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You start out with a chess board of a certain unknown size. The 64 squares are indeed square: length equal to width.
Within each square, there exists a separate chessboard, which itself has 64 perfectly square squares. So the initial chessboard is a chessboard in its own right, and contains 64 other chessboards.
But that's not the end of it. Each square of each of the 64 2nd-level chessboards ALSO contains a chessboard of 64 perfect squares. So now we have the initial chessboard, containing 64 sub-chessboards, each containing 64 sub-chessboards.
If we continue to have sub-chessboards recursively down to the 14th sublevel from the original chessboard, we would have (64 ^ 13) + 1 chessboards, or 302,231,454,903,657,293,676,545 chessboards (10 ^ 23.48033966179053322667163522547) Not even nearly enough to cover all the possible legal chess positions, which has been estimated around 7,728,772,977,965,919,677,164,873,487,685,453,137,329,736,522 or 10 ^ 45.888 ( http://homepages.cwi.nl/~tromp/chess/chess.html ).
In fact, it's so not nearly enough that you would have to multiply this number of chessboards by 10 ^ (45.888 - 23.48033966179053322667163522547), or 25,565,855,987,491,794,859,325 to cover all possible legal chess postitions. But for now, let's stay at the 14th sublevel for the following illustration of what 14 sublevels (as inadequate as that may be) means:
If the smallest (14th sublevel) chessboards have squares that are 1.0 inch square... and there are no borders between squares that make the board bigger, so that 14th level board is exactly 8.0 inches square... the top level board has squares that are 8 ^ 13 inches along each side ...
that is, 549,755,813,888 inches along each side...
45,812,984,490 feet 8 inches along each side...
/ 5280 =
8,676,701.608 miles along each side...
The entire top-level board is thus
69,413,613 miles long on each side.
The area is 4,818,249,669,713,769 square miles covered by the top level board.
If you were on a jet flying 600 mph, it would take you 13 years, 72 days and 3 hours 21 minutes to get from one side of the board to the opposite side. That's what these 14 sublevels of chessboards means.
So how many more levels would one have to go down to reach a number that would allow for all legal chess positions to be set up?
In other words, 10 ^ 45.888 equals 64 ^ ???
Well, as you may easily surmise, there isn't a nice round number. But 64 ^ 25 gives a number that is about 1/5.4 of the required number, so you'd have to go 26 levels down to be sure to have enough.
If 14 sublevels gives a board that is close to being Earth-to-Sun distance along each length, what do you think another 12 sublevels is going to give?
Well, it would be 8 ^ 26 inches along each length...
But you'll recall that our 14th sublevel of chessboards gave (64 ^ 13) chessboards (plus 1 for the top-level board). But 64 ^ 13 is exactly equal to 8 ^ 26. So the number of chessboards we would get at the 14th sublevel is exactly equal to number of inches we would need along each length to have enough levels (26) to have enough chessboards to represent every single legal chess position (plus a lot of illegal ones if we wanted).
This is just coincidence, I chose the 14th sublevel just by chance. So no magic going on here.
But 8 ^ 26 inches....
302,231,454,903,657,293,676,544 inches...
4,770,067,154,413,783,044 miles...
On that 600 mph jet, it would take you
7,950,111,924,022,972 years to get from one side of the board to the other side...
ALMOST 8 QUADRILLION YEARS...
The Universe is estimated to be 13.79 billion years old...
that is one Universenium (Age of the Universe) if you recall an earlier post of mine...
It would take 576,513 Universeniums to get from one side of the top-level board to the other on a 600 mph jet.
At the speed of light... 186,282 miles per second...
It would take 42,677,833,125 light seconds to get from one side of the board to another...
That's 1,352.38 (rounding up) light years.
With that length along each side, our all-encompassing top-level chessboard would have a distance from corner to opposite corner of about 1,912.55 light years. So it would fit within a circle of that diameter.
Our solar system is about 27,000 light years from the Milky Way's galactic center. This means our all-encompassing chessboard would fit within a circle that is (very roughly) 1/28th of the diameter of the circle with radius from galactic center out to our solar system.
This is all very approximate, especially since we can only achieve a rough estimate of the number of legal chess positions. But it does make one realize that just our Milky Way galaxy can easily contain on a flat plane all the 8 inch by 8 inch chessboards necessary to hold all the estimated legal chess positions.
Well, actually, it's not a flat plane... because that top-level chessboard, having such a large square size, would have to hold enough vertical room for one MegaGodzilla-sized chess King. The calculation of the vertical height of the 26 levels of chessboards necessary, given the height of a typical Staunton chess King that would look right on a 1 inch by 1 inch chess square, is left as an exercise for the reader.
Within each square, there exists a separate chessboard, which itself has 64 perfectly square squares. So the initial chessboard is a chessboard in its own right, and contains 64 other chessboards.
But that's not the end of it. Each square of each of the 64 2nd-level chessboards ALSO contains a chessboard of 64 perfect squares. So now we have the initial chessboard, containing 64 sub-chessboards, each containing 64 sub-chessboards.
If we continue to have sub-chessboards recursively down to the 14th sublevel from the original chessboard, we would have (64 ^ 13) + 1 chessboards, or 302,231,454,903,657,293,676,545 chessboards (10 ^ 23.48033966179053322667163522547) Not even nearly enough to cover all the possible legal chess positions, which has been estimated around 7,728,772,977,965,919,677,164,873,487,685,453,137,329,736,522 or 10 ^ 45.888 ( http://homepages.cwi.nl/~tromp/chess/chess.html ).
In fact, it's so not nearly enough that you would have to multiply this number of chessboards by 10 ^ (45.888 - 23.48033966179053322667163522547), or 25,565,855,987,491,794,859,325 to cover all possible legal chess postitions. But for now, let's stay at the 14th sublevel for the following illustration of what 14 sublevels (as inadequate as that may be) means:
If the smallest (14th sublevel) chessboards have squares that are 1.0 inch square... and there are no borders between squares that make the board bigger, so that 14th level board is exactly 8.0 inches square... the top level board has squares that are 8 ^ 13 inches along each side ...
that is, 549,755,813,888 inches along each side...
45,812,984,490 feet 8 inches along each side...
/ 5280 =
8,676,701.608 miles along each side...
The entire top-level board is thus
69,413,613 miles long on each side.
The area is 4,818,249,669,713,769 square miles covered by the top level board.
If you were on a jet flying 600 mph, it would take you 13 years, 72 days and 3 hours 21 minutes to get from one side of the board to the opposite side. That's what these 14 sublevels of chessboards means.
So how many more levels would one have to go down to reach a number that would allow for all legal chess positions to be set up?
In other words, 10 ^ 45.888 equals 64 ^ ???
Well, as you may easily surmise, there isn't a nice round number. But 64 ^ 25 gives a number that is about 1/5.4 of the required number, so you'd have to go 26 levels down to be sure to have enough.
If 14 sublevels gives a board that is close to being Earth-to-Sun distance along each length, what do you think another 12 sublevels is going to give?
Well, it would be 8 ^ 26 inches along each length...
But you'll recall that our 14th sublevel of chessboards gave (64 ^ 13) chessboards (plus 1 for the top-level board). But 64 ^ 13 is exactly equal to 8 ^ 26. So the number of chessboards we would get at the 14th sublevel is exactly equal to number of inches we would need along each length to have enough levels (26) to have enough chessboards to represent every single legal chess position (plus a lot of illegal ones if we wanted).
This is just coincidence, I chose the 14th sublevel just by chance. So no magic going on here.
But 8 ^ 26 inches....
302,231,454,903,657,293,676,544 inches...
4,770,067,154,413,783,044 miles...
On that 600 mph jet, it would take you
7,950,111,924,022,972 years to get from one side of the board to the other side...
ALMOST 8 QUADRILLION YEARS...
The Universe is estimated to be 13.79 billion years old...
that is one Universenium (Age of the Universe) if you recall an earlier post of mine...
It would take 576,513 Universeniums to get from one side of the top-level board to the other on a 600 mph jet.
At the speed of light... 186,282 miles per second...
It would take 42,677,833,125 light seconds to get from one side of the board to another...
That's 1,352.38 (rounding up) light years.
With that length along each side, our all-encompassing top-level chessboard would have a distance from corner to opposite corner of about 1,912.55 light years. So it would fit within a circle of that diameter.
Our solar system is about 27,000 light years from the Milky Way's galactic center. This means our all-encompassing chessboard would fit within a circle that is (very roughly) 1/28th of the diameter of the circle with radius from galactic center out to our solar system.
This is all very approximate, especially since we can only achieve a rough estimate of the number of legal chess positions. But it does make one realize that just our Milky Way galaxy can easily contain on a flat plane all the 8 inch by 8 inch chessboards necessary to hold all the estimated legal chess positions.
Well, actually, it's not a flat plane... because that top-level chessboard, having such a large square size, would have to hold enough vertical room for one MegaGodzilla-sized chess King. The calculation of the vertical height of the 26 levels of chessboards necessary, given the height of a typical Staunton chess King that would look right on a 1 inch by 1 inch chess square, is left as an exercise for the reader.
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