hex statistics

what is the average # of moves in a hex game per board size?
does anyone have the empirical data for this for, let's say, games played at littleGolem?

Based on the data from my database of games here.

Note that this isn't all games, it's supposed to consist of games played by "good" players.

Size 11 (118 games): 24.0 moves

Size 13 (6797 games): 36.1 moves

Size 15 (30 games): 44.5 moves

Size 19 (368 games): 76.2 moves

Seems that about 1/5th of the number of hexes is a reasonable heuristic

to be clear that's total moves, not moves per player?

Yes.

Your post reminded me to update the database again, so I went ahead and did that. Might as well give you the updated stats:

Size 11 (138 games): 23.9 moves

Size 13 (6957 games): 36.0 moves

Size 15 (37 games): 46.5 moves

Size 19 (376 games): 75.9 moves

Seems very consistent that about 21% of the board is filled.

Right. Makes sense. Thanks!

By the way do you have similar statistics for any other connection game?
Havannah, Y, Unlur?

You're welcome
Unfortunately no.

spartacu5, I used to think it was 1/3 of the board (33%) but maybe 21% is a better number indeed.

I'd say if the losing side defends in the most reasonable way then it should be about 33%.

that's interesting. by default I would assume the closest possible hex game (perfect play by both sides) would fill up as much of the board as possible.

There's a related question, how many stones should a Hex set have? Enough that the chance of running out (assuming both players are trying to win) is negligible, but the publisher may prefer to keep costs down, so no more than that. So the longest game on record for a specific size board, with a few more stones added to that? How many stones does Nestorgames include with their 14x14 set?

Similarly as lazyplayer I've used 30% in the past as an indicator of a game that's a close battle between the sides. Ofcourse many games end much quicker as there often is a forced win that used just a part of the board instead of it's whole area.

https://littlegolem.net/jsp/game/game.jsp?gid=2216237

Here is a nice example, even 34% taken!

https://littlegolem.net/jsp/game/game.jsp?gid=2157682

This is 40% filled but there are a few filling moves that I played to recharge the timer

So yeah overall ideally it should be near 100% but practically it'll be about 33%

David, enough stones to fill the board? Surely you don't want the risk of running out of stones just to save a few grams of weight in your toy box... :D

The problem may be more interesting in Go where the worst case scenario is really really unlikely.

David, in fact this suggests an hex variant: both players have a limited number of stones and, when they run out of stones, they can redeploy the stones that they already have on the board.

It's not so much about a lighter game box as it is an economic decision. Although yes, shipping costs are also a factor. There are plenty of real-world examples: Yinsh, Twixt, etc. Yinsh addresses the limited supply in the rules. Go has a VERY long-established tradition that there should be enough stones to fill the board. But Hex set manufacturers are under no obligation to follow this tradition. Certainly, a few stones could make a big difference to the bottom line.

For whatever is worth, this is a summary of ALL the 78306 Hex tables since the beginning of the world (i.e. LG! ;) ) up to mid-2021 (2021-06-30). So, not the "good" matches as reported by David, but all of them.

(Note that my algorithm counts swap and resign both as turns; The [SWAP_PIECES] used by LG actually involves another stone, but of course [RESIGN] does not, so the number of stones may be 1 lower than the number of turns.)

Size: Number of tables - Total number of turns -> Turns per table on average:

11x11: 78306 - 4267408 -> ~54.5 turns per table

13x13: 95231 - 2896114 -> ~30.4 turns per table

15x15: 382 - 11320 -> ~29.6 turns per table

19x19: 24971 - 1142308 -> ~45.7 turns per table

---

The max numbers of turn for each size are:

11x11: 105 (only 1: this, which seems a 'real' game)

13x13: 149 ( this ), to then jump at 170 ( this, which is obviously a joke and this which seems very likely to be contrived)

15x15: 100 ( this ) to then jump to 112 ( this and this both of them quit reasonable)

19x19: 261 ( this ) to then jump to 300 ( this ) also both reasonable.

---

The distribution of the number of tables per number of turns in 11x11 is quite peculiar, with its alternation of odd number of turns more frequent than even number of turns:

Of course, the TOTAL number of Hex tables is 78306 + 95231 + 382 + 24971 = 198890; and not 78306 as stated above!!

I think this dataset includes too many games that lasted 0,1, and 2 moves, but maybe i'm wrong.
I couldn't open the turn distribution image...could you represent the histogram in numeric format?

I should note that my database doesn't include games which timed out, unless I could prove that the player who timed out had lost.
There must also be at least 5 moves in the game for it to be included.
I didn't count resignation as a move, but I did count swapping as a move.

@HappyHippo: of course your points make a lot of sense as far as collecting the most relevant matches and studying Hex is concerned; my approach was to collect everything and see what "peoples actually do" before filtering anything out (which often involves some degree of subjectivity). I am not saying any approach is more correct than the other, simply they may yield different informations; for instance, this is how I found the longest matches for each size.

@spartacu5: to open the image right click on its description ("Hex 11x11 Turn...") and choose "Open image in new tab" (or similar); sorry for the inconvenience, I think this is how this web site works...

And yes, the number of matches with a very low number of turns is surprising large; as I said above, they still have been played and may be of some interest under some kind of perspective, for instance to understand something of how the site is used (and Hex is actually played?).

Just in case, here is the LibreOffice Calc spreadsheet with all the data (and a few charts).

Nice analysis! Where can I download the dataset of all hex games played on LittleGolem?

Answering my own question, I found https://littlegolem.net/jsp/info/player_game_list_txt.jsp?plid=149530>id=hex (removing the plid gives all games)

@Happy Hippo

Do those numbers include a large proportion of games that were resigned before the game "officially" ended? I know its ridiculous to play a lot of those games, but for a game tree complexity estimate the avg. game length would be longer? Wiki lists the avg game length of 11x11 hex at around 50 plies. Which is twice as much as your recorded number. Which I assume is somewhat filtered for "true" games and excludes timeouts etc.

Also does anyone have any statistics on win percentage of the first and second player by board size?

Yes, in fact I would say the vast majority were not played to the final conclusion.
I have no way of knowing how long these games would take if played out to the bitter end, but if one is interested in how long a game takes in practice then these numbers are helpful. Of course, different players have different thresholds as to when they resign, so the numbers also reflect the community of players involved.

The winning percentages for Black are:
Size 11: 66/138 47.83%
Size 13: 3491/6957 50.18%
Size 15: 16/37 43.24%
Size 19: 194/376 51.60%

With perfect play, White should win (because she has the swap). You can see that Black's win percentage is slightly lower for size 11. I would ignore size 15 (because there aren't many games included yet), but with size 13 and 19 the swap rule is keeping things pretty evenly matched.

If we restrict ourselves to games played between the top computer players (gzero_bot, mootwo, and leela_bot), for size 13 black wins 47/107 (43.93%) of games, so we can start to see White's theoretical advantage show up here. There aren't a sufficient number of games at other sizes between these players to report.

@Hippo

Thank you. Those are exactly the kind of stats I've been looking for.

Have always been curious as to exactly effective the pie rule is.

size 11 from Miwarre's post looks wrong. There is some parity issue going on there.