Why do all Go board sizes have an odd number of lines? Why is a 19x19 board so much better than a 20x20 board?

Question

Here are the number of games played on every different board sizes on DGS (Dragon Go Server):

| Size | CNT    |
+------+--------+
|    5 |   2464 |
|    6 |    466 |
|    7 |  11015 |
|    8 |    434 |
|    9 | 101230 |
|   10 |   1256 |
|   11 |   3166 |
|   12 |    716 |
|   13 |  65205 |
|   14 |    455 |
|   15 |   4656 |
|   16 |    315 |
|   17 |   1145 |
|   18 |     91 |
|   19 | 639319 |
|   20 |    293 |
|   21 |    371 |
|   22 |    104 |
|   23 |    265 |
|   24 |     79 |
|   25 |   3657 | 

As you can see boards of odd size are much much more popular than boards of even size.

In total: 99.5% of games have been played on boards of odd size, while only 0.5% of games have been played on boards of even size.

There seems to be something severely wrong with boards of even size...

So my first question is: what is the problem with boards of even size?

I also wondered why 19x19 is the number one most popular size for Go boards...

Just like the accepted answer to this question says, the principal reason that the 19x19 board is considered to be the best is that:

it is the largest board size on which there are more points on the sides (under the third line) than in the center (above the fourth line).

But I found that to be inaccurate! The board size with the best balance between territory (on the sides) and influence (in the center) is not 19x19, it's 20x20.

If Black puts all his stones on the third line, and White puts all his stone on the fourth line, then:

On a 19x19 board:

• Under area scoring: Black has 192 points, White has 169 points.
• Under territory scoring: Black has 136 points, White has 121 points.

On a 20x20 board:

• Under area scoring: Black has 204 points, White has 196 points.
• Under territory scoring: Black has 144 points, White has 144 points.

So my second question is: why not a 20x20 board?

Answer 1

Almost all go boards are x by x, where x is an odd number (e.g. 9, 13, or 19).

The point is to have the product yield an odd number of potential points. So that there would be an odd number of points in the game and a clear winner.

(In the Japanese counting style of "points," it is possible for both players to have the same number, but under the Chinese counting style of points plus stones played, the sum of the two players' results will be an odd number.)

Answer 2

On an even sized board it is much easier to mirror your opponent's moves (this special tactic is known as "mane go"). Playing the tengen point on an odd sized board is a move that cannot be mirrored.

Answer 3

The 19x19 board is considered the most interesting configuration because of the strategic and tactical scenarios it enables.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . a . . x . b . x . b . x . . a . . |
$$ | . . . B . B . b . B . b . B . B . . . |
$$ | . . . . . . . . . . . . . . . . . . . |

In the above diagram for example, K16 is a 3 space extension from both F16 and O16. It gives black a strong option on the top side, but still allows white invasions, at b for example. K17 could be considered too. Details on the possible invasions can be found here.

It is also true that the 19x19 offers a good balance between center and sides. Beyond the simple math in your question, you need to factor in that in a real game of go:

• black and white are supposed to play the same number of stones
• a central area is harder to enclose (4 borders) than a side area (3 borders) or a corner area (2 borders).

Now, once a size has been chosen as the official reference, it becomes de facto the most popular. Same applies to gomoku, ping pong or swimming... Humans might reconsider their decision to trump computers (cf. AlphaGo's achievement), but that's another story.

Historically, I think the board size also has an astrological meaning (e.g. 19x19 is the closest to the number of days in the year), but I am not an expert here...

Answer 4

Just speculation: Boards with even lengths are rarely used because there is no tengen and because handicap stones cannot be placed point symmetrically in a 9x9 grid.