morleys experiment, draw the conclusion that either is the Earth : not moving with respect to the ether or there is no ether, and then exclude : the first alternativ.
: This is due to the fact that the physicists and mathematicians that : discussed these matters did it on suburban-trains on the way to and from [quoted text clipped - 3 lines] : in the surrounding media to measure speed. No wind ever blows on board a : ballon.
: And after all: Is it not obvious that the Earth hovers like a ballon in the : ether rather than it moves like a train through the ether? Does the moon also hover like a balloon in the ether? How about the Sun? How do all three of these objects manage to hover in the ether but at the same time move with respect to each other the way that they do? I have never seen a balloon "orbit" another balloon.
Stephen
mtm - 25 Jan 2005 03:42 GMT Hi Stephen,
> Does the moon also hover like a balloon in the ether? How about > the Sun? How do all three of these objects manage to hover in the > ether but at the same time move with respect to each other the > way that they do? I have never seen a balloon "orbit" another > balloon. Keep an open mind (well, maybe not as open as Leif :-))
What if Leif aether form a buble around a mass direct proportional with body gravity? This can happen into a LeSage model where aether it is consumed by mater. Then, you will have a baloon orbiting another baloon orbiting a bigger baloon alltogether, under the influence of the "wind". See that it is possible with a little immagination !!!
Well, I have to stop here the openess since I am not prepared yet to take the road "From creation to Nirvana" :-))
Bill Hobba - 25 Jan 2005 02:52 GMT > The usual way of describing special realativity is to describe > Michelsson-Morleys experiment, draw the conclusion that either is the Earth [quoted text clipped - 4 lines] > discussed these matters did it on suburban-trains on the way to and from > work. I am not so sure Einstein spent a lot of time discussing these matters on suburban trains while going to work as a patent clerk - but I suppose it is one of those things we probably will never know for sure. But then again we do not know for sure he did either - but fine (or not so fine) points of logic seem to be lost on thus making such inane comments.
> For a train-passenger it is natural to think of the Earth running > through the ether like a train running through the air. But if they had > traveled by balloon they had never even come up with the idea of using waves > in the surrounding media to measure speed. No wind ever blows on board a > ballon. Ever hear of the aether drag hypothesis? I wonder why is was proposed and rejected? For the real reason aether theories were rejected see the following ancient but IMHO excellent posts by Tom Roberts. http://www.google.com/groups?selm=3838AC00.87B78404%40lucent.com http://www.google.com/groups?selm=3838A838.81CE8090%40lucent.com http://www.google.com/groups?selm=3838AA2A.829F46AD%40lucent.com
Bill
> And after all: Is it not obvious that the Earth hovers like a ballon in the > ether rather than it moves like a train through the ether? [quoted text clipped - 9 lines] > > Leif Andersson Joe Fischer - 25 Jan 2005 06:44 GMT >The usual way of describing special realativity is to describe >Michelsson-Morleys experiment, draw the conclusion that either is the Earth [quoted text clipped - 8 lines] >in the surrounding media to measure speed. No wind ever blows on board a >ballon. Of course passengers can feel wind blow on balloon, if there are gusts, but balloonists try very hard to not have the balloon inflated if there is very strong wind, or if there is a gusty wind.
>And after all: Is it not obvious that the Earth hovers like a ballon in the >ether rather than it moves like a train through the ether? [quoted text clipped - 3 lines] >Galilean transformation to the Lorentz-transformation but the interpretation >would be different from the usual way of handling relativity. So please move on to General Relativity, and move into the 21st century.
Joe Fischer
Tom Roberts - 25 Jan 2005 14:58 GMT > The usual way of describing special realativity is to describe > Michelsson-Morleys experiment, draw the conclusion that either is the Earth > not moving with respect to the ether or there is no ether, and then exclude > the first alternativ. Naive people might do that, but not knowledgeable physicists. Physicists know that this experiment is consistent with SR (which has no ether), and several variations of ether theories; they also know that several other experiments, specifically Kennedy-Thorndike and Ives-Stillwell, refute the usual variations of ether theories. Physicists who have studied ether theories know there remains a class of ether theories not refuted by these experiments, but all theories in this class are experimentally indistinguishable from SR and there is no possibile way to ever observe their ether; as such inherently unobservable quantities are out of favor in science, these ether theories are generally ignored.
> And after all: Is it not obvious that the Earth hovers like a ballon in the > ether rather than it moves like a train through the ether? This is not "obvious" at all. Specifically, it is difficult to reconcile such a model with the observed stellar aberration. Or with the motions of planets and moons....
> [...] Tom Roberts tjroberts@lucent.com
shevek - 26 Jan 2005 17:32 GMT > > The usual way of describing special realativity is to describe > > Michelsson-Morleys experiment, draw the conclusion that either is the Earth [quoted text clipped - 3 lines] > Naive people might do that, but not knowledgeable physicists. Physicists > know that this experiment is consistent with SR (which has no ether),
> and several variations of ether theories; A very good point.
> they also know that several > other experiments, specifically Kennedy-Thorndike and Ives-Stillwell,
> refute the usual variations of ether theories. Physicists who have > studied ether theories know there remains a class of ether theories not > refuted by these experiments, but all theories in this class are > experimentally indistinguishable from SR and there is no possibile way > to ever observe their ether; as such inherently unobservable quantities > are out of favor in science, these ether theories are generally ignored. But if you are right, that these theories are experimentally indestinguishable from one another, than they are the same theory. Just as wave and matrix versions of QM are shown to be the same. To my knowledge such a proof has not been claimed.
Also, you forgot your usual caveat of living in the error bars. Perhaps some anisotropic round trip light speed theory exists in those tiny error bars. A LET type theory also exists in the large error bars of Krisher et al., predicting null results for Kennedy-Thorndike, Ives-Stillwell, and MMX.
Cheers -
Bill Hobba - 27 Jan 2005 00:56 GMT > > > The usual way of describing special realativity is to describe > > > Michelsson-Morleys experiment, draw the conclusion that either is [quoted text clipped - 27 lines] > But if you are right, that these theories are experimentally > indestinguishable from one another, than they are the same theory. Not so. LET for example postulates the existence of an actual aether and motion wrt to that aether causes an actual shortening of rods. SR says it is simply how space-time is - no aether required. The two theories are based on different assumptions. One reason SR is preferred to LET is because it does not contain an inherently unobservable aether.
Say I hit your car and damage it. You naturally want me to pay for the damage. However I claim it was not my hitting that caused the damage - it was an invisible angel that actually caused the damage and you should find the angel and get the money from it. I think you will be unable to prove me wrong eg you may claim the force applied by my hitting your car did it - I can counter - no - the force made the invisible angel do it - I am sure you get the idea. The two theories are based on different assumptions and lead to different outcomes but are indistinguishable from each other. But why to we prefer the non angel theory? - it is simply more reasonable eg no one has ever actually found this tricky little angel, why would it act in such a capricious way etc. The same applies to flat earthers - I can not prove them wrong - but it is an entirely different theory than the round earth. It is just mush moor reasonable to believe the earth is round.
> Just as wave and matrix versions of QM are shown to be the same. To my > knowledge such a proof has not been claimed. At one time it was thought Bohmian mechanics was experimentally indistinguishable from standard QM. However recent work showed it was; with standard QM winning out. But even before that it was well known they were different theories resting on different assumptions. Just because two theories make the same predictions it does not mean they are the same theory. We may not fully understand a theory at this stage or it may be a theory that gives exactly the same results but based on different assumptions - sometimes it is hard to tell. Another example is the Feynman Wheeler theory of direct interparticle action. It might seem they are the same theory - but a quantum version of the Wheeler Feynman theory has never been found.
Thanks Bill
> Also, you forgot your usual caveat of living in the error bars. > Perhaps some anisotropic round trip light speed theory exists in those [quoted text clipped - 3 lines] > > Cheers - shevek - 27 Jan 2005 22:12 GMT > > > > The usual way of describing special realativity is to describe > > > > Michelsson-Morleys experiment, draw the conclusion that either is [quoted text clipped - 45 lines] > ever actually found this tricky little angel, why would it act in such a > capricious way etc. The difference here is only in your language; i.e. it is not a structural difference. Sure, you could claim the damage was done by virtual photons that were mediating the electro-weak force brought in to play by the proximity of the two cars.. same thing. You used the word "indistinguishable". If LET and SR are indeed indistinguishable, than I want to see a proof. Otherwise I don't buy it. Proof by exaustive test of all possible experiments is not an option.
> The same applies to flat earthers - I can not prove > them wrong - but it is an entirely different theory than the round earth. > It is just mush moor reasonable to believe the earth is round. Very different, unless of course you are talking about people who call a spherical earth flat. It is extremely flat, by the way - the radius of curvature can often be treated as infinite. If you are talking about people who think it is perfectly flat, you can prove them wrong by showing them a mountain.
> > Just as wave and matrix versions of QM are shown to be the same. To my > > knowledge such a proof has not been claimed. [quoted text clipped - 5 lines] > theories make the same predictions it does not mean they are the same > theory. An excellent point. They must make the same prediction -for all of the elements for which they apply-, i.e. for all physical systems to which the theory applies.
> We may not fully understand a theory at this stage or it may be a > theory that gives exactly the same results but based on different > assumptions - sometimes it is hard to tell. Another example is the Feynman > Wheeler theory of direct interparticle action. It might seem they are the > same theory - but a quantum version of the Wheeler Feynman theory has never > been found. I'm not familiar with that theory, but I'm sure the phenomenon is ubiquitous. Many different interpretations often wind up being equivalent.
Cheers -
Bill Hobba - 28 Jan 2005 00:21 GMT > > > > > The usual way of describing special realativity is to describe > > > > > Michelsson-Morleys experiment, draw the conclusion that either [quoted text clipped - 68 lines] > The difference here is only in your language; i.e. it is not a > structural difference. That would seem to be the main issue. You are claiming if they make the same experimentally testable predictions then it is only a difference in language. I claim otherwise. I claim if they make any different prediction - experimentally testable or otherwise - then they are a different theory. The case with the angel is just an illustration. I think Tom Roberts expressed it even better than I did:
'No. They make different interpretations. Theories are the "language" of humans interpreting Nature, and your claim is like saying "since any language can say 'the sky is blue' then they are the same language". There is somply no way of getting around both the anngel and the aether are diffenrt interpretations.
Making the same experimentally testable predictions is not the same as making the same predictions. Predicting (or assuming) motion through an aether causes rod shortening is not the same as testing it.
> Sure, you could claim the damage was done by > virtual photons that were mediating the electro-weak force brought in > to play by the proximity of the two cars.. same thing. No it is not the same thing. In one case you are claiming the force causes it directly in the other you are claiming it was caused by virtual particles. They are two different theories. In the case of virtual particles they are able to account for the casimir effect - Newton's laws can not account for that. So if one wishes to be exact the explanation involving virtual particles is to be preferred. But then again I am not so sure it is caused by virtual particles either. Many people think that solidity is a result of the electrons in outer shells repelling each other. I think it was Dyson in a famous paper that showed such is not the case - it is the Pauli exclusion principle that is responsible for it.
> You used the > word "indistinguishable". I used the word indistinguishable to mean experimentally indistinguishable - it was meant to convey predictions of a theory may be the same in a certain domain of applicability not that the interpretation is the same.
> If LET and SR are indeed indistinguishable, > than I want to see a proof. Otherwise I don't buy it. Proof by > exaustive test of all possible experiments is not an option. I have not studied the matter is enough detail to give a definite answer if they are experimentally indistinguishable. Indeed one would need to decide exactly what one would accept as refutation. For example AFAICS LET attributes rod shortening to EM effects due to an interaction with the aether and a simple model of an electron we now know to be incorrect - this comes from Lorentz's 1904 paper on the matter. So I would say it has been disproven. But hey that is just my view - I know aether theorists would argue that one - in fact when I pointed it out they did. I mentioned previously about being able to tell the difference between standard QM and Bohmian mechanics. Scholarly papers that I agree with have been written that purport to show there is an experimental difference but we have knowledgeable posters on this group that disagree. These issues can be subtle an difficult to pin down.
> > The same applies to flat earthers - I can not prove > > them wrong - but it is an entirely different theory than the round [quoted text clipped - 3 lines] > Very different, unless of course you are talking about people who call > a spherical earth flat. Be careful here. Have you actually be out in space and seen a round earth? And even if you did how do you know it was not an elaborate hoax to convince you it was round and you remained right here on a flat earth? I think if you examine the actual evidence you will find it deepends on believing things people say as you will find out if you actually argue with a flat earther - see http://www.newton.dep.anl.gov/askasci/gen99/gen99427.htm.
> It is extremely flat, by the way - the radius > of curvature can often be treated as infinite. If you are talking > about people who think it is perfectly flat, you can prove them wrong > by showing them a mountain. It is obvious what the flat earthers mean by flat.
> > > Just as wave and matrix versions of QM are shown to be the same. > To my [quoted text clipped - 13 lines] > elements for which they apply-, i.e. for all physical systems to which > the theory applies. You may be confused between and experimentally testable prediction and a prediction in general. To say in LET, for example, motion though the aether causes rod shortening is a prediction (or maybe even an assumption) of the theory - but if you can test it is another matter.
> > We may not fully understand a theory at this stage or it may be a > > theory that gives exactly the same results but based on different [quoted text clipped - 9 lines] > ubiquitous. Many different interpretations often wind up being > equivalent. As far as I can tell they are not equivalent - at least no one has ever shown they are. And the fact a quantum version has never been worked out indicates it probably is a different theory that just happens to classically give the same results.
Bill
> Cheers - reany@asu.edu - 28 Jan 2005 14:49 GMT > > > > > > The usual way of describing special realativity is to describe > > > > > > Michelsson-Morleys experiment, draw the conclusion that either [quoted text clipped - 78 lines] > 'No. They make different interpretations. Theories are the "language" of > humans interpreting Nature, Somewhat poetic for you, Bill. A theory is an explanation in the form of a deductive system, hence theories can be compared on the basis not only of their empirical predictions (theorems and postulates of empirical content) but also by their principles, models, and ontologies. For example, LET has the ontology of absolute rest space and even claims to have a formal instance of it in the rest space of the ether. However, SR not only has no instance of an absolute rest space, it also has no ontological provision for one, since it was invented specifically to deconstruct the one used by Lorentz.
> and your claim is like saying "since any > language can say 'the sky is blue' then they are the same language". There [quoted text clipped - 108 lines] > indicates it probably is a different theory that just happens to classically > give the same results. Funny how the basis of most arguments on this NG continue to be at heart in the nature of the philosophy of physics, which science education does NOT teach properly. For if it had, we would surely not need to continually bandy the meaning of such basic terms as "theory," model," "hypothesis," and "law" here. If the FAQ could stipulate the meanings of these terms, that might help reduced the perenniel pointless arguments over these basics.
Patrick
Tom Capizzi - 28 Jan 2005 15:44 GMT > Funny how the basis of most arguments on this NG continue to be at > heart in the nature of the philosophy of physics, which science [quoted text clipped - 5 lines] > > Patrick Wouldn't help. People around here have their own pet definitions for every term. Either they would all have to be included, which would solve nothing, or one would have to be selected, which would just trigger an endless series of arguments about the FAQ's. People have to stop nit- picking the choice of word and ask for clarification about what was meant. You aren't going to change most people's personal dictionary. I realize this results in a certain imprecision in the usage, but endless arguments about definitions lead nowhere and are rarely resolved. We could be arguing about the idea instead of the semantics.
reany@asu.edu - 28 Jan 2005 17:02 GMT > > Funny how the basis of most arguments on this NG continue to be at > > heart in the nature of the philosophy of physics, which science [quoted text clipped - 8 lines] > Wouldn't help. People around here have their own pet definitions for > every term. Assuming that we all had a "standard" science education, how could this state of affairs have arisen in the first place.
> Either they would all have to be included, which would solve > nothing, or one would have to be selected, which would just trigger an > endless series of arguments about the FAQ's. The FAQ should offer one definition for each one. Posters are free to accept them or reject them, of course, and if so, to stipulate their own definitions.
> People have to stop nit- > picking the choice of word and ask for clarification about what was > meant. You aren't going to change most people's personal dictionary. > I realize this results in a certain imprecision in the usage, but endless > arguments about definitions lead nowhere and are rarely resolved. We > could be arguing about the idea instead of the semantics. But it is wrong to characterize semantical debates as nitpicking. Your attitude is a part of the reason why science education isn't taught right in the first place.
Cooking up sentences is like cooking up a meal. It's like saying that if I happen to like cilentro, everyone must.
If my (hypothetical) opponent thinks that a model is a mere rescaling of something by size, we are never going to get each other when we use the term "model" because each of us will understand its meaning differently.
Patrick
AllYou! - 28 Jan 2005 18:52 GMT [snip]
> > Wouldn't help. People around here have their own pet definitions for > > every term. > > Assuming that we all had a "standard" science education, how could this > state of affairs have arisen in the first place. Why don't you tell us. How did you ever come to use circular definitions?
reany@asu.edu - 28 Jan 2005 19:25 GMT > [snip] > [quoted text clipped - 5 lines] > > Why don't you tell us. How did you ever come to use circular definitions? I my vain effort to define what kind of intellectual lifeform you are. Patrick
AllYou! - 28 Jan 2005 19:56 GMT > > [snip] > > [quoted text clipped - 10 lines] > > I my vain effort to define what kind of intellectual lifeform you are. So you never really believed that Physical meant that which lives in a physical theory?
reany@asu.edu - 29 Jan 2005 17:15 GMT > [snip] > [quoted text clipped - 5 lines] > > Why don't you tell us. How did you ever come to use circular definitions? I blame the topology of the universe for that. Any definition that is long enough has to come back on itself!
Patrick
Tom Capizzi - 29 Jan 2005 00:00 GMT >> > Funny how the basis of most arguments on this NG continue to be at >> > heart in the nature of the philosophy of physics, which science [quoted text clipped - 14 lines] > Assuming that we all had a "standard" science education, how could this > state of affairs have arisen in the first place. Because "standard" is defined by hundreds of different local school boards.
>> Either they would all have to be included, which would solve >> nothing, or one would have to be selected, which would just trigger [quoted text clipped - 14 lines] > > But it is wrong to characterize semantical debates as nitpicking. Why? This is sci.physics.relativity, not some etymology newsgroup.
Your
> attitude is a part of the reason why science education isn't taught > right in the first place. I'm not teaching science education, just trying to cope with the results of hundreds of different curricula.
> Cooking up sentences is like cooking up a meal. It's like saying that > if I happen to like cilentro, everyone must. Where did you get that idea? I said it is a waste of time to try to change someone's definition if they don't get it the first time it is pointed out to be faulty. As long as their use or misuse is consistent we can focus on what they meant and move on.
> If my (hypothetical) opponent thinks that a model is a mere rescaling > of something by size, we are never going to get each other when we use > the term "model" because each of us will understand its meaning > differently. Until each of us clarifies the meaning of the word. Then we can agree to disagree on our choice of words. It's a little like speaking different languages. We just need a phrase book to communicate.
> Patrick reany@asu.edu - 29 Jan 2005 00:42 GMT > >> > Funny how the basis of most arguments on this NG continue to be at > >> > heart in the nature of the philosophy of physics, which science [quoted text clipped - 63 lines] > languages. > We just need a phrase book to communicate. If we do it your way that will make if virtually impossible for more than two people per discussion to discuss anything which uses controversial terms! This is a rational basis for discussing science?
We can either complain about problems and do nothing to fix them, or we can try to do something positive to fix them. What's your pleasure? Patrick
Tom Capizzi - 29 Jan 2005 02:53 GMT >> >> > Funny how the basis of most arguments on this NG continue to be > at [quoted text clipped - 88 lines] > can try to do something positive to fix them. What's your pleasure? > Patrick I don't think that arguing about correct usage of language actually fixes anything. These diversions go on endlessly and little science is actually discussed. Besides, why should it be impossible for more than two people per discussion? Unless each one uses a different definition. In which case the argument about which definition to use would just go on longer anyway. It would be nice if we all used terms the same way. It just seems that too much time is wasted trying to accomplish it.
reany@asu.edu - 29 Jan 2005 12:26 GMT [snip]
> > If we do it your way that will make if virtually impossible for more > > than two people per discussion to discuss anything which uses [quoted text clipped - 7 lines] > anything. These diversions go on endlessly and little science is actually > discussed. No science can be discussed without a COMMON language to do the discussing in.
> Besides, why should it be impossible for more than two > people per discussion? Unless each one uses a different definition. Almost a given at this point. If you don't believe me, ask them.
> In > which case the argument about which definition to use would just go > on longer anyway. It would be nice if we all used terms the same way. Tom, that's been my point all along.
Patrick
Bill Hobba - 29 Jan 2005 10:55 GMT > > > > > > > The usual way of describing special realativity is to > describe [quoted text clipped - 102 lines] > > Somewhat poetic for you, Bill. I presume you mean of Tom - I did not write it - of course I agree with it - which is why I quoted it.
> A theory is an explanation in the form > of a deductive system, hence theories can be compared on the basis not > only of their empirical predictions (theorems and postulates of > empirical content) but also by their principles, models, and > ontologies. Now correct me if I am wrong but are not the primitives of that deductive system different for different theories? eg LET has an aether SR does not. And would not those different primitives form part of a different interpretation?
> For example, LET has the ontology of absolute rest space > and even claims to have a formal instance of it in the rest space of > the ether. However, SR not only has no instance of an absolute rest > space, it also has no ontological provision for one, since it was > invented specifically to deconstruct the one used by Lorentz. And that different 'ontology' is part of what makes it a different interpretation.
Rest of Patrick's usual philosophy rant snipped.
Thanks Bill
Tom Roberts - 27 Jan 2005 05:20 GMT >> all theories in this class are >>experimentally indistinguishable from SR [...] > > But if you are right, that these theories are experimentally > indestinguishable from one another, than they are the same theory. No. They make different interpretations. Theories are the "language" of humans interpreting Nature, and your claim is like saying "since any language can say 'the sky is blue' then they are the same language".
> Also, you forgot your usual caveat of living in the error bars. > Perhaps some anisotropic round trip light speed theory exists in those > tiny error bars. I did not feel it was needed here, as the statements I made needed no such qualification.
> A LET type theory also exists in the large error bars > of Krisher et al., predicting null results for Kennedy-Thorndike, > Ives-Stillwell, and MMX. LET predicts a null result for Krisher et al.
Tom Roberts tjroberts@lucent.com
shevek - 27 Jan 2005 22:02 GMT > >> all theories in this class are > >>experimentally indistinguishable from SR [...] [quoted text clipped - 5 lines] > humans interpreting Nature, and your claim is like saying "since any > language can say 'the sky is blue' then they are the same language". Many linguists would say all languages are fundamentally the same, that is the Chomsky approach. Mathematicians say that a game of choosing single digits between 1 and 9 until someone can make 15 is the same game as tick tack toe.
> > Also, you forgot your usual caveat of living in the error bars. > > Perhaps some anisotropic round trip light speed theory exists in those > > tiny error bars. > > I did not feel it was needed here, as the statements I made needed no
> such qualification. I guess you didn't really. Sorry for nitpicking.
> > A LET type theory also exists in the large error bars > > of Krisher et al., predicting null results for Kennedy-Thorndike, > > Ives-Stillwell, and MMX. > > LET predicts a null result for Krisher et al. And this is my real problem, I don't see that. Do you have any references where that is derived?
THanks a lot - shevek
Tom Capizzi - 28 Jan 2005 01:10 GMT >> >> all theories in this class are >> >>experimentally indistinguishable from SR [...] [quoted text clipped - 11 lines] > single digits between 1 and 9 until someone can make 15 is the same > game as tick tack toe. It doesn't take a mathematician to understand why, either. Just construct a 3x3 magic square: first row (8, 1, 6), second row (3, 5, 7), third row (4, 9, 2). It is easy to verify that the sum of 3 elements in each row is 15, the sum of 3 elements in each column is also 15, and the sum of 3 elements in each of the two major diagonals is also 15. If any one of those combinations is all X's or O's, that's a winner. These are the only combinations of 3 elements which add up to 15. So each digit is simply a label for one of the squares in the tic-tac-toe grid.
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