How many possible chess games ?

 

On the YouTube there is a video about this question and it would bebest to see that one before reading a solution to this question. 

www.youtube.com/watch?v=Km024eldY1A

 

In order to answer this with armed nothing more then excel I came up with the next solution:

 

At first I had only the numbers that this man dr James Grime gave us via this video but on searching on internet I son came up with more real known numbers. 

but my first solution was this one:

1     20

2     400

3     8902

4     197742

 

based on this I constructed a exponential formula in excel as a trend line:

y = 8,995193914985120E-01e3,069952517088020E+00x    with an R² = 9,999290244920450E-01

 

Based on that I found that it takes 60-61 moves to meet with the estimated numbre of atoms in the entire universe approx 10^80 

By this model It would take 91 moves to meet with Shannons number 10^120.

It would take 151 moves for 10^200 posibilities

 

And excel breaks down with its calculations in this exponential formula at 231 moves. leaving us with 8,67E+307 posibilities. 

 

 

Now in a later newer approach I found some more real numbers that were found by other researchers.

https://en.m.wikipedia.org/wiki/Shannon_number

 

No

Moves      posibilities             Log (posibilities)

1                 20                            1,3010299956639800

2                 400                         2,6020599913279600

3                8902                        3,9494875899465000

4                197281                     5,2950852606576800

5                4865609                 6,6871372061227100

6                119060324              8,0757670600670800

7                3195901860            9,5045934344983800

8                84998978956       10,9294137088150000

9                2439530234167    12,3873062048834000

10              69352859712417   13,8410643735811000

 

I Used both the number series "Posibilities" and "Log(Posibilities)" to construct a graph.

 

Now I used both these Log(posibilities) to make up a line which turned out to be lineair to some extend. 

y = 1,39559889615728000000x - 0,21849944630866600000    with a R² = 0,99970112927886700000 

 

Now I tried of course some higher order polynomials, but they did not model this math problem in a predictive way as good as this lineair model did. 

using this lineair model.

 

It turns out that the lineair model is way much better to work with in the high number ranges and matches perfectly with the expenential function as far as that one goes. 

 

Based on this it takes 10^80 is reached in 58 moves (10^80 is the estimated numbero of atoms in the universe) 

10^120 (Shannons number) is reached in 87 moves

10^200 is met at 144 moves

and excel breaks down at 220 moves leaving us with 6,505 * 10^306 posibilities.

 

Now the lineair log (posibilities) model goes on from there.

 

In the movie with James Grime he talks about a max of 11800 moves. 

 

The lineair model predicts a number of posibilities of 

 

10^16467,84848 posibilities with Chess in case 11800 moves are performed. 

 

Now if every posibility would take just a yocto second or 10^ (-24) second this would still take about 10^16444 seconds 

 

this is an unimaginable number and amount of time. 

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