According to John Tromp et al at http://tromp.github.io/go/legal.html the number of legal 19x19 go positions is

P19 =
2081681993819799846
9947863334486277028
6522453884530548425
6394568209274196127
3801537852564845169
8519643907259916015
6281285460898883144
2712971531931755773
6620397247064840935

P19 ~ 2.081681994 * 10^170

***

The number G19 of legal games under a given go ruleset is unknown. We only know lower bounds and the following upper bound, where N = 19x19 = 361 is the number of intersections on the board: For positional superko (prohibition of recreation of the same position after completion of a move on the board), no passes, and no resignation, the number of possible games is smaller than N^P19 because P19 also restricts the maximal number of moves per game and there are at most N possible intersections per move. So approximately the upper bound for the number of legal games under the mentioned rules is

G19 < 361^(2.081681994 * 10^170)

Note that, changing the ko rules from positional superko to Japanese / Korean / Chinese style no-result ko rules, we get an infinite number of legal games with equivalence classes for as many different games as under positional superko. (For the sake of simplicity, let us ignore the impact of passes and resignations as technicalities for now.)

***

Citation from my rec.games.go Rules FAQ: "Extremely modest estimates look like 10^N, which is based on an assumption of 10 reasonable intersections per move. A popular estimate for 19x19 go is 10^761, which must have originated from a typo and should be 10^361, if at all... These types of estimates are so popular because humans cannot even imagine the suggested number of atoms in the universe, 10^80. For comparison, 3^361 is ca. 10^172."

In the AlphaGo research paper
https://storage.googleapis.com/deepmind ... ing-go.pdf
the number of possible go games is estimated as approximately b^d, where b is the breadth (number of available intersections per move) and d is the depth (length of a game as its number of moves) and stated as approximately 250^150 by referencing to an earlier research article.

This number is not completely meaningless: it is somehow related to empirical data. Even so, the number would be wrong because practically occurring games have been reported to last up to ca. 450 moves. Supposing 250 would be a reasonable average number of available intersections, we would get 250^450. I.e., already by correcting a minor mistake in the referenced number, we get a more meaningful number that already is astronomically larger than the stated 250^150.

If we pretend the 250 in 250^450 to be reasonably correct, this more correct empirical number serves as an empirical lower bound.

***

Therefore, under the made assumptions (such as permitting an empirical estimate for the lower bound), we know that the number G19 of legal go games is

250^450 < G19 < 361^(2.081681994 * 10^170).

***

In various go blog messages and media articles, the same kinds of wrong (by astronomic factors too small) numbers are circulated for "the" complexity of go: 10^361, 10^170, 10^171, 250^150.

The AlphaGo research paper, go players feeding journalists with information and journalists make the same mistake: they copy or cite without understanding the meaning of a particular number.

The worst example has been a popular science journal's statement that ca. 10^170 would be the number of possible go games (when confusing it with the extremely smaller number of possible go positions). This leads us to types of complexities. There are the complexity of the number of legal positions, the complexity of the number of legal games (which is the actual complexity of go as a game because great applicable shorthands to a solution of perfect play are unknown) and the computational complexity of the generalised game on boards with N intersections (but go as a game is played on the 19x19 board with its fixed N = 361).

Thanks, I always thought the number of legal 19x19 go position is around 10^170 (or 2x10^170 according to the new research)
As a non-mathematician/scientist, we need just one number, can you tell me that?

Ironically, I had just come to L19 to comment on the apparent...

...Hmm...

...For a word that best describes...

I have heard many quotes which either may imply or do imply that Fan Hui 2p is one of the "strongest" Go players per se. Of course, he is one of the "strongest" Go players in the world by almost any sense, far stronger than most of us could begin to conceive (traing far harder from far younger than many would have the power to do). But it's in the popular sense of being lethal in the highest forms of international competition this is sometimes implied, slightly extreme. I have also heard the quote that it was the "kill level" in particular leading to the difficulty in pros distinguishing whether it was the human or the bot.

In part, this may be because there is much conviction that Lee Sedol will lose the match in march (or at least one game), so it doesn't really matter too much. However, I am sure that it's probably the routine, happy experience of many physicists, scientists in general, or experts in most other advanced fields where the gap in understanding to an intuitive level is so great, yet being greatly opinionated with limited intuitive information is so easy. In fact, this whole text is an example.

But when it comes to mathematics, maybe fortunately or not, it is not so easy to do so being unskilled, and we have no concept, rather than a possibly erroneous one. Accepting the numbers. Thanks for your insights.

_________________
On Go proverbs:
"A fine Gotation is a diamond in the hand of a dan of wit and a pebble in the hand of a kyu" —Joseph Raux misquoted.

I just wanted to point out that both this thread and the thread on Go being called "The Mentos game," are both considered "General Go Chat."

That is all, I'll see myself out now...