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ZeN» Forums » Rules

Subject: A more complete rule set rss

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Craig Duncan
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At the time of this writing (3/13/2018), the game description on ZeN's BGG site is pretty bare bones.

I've played a number of PBEM games of this with the designer, Bill Taylor. He now calls the game just "NZ" rather than "ZeN"(a joke of sorts, since Bill is from New Zealand). Here are the rules, succinctly stated, that we used in our games:

NZ
==
Played on a torus represented as a square grid board. |
Players place one stone per turn on any empty space, |
subject to a permanent ban on making a cross-cut. |
- |
The 2nd player, Z, has the option of swapping colours |
immediately after 3 placements. |
- |
When the board is full, N wins if there is a global path |
in the vertical or NW/SE directions; Z wins if there is |
one in the horizontal or NE/SW directions. |


Note that this is played on an ordinary square grid, not the Quax grid that ZeN is played on, according to the BGG ZeN site.

One player is called "N" because he wins by making a vertical global loop or a falling loop in the "\" direction):

. x . . .
. x . . .
. x . . .
. x . . .
. x . . .


. . x . .
. . . x .
x . . . x
. x . . .
. . x . .


By contrast, Z wins by making a horizontal global loop or a rising loop in the "/" direction.

. . . . .
x x x x x
. . . . .
. . . . .
. . . . .


. . . x .
. . x . .
. x . . .
x . . . x
. . . x .


That's the basic idea. Now, a few notes:

Note that as in Looper, a "global loop" = non-contractible loop. But note that unlike in Looper, diagonally adjacent stones count as connected.

One technicality:

Suppose the N player were to make this loop:

. . x . .
. . . x .
x . . . x
. x . . .
. . x . .


You might think that that is a winning loop for N -- after all, it contains a stone in each row, just as the following does:

. x . . .
. x . . .
. x . . .
. x . . .
. x . . .


So shouldn't it count as a vertical loop?

The answer is No, it counts as a falling spiral global loop, which only is a win for the Z player.

In other words, a global vertical loop is a non-contractible loop which includes a stone on every row, but NOT one on every column. A global horizontal loop is a non-contractible loop which includes a stone on every column, but NOT one on every row.

The reason for this definition is to guarantee "win/loss complementarity" -- that is, to guarantee that a winning position for one player excludes any possibility for a win by the opponent.

If we were to count the rising loop as a vertical win for N, then winning groups for both N and Z (playing o's) could exist simultaneously:

. . x o .
o . . x o
x o . . x
. x o . .
. . x o .


Of course, one player would be the first to make the winning group and thereby win the game, but the game is less interesting because it could be a "mere race" to complete the winning group. Games with win/loss complementarity won't be mere races.

Finally, one last technicality:
Note that a winning group can be a subset of an existing group.

E.g. suppose the board prior to N's turn is:

. . x . .
. . x . .
x x . x x
. . x . .
. . x . .


Obviously, N will play to the center:

. . x . .
. . x . .
x x x x x
. . x . .
. . x . .


Technically this group has a stone in every column, so as a whole it is not a vertical global loop. However, a subset of it meets the definition a vertical global loop. So in fact N has won the game.

Contrast this with the loop we started with above:

. . x . .
. . . x .
x . . . x
. x . . .
. . x . .


Removing any stone breaks the loop, so there is no subset of this that makes a global vertical loop.

These technicalities can make the game seem complicated but in practice it is quite straightforward.

-------------------------------------------

FWIW, here is a game record of a short game we played on a 6x6:

a b c d e f __ __N____Z__
| . . . n . . 1. e3 c2
| . z z z n . 2. c5 d4
| z . n n n . 3. d1 f4
| . n . z . z 4. b4 a3
| . . n . . . 5. c3 b2
| . . n z . . 6. d3 d6
7. c6 d2
8. e2


N's d1 stone connects diagonally to the c6 stone to complete a vertical loop.


[Note: this post is a cut-and-paste from a separate thread here.]
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Interesting!

cdunc123 wrote:
Players place one stone per turn on any empty space, |
subject to a permanent ban on making a cross-cut.

So, in a nutshell NZ is to Crossway what ZeN is to Quax. We could thus also play the Quadrex and the Quickway versions, or the Query version (or even the Bløctagøns one)
cdunc123 wrote:
In other words, a global vertical loop is a non-contractible loop which includes a stone on every row, but NOT one on every column.

What do you mean? Your diagrams precisely show a global vertical loop and a horizontal one that both have a stone on every row and on every column. It even looks like it's necessary, except for the two loops that are not spirals.

cdunc123 wrote:
in Looper

Is it 'Looper' or 'Loopus'? You wrote both in the same post previously.
 
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Craig Duncan
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eobllor wrote:
So, in a nutshell NZ is to Crossway what ZeN is to Quax. We could thus also play the Quadrex and the Quickway versions, or the Query version (or even the Bløctagøns one)



eobllor wrote:
cdunc123 wrote:
In other words, a global vertical loop is a non-contractible loop which includes a stone on every row, but NOT one on every column.

What do you mean? Your diagrams precisely show a global vertical loop and a horizontal one that both have a stone on every row and on every column. It even looks like it's necessary, except for the two loops that are not spirals.

Below is a winning vertical loop. It has a stone on every row, but does not have a stone on every column.

. x . . .
. x . . .
. x . . .
. x . . .
. x . . .


Below is a falling loop. It has a stone on every row; it shares that property in common with the winning vertical loop above.

. . x . .
. . . x .
x . . . x
. x . . .
. . x . .


A rather natural definition of "vertical loop" in principle could be: a vertical loop = a loop that traverses the entire board in a vertical direction, i.e. a loop that contains a stone on every row.

On that definition, the falling loop above would count as a vertical loop. Note that a rising loop would also count as a vertical loop:

. . . x .
. . x . .
. x . . .
x . . . x
. . . x .


The problem with that definition of "vertical loop," however, is that one would also want a parallel definition of "horizontal loop" as a loop containing a stone on every column. Rising and falling loops would meet that criterion, and thus they would count as horizontal loops as well as vertical loops.

That in turn would mean that rising and falling loops would be winning loops for both the N and Z players. And each such loop permits simultaneous parallel loops in different colors, e.g.

. . x o .
o . . x o
x o . . x
. x o . .
. . x o .


The fact that one player's winning loop would not necessarily exclude the possibility of an opponent's winning loop makes for a less interesting game, for reasons explained in the original post above. (Games could be "mere races.")

So, the designer made the decision that a rising loop is a distinct kind of loop, i.e. neither a vertical or horizontal loop, and is a win exclusively for Z. Likewise for a falling loop: it's distinct from vertical and horizontal loops, and is a win exclusively for N.

The question then becomes: how to define vertical and horizontal loops so as to exclude rising and falling loops?

The basic idea = a vertical loop is a global loop that traverses the vertical without at the same time traversing the horizontal; a horizontal loop is the reverse of this.

This basic idea produces the definitions from the original post above:

Quote:
In other words, a global vertical loop is a non-contractible loop which includes a stone on every row, but NOT one on every column. A global horizontal loop is a non-contractible loop which includes a stone on every column, but NOT one on every row.

Does that make things clear? If not, let me know!

eobllor wrote:
cdunc123 wrote:
in Looper

Is it 'Looper' or 'Loopus'? You wrote both in the same post previously.

It's Looper. Loopus was the original name but I decided to change it to Looper. In composing my original post about Looper yesterday in the Abstract Games forum, I accidentally reverted to the original name.
 
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cdunc123 wrote:
The basic idea = a vertical loop is a global loop that traverses the vertical without at the same time traversing the horizontal; a horizontal loop is the reverse of this.

This basic idea produces the definitions from the original post above:

Quote:
In other words, a global vertical loop is a non-contractible loop which includes a stone on every row, but NOT one on every column. A global horizontal loop is a non-contractible loop which includes a stone on every column, but NOT one on every row.

Does that make things clear? If not, let me know!

Hmm the difference between the four types of loop is (and was already) perfectly clear for me.
But I'm sorry, your way of stating it doesn't make sense to me. Both the falling and rising loops have a stone on every row and every column, and they likewise both traverse simultaneously the vertical and the horizontal. The only difference I reckon is their chirality, as one might say: they are the two enantiomorphs of the same type of loop.
Besides, I can't see why we should use the pairs (vertical or falling) and (horizontal or rising) rather than the pairs (vertical or rising) and (horizontal or falling), it seems to me the choice is completely arbitrary.

cdunc123 wrote:
It's Looper. Loopus was the original name but I decided to change it to Looper. In composing my original post about Looper yesterday in the Abstract Games forum, I accidentally reverted to the original name.

OK
 
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eobllor wrote:
I'm sorry, your way of stating it doesn't make sense to me. Both the falling and rising loops have a stone on every row and every column, and they likewise both traverse simultaneously the vertical and the horizontal. The only difference I reckon is their chirality, as one might say: they are the two enantiomorphs of the same type of loop.

You lost me with those technical terms!

In any case, let's try this as an alternative explanation:

If a global loop traverses the vertical but not the horizontal, then it is a vertical loop.

If a global loop traverses the horizontal but not the vertical, then it is a horizontal loop.

If a global loop traverses both the vertical and horizontal, then it is either a rising or falling loop depending on its direction, but (given the definitions above) it is neither a vertical or horizontal loop.

The N-player wins by making either a vertical or falling loop; the Z-player wins by making either a horizontal or rising loop.

Does that help?

Technical footnote 1: We could invent a group term for a global loop that traverses both the vertical and horizontal, e.g. a "spiral loop." Then that last statement could become, "If a global loop traverses both the vertical and horizontal, then it is a spiral loop. A spiral loop is either a rising spiral loop or falling spirial loop (depending on its direction)."

Technical footnote 2: Implicit in the last statement is that we are talking about groups that traverse the vertical and horizontal without containing a proper subset of stones that traverses the vertical and without containing a proper subset of stones that traverses the horizontal.

E.g.

. . x . .
. . x . .
x x x x x
. . x . .
. . x . .


This is a group of stones that traverses the vertical and the horizontal. However, a proper subset of stones within this group traverses the vertical, and a proper subset of stones within this group traverses the horizontal. Thus, this group is the union of a vertical loop and a horizontal loop, rather than a spiral loop.

--------------------------------------------

Here's yet another alternative explanation -- quite different in its lingo, but leading to the same game.

A loop is a vertical loop if it traverses the vertical; a loop is a horizontal loop if it traverses the horizontal.

A loop is a non-composite-vertical loop if it traverses the vertical without traversing the horizontal, that is, is a vertical loop only (not also horizontal loop).

A loop is a non-composite-horizontal loop if it traverses the horizontal without traversing the vertical, that is, is a horizontal loop only (not also a vertical loop).

A loop is a composite loop if it traverses both the vertical and horizontal, i.e. if it is both a vertical loop and a horizontal loop). A composite loop will be either a rising composite loop or a falling composite loop, depending on its direction.

[The same technical point about proper subsets, mentioned above, applies as well to the definition of composite groups.]

The N-player wins by making either a non-composite-vertical loop or a falling-composite loop; the Z-player wins by making either a non-composite-horizontal loop or a rising-composite loop.

eobllor wrote:
Besides, I can't see why we should use the pairs (vertical or falling) and (horizontal or rising) rather than the pairs (vertical or rising) and (horizontal or falling), it seems to me the choice is completely arbitrary.

Yes, it is arbitrary. The game would work just as fine with the opposite pairing. The actual pairings used were chosen simply because there is a nice mnemonic device to remember them: vertical is paired with falling just like the letter N has vertical and falling lines. Horizontal is paired with rising just like the letter Z has horizontal and falling lines. Quite clever, I think!
 
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cdunc123 wrote:
You lost me with those technical terms!

Sorry!

cdunc123 wrote:
If a global loop traverses the vertical but not the horizontal, then it is a vertical loop.

If a global loop traverses the horizontal but not the vertical, then it is a horizontal loop.

If a global loop traverses both the vertical and horizontal, then it is either a rising or falling loop depending on its direction, but (given the definitions above) it is neither a vertical or horizontal loop.

The N-player wins by making either a vertical or falling loop; the Z-player wins by making either a horizontal or rising loop.

Does that help?

Well, this time it seems correct to me, whereas the previous definitions do not.

cdunc123 wrote:
Horizontal is paired with rising just like the letter Z has horizontal and falling lines. Quite clever, I think!

Neat
 
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eobllor wrote:
cdunc123 wrote:
You lost me with those technical terms!

Sorry!

cdunc123 wrote:
If a global loop traverses the vertical but not the horizontal, then it is a vertical loop.

If a global loop traverses the horizontal but not the vertical, then it is a horizontal loop.

If a global loop traverses both the vertical and horizontal, then it is either a rising or falling loop depending on its direction, but (given the definitions above) it is neither a vertical or horizontal loop.

The N-player wins by making either a vertical or falling loop; the Z-player wins by making either a horizontal or rising loop.

Does that help?

Well, this time it seems correct to me, whereas the previous definitions do not.)


Glad things seem correct now.

But I'm still curious as to the problem you see with the previous definitions.

The previous definitions were...

Quote:
In other words, a global vertical loop is a non-contractible loop which includes a stone on every row, but NOT one on every column. A global horizontal loop is a non-contractible loop which includes a stone on every column, but NOT one on every row.

If we define "traverses the vertical" as "includes a stone on every row" and "traverses the horizontal" as "includes a stone on every column," then aren't the two quoted definitions (i.e. in the blue box immediately above, and in the other blue box at the start of this post) the same?

 
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cdunc123 wrote:
If we define "traverses the vertical" as "includes a stone on every row" and "traverses the horizontal" as "includes a stone on every column," then aren't the two quoted definitions (i.e. in the blue box immediately above, and in the other blue box at the start of this post) the same?

Ah, I'm really sorry, I had read you too quickly (I just read your previous posts again).
I thought that instead of distinguishing vertical and horizontal loops from the spirals, you were formulating a definition of vertical (resp. horizontal) loops that also included falling (resp. rising) spirals while excluding the other type of spiral. My mistake! Indeed, we're saying exactly the same thing.
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eobllor wrote:
cdunc123 wrote:
If we define "traverses the vertical" as "includes a stone on every row" and "traverses the horizontal" as "includes a stone on every column," then aren't the two quoted definitions (i.e. in the blue box immediately above, and in the other blue box at the start of this post) the same?

Ah, I'm really sorry, I had read you too quickly (I just read your previous posts again).
I thought that instead of distinguishing vertical and horizontal loops from the spirals, you were formulating a definition of vertical (resp. horizontal) loops that also included falling (resp. rising) spirals while excluding the other type of spiral. My mistake! Indeed, we're saying exactly the same thing.


Ah, that explains it. No worries. Now I get to enjoy the relief that I'm not losing my mind after all.
 
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