#1242
Yes, Stirling's Formula approximates the factorial or more generally the Gamma Function.
m! is the number of permutations of m objects, in this case m moves assumed interchangeable.
For example there are 52! ways to shuffle a deck of 52 cards, that is 8*10^67.
That is way more than there a chess positions...
Chess will never be solved, here's why
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Feb 18, 2022
0
#963
tygxc wrote:
#1242
Yes, Stirling's Formula approximates the factorial or more generally the Gamma Function.
m! is the number of permutations of m objects, in this case m moves assumed interchangeable.
For example there are 52! ways to shuffle a deck of 52 cards, that is 8*10^67.
That is way more than there a chess positions...
Again - now you're talking !
The world will probably never see all of those shufflings.
Nor even a small percentage of them.
But in the case of cards - there's 52 identities. And 52 positions.
There is no division there to be done.
One of the reasons - the suitings of the cards make repetitions impossible.
With chess - there's only 13 identities although up to 64 positions.
Division can be done for the purpose of compacting the expressions in a more manageable form.
Such as for example - a number to express the number of ways of having 32 squares empty. Which will always be the case in chess.
The factorials will always have an integer value.
And the bottom factorial will always potentially divide into the numerator to produce an integer result. It 'cancels' as it were.
But - no need to do that till its time to present all results in powers of ten. Many of the expressions are constants.
They could be presented algebraically with letter codes.
Factorial division could or would occur in presenting the number of ways for x pawns of one color to be present on 48 squares.
#1245
As I said before dividing by the factorial would only be correct if the moves were interchangeable but they are not always.
Example:
1 e4 e5 2 Nf3 Nc6 3 Bb5 Nf6
This can also be reached with 1 Nf3 Nc6 2 e4 e5 or 1 e4 Nc6 2 Nf3 e5 or 1 e4 e5 2 Nf3 Nf6 3 Bb5 Nc6 or 1 e4 Nf6 2 Nf3 e5 3 Bb5 Nc6 or 1 Nf3 Nf6 2 Nc3 Nc6 3 e4 e5 4 Bb5 Nb8 5 Nb1 Nc6 or 1 e3 e6 2 Nf3 Nc6 3 e4 e5 4 Bb5 Nf6 and much much more ways, however, Bb5 can never happen before e4 or e3.
The right approach is not to think of games, but only of positions.
The game of chess unlike e.g. checkers has much, much more transpositions: different games reaching the same position.
Feb 18, 2022
0
#965
From #1245
"As I said before dividing by the factorial would only be correct if the moves were interchangeable but they are not always."
No. Factorial division would occur in the ways of having 32 or more squares empty.
It would also occur in the instance of number of ways to arrange x number of pawns of one color on 48 squares.
Regarding factorials and permutations regarding moves to be made or possible games - there's no improvement there - the number of potential chess games is much bigger than the number of ways 52 cards can be arranged in a deck.
#1250
Think of positions, not of games.
That is another reason why it is smart to start from tabiya, not from the initial position.
m! progresses faster than exponential. lim exp(m) / m! = 0
The Big Bang has been experimentally verified by the WMAP probe.
Feb 18, 2022
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#967
I keep coming here to see if @tygxc has solved chess yet.
I thought he had for a moment there, but unfortunately his solution failed entirely to deal with this position in the white strategy (and untold numbers of others) because they are not in the human-selected tabiya.
Feb 18, 2022
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#968
Optimissed wrote:
tygxc wrote:
[snip]
The Big Bang has been experimentally verified by the WMAP probe.
Yes. Further verified - it was already an established scientific fact.
It hasn't been verified and many physicists think that the universe is steady state.
That is not an accurate description of the state of knowledge. It is difficult to find a physicist who would deny the Big Bang exists without first redefining the terms to mean something different to what that phrase means to other scientists.
Are you capable of making an objective scientific statement that avoids considerable ambiguity? (This is necessary because I genuinely don't know what you mean. The Big Bang is a concept that has developed greatly over time since it was first hypothesised and confirmed. You could be denying any aspect of the model).
As an example, would you say:
"the Universe was never as hot as [...] Kelvin"
or
"the Universe was never as dense as [...] kg/m^3"
Basically, the existence of the Big Bang is the same as saying the observable Universe originates in a region that was once extremely hot and extremely dense (with the values being determined by the point at which known physics is no longer reliable).
[There is more to the model these days, but that is separate to the existence of the Big Bang]
Very interesting fact: the highest temperatures reached in particle accelerators (current record 5.5 trillion Kelvin at the LHC) are just hot enough to investigate the phase change where the Universe changed from a quark plasma to a hadron plasma.
The above position makes an assumption-
Anything is possible.
And concludes it must be considered as a viable possibility. The conclusion is grounded/based in personal beliefs, beliefs as to how the world/universe operates.
Pigs will never fly- irregardless of any imagined multi-verse.
Rational thinking dismisses the possibility, is intelligent enough to comprehend that zero possibility exits for such events and does not waste it’s time on such consideration.
Feb 18, 2022
0
#970
tygxc wrote:
#1250
Think of positions, not of games.
That is another reason why it is smart to start from tabiya, not from the initial position.
m! progresses faster than exponential. lim exp(m) / m! = 0
The Big Bang has been experimentally verified by the WMAP probe.
The Big Bang is verified by at least two things.
Something called cosmic background radiation.
And something called 'Red Shift'.
But what many scientists try to argue is that the Big Bang is 'the universe'.
While they fail to acknowledge that significant violations of Red Shift would not be observable from earth for so many reasons.
I pointed this out to @Elroch a long time ago - and he eventually conceded the point. Took him a while.
The 'Big Bang' could be terminology to refer to the 'observable universe'.
It also has not been proved the universe is finite either.
Neither in mass nor size. Nor in age.
And those will probably Never be proven. Ever.
But nonetheless many like to say 'the universe is expanding'.
How does an infinite object expand?
Also argued is that 'matter creates space'.
Its like its argued there is some kind of 'envelope' of 'non-space' around 'real' space. 
For the 'real space' to expand into ...
Matter alters the space - that doesn't mean it creates it.
I also pointed out to @Elroch several years ago -
and he also eventually conceded -
that if two flashlights are shined away from each other in a vaccuum - then the fronts of their light beams head away from each other at 2c.
Not c.
At twice the speed of light.
That got him very excited. He eventually conceded. But didn't like it.
When this is pointed out to various people -
they often sputter and you get something like:
'but but but if you're riding the head of the light beam then passage of time ...' ('Larentz transformation)
And what could possibly 'ride the lightbeam' ? Lol.
(that would be a hypothetical and special case - as opposed to the main situation)
One can look it up if one wishes.
The two beams head away from each other at Twice the speed of light.
You can think about it too. How would it Not be so? 
!!
(Relevance: somebody on the internet reads that 'chess can be solved in five years using Cloud computers' - and simply chooses to believe it.
Its the same as believing the two beams can't head away from each other at 2c. But they can.)
Feb 18, 2022
0
#971
Do the two lightbeams make a deal?
And somehow say to each other (using tachyons)
"Hey - you know we're not supposed to head away from each other at 2c right? c isn't supposed to be exceeded."
Other lightbeam - "Yeah - you're right. We could get in trouble ...
if you slow down to half-c - I will too. But I'm not going to sign any papers."
Electron and proton walk into a bar.
Bartender to proton: "You can stay. But your friend has to leave. We don't like negative attitudes here."
The two flashlights thing is related to another idea ...
"If a tree falls in the forest - it makes no sound unless somebody's around to hear it"
Of course it makes sound !
But for those who want to put science in a kind of box - they might end up believing any old thing ...
(Relevance) including believing that chess could be 'solved' in 5 years.
The facts support the big bang and relativity theory and chess being weakly solvable in 5 years.
Feb 18, 2022
0
#973
playerafar wrote:
tygxc wrote:
#1250
Think of positions, not of games.
That is another reason why it is smart to start from tabiya, not from the initial position.
m! progresses faster than exponential. lim exp(m) / m! = 0
The Big Bang has been experimentally verified by the WMAP probe.
The Big Bang is verified by at least two things.
Something called cosmic background radiation.
And something called 'Red Shift'.
The cosmic abundance of elements is a key piece of evidence. Especially the distribution in primordial gas clouds that haven't been affected by stars. See more details
But what many scientists try to argue is that the Big Bang is 'the universe'.
They really don't. I bet you can't exhibit a single example of that.
The Big Bang is the first part of the history of our observable Universe. That's it. No knowledgeable person could claim there is nothing beyond what we have observed - it's more a matter of how much there could be, from quite a lot to an infinite amount.
While they fail to acknowledge that significant violations of Red Shift would not be observable from earth for so many reasons.
I pointed this out to @Elroch a long time ago - and he eventually conceded the point. Took him a while.
I don't even know what I am meant to have conceded, never mind recall such a discussion. I don't know what you think you mean by "a violation of red-shift". Red shift is an observational phenomenon resulting from general relativity. Violating it would seem to mean proving GR wrong.
The 'Big Bang' could be terminology to refer to the 'observable universe'.
It also has not been proved the universe is finite either.
It is unknown if the spatial extent of the Universe is infinite. Simple as that.
Neither in mass nor size. Nor in age.
There is no knowledge of anything beyond the start of the Big Bang 13.8 billion years ago. There are speculatory ideas of a larger Universe, but nothing testable.
And those will probably Never be proven. Ever.
But nonetheless many like to say 'the universe is expanding'.
How does an infinite object expand?
Forget that. Just remember that things are moving apart on a large scale. This means the distance between two things increases. Nothing tricky about that, nor does it matter a damn if the Universe is infinite, since it's about two things at a time.
Also argued is that 'matter creates space'.
Its like its argued there is some kind of 'envelope' of 'non-space' around 'real' space.
For the 'real space' to expand into ...
Matter alters the space - that doesn't mean it creates it.
I think you are a bit mixed up here. Energy-momentum is the source of space-time curvature. To my knowledge, no-one worth listening to has ever said "matter creates space".
I also pointed out to @Elroch several years ago -
and he also eventually conceded -
that if two flashlights are shined away from each other in a vaccuum - then the fronts of their light beams head away from each other at 2c.
Not c.
At twice the speed of light.
It is a fact that when you add up the two speeds, each of which is c, you get the value 2c. However, this value is not an observed quantity, it is the sum of two observed quantities.
One reason not to think of this as a single speed is that none of the machinery of special relativity works with it. Rather, do calculations with the individual speeds and they will work.
In Galilean physics, there is no reason to be so careful. That is because the quantity you create is exactly the same as the speed of the second object w.r.t to the first. If something moves one way at speed s and something else moves the other way at a speed -s, then the relative speed is 2s, as observed from one of the moving points.
In special relativity, the speed of one light beam relative to the other is:
(c + c) / (1 + c*c/(c^2)) = c
This is an extreme example of the fact that the speed of light is constant. It is not the same as the Galilean physics, where it would be 2c.
The problem with your quantity 2c is there is no way to transform it if you move to another Lorentz frame. Every quantity used in relativity transforms in a well-behaved way when you move from one frame to another.
It's best to deal only with quantities that transform correctly. These are known as 4-scalars (which remain constant) 4-vectors (which transform according to Lorentz transformations) and 4-covectors (which transform according to inverse Lorentz transformations).
4-scalars you can simply add up, but unfortunately the two speeds of the light in different directions are not 4-scalars. Rather they are the magnitudes of 3-vectors.
In your scenario there are two 4-vectors corresponding to the velocity of light to the left and the velocity of light to the right.
To understand these 4-vectors and how to combine them, it is best to start with the simplest 4-vector which consists of position and time, the 4-position. The complication is that the 4-velocity is the derivative of this with respect to time, but time is a frame-dependent quantitity subject to time dilation. Following on from 4-positions it is time to learn about 4-velocities. Unfortunately, you never get to a quantity corresponding to your 2c, because this is really two quantities combined in an inconsistent way.
Sorry, but that's the way it is.
That got him very excited. He eventually conceded. But didn't like it.
No. Read about 4-positions and 4-velocities to see.
playerafar wrote:
... if two flashlights are shined away from each other in a vaccuum - then the fronts of their light beams head away from each other at 2c.
When this is pointed out to various people -
they often sputter and you get something like:
'but but but if you're riding the head of the light beam then passage of time ...' ('Larentz transformation)
And what could possibly 'ride the lightbeam' ? Lol.
If I recall well, it was Einstein himself who made this mental experiment in his youth. He imagined himself riding a light wave, and understood that the wave would have to be static; but if one accepts that all the physical laws (including electromagnetism) have to be the same in all inertial frames of reference, that would not be possible, so the observer must see the light moving at c with respect to him, no matter what his speed is (but it has to be less than c, to preserve causality). That is possible only if the observer's time slows down as his speed increases. This idea led to the special relativity and as @Elroch already said, you have to use the Lorentz transformations, not the classical ones, if the observers' speeds with respect to the same origin O are comparable to c.
Please stay on topic.
Here is an interesting paper how losing chess was solved.
http://magma.maths.usyd.edu.au/~watkins/LOSING_CHESS/ICGA2016.pdf
It took them to visit 900 million positions.
You went a lot OT in this other thread, where you started this "5 years to solve chess" campaign (we have already argued about that); now you complain because of few OT posts?
Feb 18, 2022
0
#977
tygxc wrote:
Please stay on topic.
Here is an interesting paper how losing chess was solved.
http://magma.maths.usyd.edu.au/~watkins/LOSING_CHESS/ICGA2016.pdf
It took them to visit 900 million positions.
There's no merit in "staying on topic" when you blatantly ignore huge holes in your own viewpoint. Such as (in the case where 1. d4 forms part of an optimal strategy for white, as it probably does) dealing with this position and the vast number of positions that can be reached from it but which do not appear in a set of optimal strategies from tabiya. Sorry to disappoint you, but you cannot stop black playing this. (The fact that it is also unknown if white can force a win here is not crucial).
Given my respect for Sveshnikov, it is clear to me that when he made his comment about the solution of chess, he was talking about doing so to a certain level of confidence for knowledgeable chess players who view that it is impossible for some tabiya to be reached by some pair of optimal strategies.
It is worth pondering on this assumption and its consequences.
If there exists an optimal strategy for white and an optimal strategy for black that reaches a position P, then solving that single position P will determine the value of the game of chess.
This would therefore be an ultraweak solution of chess (with the assumptions) but not a weak solution of chess because a substantial part of the strategies would be missing (eg my position above). This would remain so after every one of a list of tabiya was solved.
The failing of Sveshnikov's proposed program for the genuine solution of chess (even ultraweak solution) is that, even given a long list of tabiya not including the initial position, there is no reliable reasoning that any of them appears in a pair of optimal strategies. Rather, thinking at least one of the tabiya must be in some optimal strategy is an inductive guess based on experience.
Feb 18, 2022
0
#978
Optimissed wrote:
But red shift indicates the contrary. We see light moving at different velocities, dependant on our velocity relative to the light wave origination.
Sorry, but that is simply false.
For example, suppose you are observing some very distant, massively redshift supernova. You observe it both from a telescope far from Earth (which happens to have the initials J.W.) and from Earth and you observe the delay in seeing it from those points. You will find the (very red-shifted) light travelled at the speed of light between the two telescopes.
Elroch wrote:
Optimissed wrote:
But red shift indicates the contrary. We see light moving at different velocities, dependant on our velocity relative to the light wave origination.
Sorry, but that is simply false.
For example, suppose you are observing some very distant, massively redshift supernova. You observe it both from a telescope far from Earth (which happens to have the initials J.W.) and from Earth and you observe the delay in seeing it from those points. You will find the (very red-shifted) light travelled at the speed of light between the two telescopes.
Exactly, the red shifting changes the wavelength, not the speed of light.
#1263
I always stay on topic.
Big bang and relativity are interesting topics, but have nothing to do with solving chess.
So far nobody has found any hole in my arguments.
After 1 d4 black tries to draw with either 1...d5 or 1...Nf6. If either of these draws, then 1...a5 is not relevant. If neither of these draw, then it is sure that 1...a5 cannot draw either.
We can debate the 10^36: maybe more, maybe less.
We can debate the square root: maybe more, maybe less.
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