(Probably not a) New game: Yex
luigi
the discussion on the HEXA-Y problem...
YEX
Yex is a two player connection game played on a hex-hex board. Before
starting the game, three non-consecutive corner cells (A, C and E, for
instance) are marked in some way, thus dividing the board perimeter
into three "sections". The marked cells belong to its two adjacent
sections.
The game starts on an empty board. Each player has stones of his
allocated color: white or black. In their turn, starting with Black,
players put one stone on a vacant cell. The player who connects the
three sections with a single chain of his stones wins. Draws are not
possible in Yex, and, as in Y (I hope I will be right this time) a
randomly filled board always produces one, and only one, winner.
Of course, this is simply Y played on a board whose corners have been
truncated to make it hexagonal. The advantage of this game over Y is
the more, say, harmonic board; the disadvantage, the artificial
marking of three corner cells.
Has anyone ever tried this game?
Nick Bentley
proposing, my first thought is that there are cases where neither
player satisfies the win condition when the board is full (e.g. try
filling one half of the board with one color, and the other half of
the board with the other. in that case, one player will get two
sections and his opponent will get the 3rd.)
Bill Taylor
> I'm probably not the first person to suggest this,
Indeed not, but still YEX is a good name!
> The player who connects the
> three sections with a single chain of his stones wins. Draws are not
> possible in Yex, and, as in Y (I hope I will be right this time) a
> randomly filled board always produces one, and only one, winner.
Yes, you are right, (and Nick's comment about half the board each
is mistaken, as I'm sure he will see if he sketches a tiny pic.)
> Of course, this is simply Y played on a board whose corners have been
> truncated to make it hexagonal.
You are right again. There is, OC, a whole family of such Y games,
played on any-shaped board, with any three marked exterior sections
that overlap only at one cell per adjacency, and the board interior
is locally hex, (or more generally of constant valence 3.)
The three sections need not be of equal length. As it is still
an equi-tasked game, neither player has an advantage,
(except the first player, as always eliminable by PIE).
> The advantage of this game over Y is
> the more, say, harmonic board;
I take it you mean more symmetrical.
> the disadvantage, the artificial marking of three corner cells.
Barely; and these are aesthetic matters only.
> Has anyone ever tried this game?
Yes, Joao Neto and I have played it, but in the end regular Y
seems more natural.
There is, OC, a whole super-family of such games, consisting
of any ODD number of sides, (the PARITY CURSE again!)
We have played 5-Y, (on a jewell-shaped board, a truncated
rhombus), which is quite fun, and I think once or twice, 7-Y.
5-Y is possible to play on a regular penatagonal board,
still with (mostly) interior hex cells, but this requires drawing
a special board, which is a pain.
The winning criterion for 5-Y is obvious, but for 7 and above,
one needs to be careful, as there are conflicting Y shapes.
The proper method is to regard each of the 7 sections as
a single blob, then the winner is whoever makes a Y whose
3 blobs mark off 3 blob-sections that are each less than
half the blobby-circumference.
The ultimate form of this, OC, is the case where EVERY
exterior point is a blob of its own, in which case the wnning
criterion is my favourite:- two nearly-opposite points OR
a Y with convex hull enclosing the centre.
This is, in fact, the same criterion as the blobby thing
mentioned above!
There must be an ODD number of exterior cells, (or blobs),
respectively. e.g. an n:(n+1):n:(n+1):n:(n+1) hexagon
is nice! But the PARITY CURSE cannot be avoided!!!
-- Oddly-even-tempered Taylor
** Dogs believe that they are Men.
** Cats believe that they are God.
Nick Bentley
luigi
> On Oct 27, 7:50 am, luigi <luis.9.8...@gmail.com> wrote:
> The winning criterion for 5-Y is obvious, but for 7 and above,
> one needs to be careful, as there are conflicting Y shapes.
> The proper method is to regard each of the 7 sections as
> a single blob, then the winner is whoever makes a Y whose
> 3 blobs mark off 3 blob-sections that are each less than
> half the blobby-circumference.
I understand your point, though your 'blobs' wording, not being myself
a very good English speaker, somewhat confuses me.
Anyway, it is a wonderful generalization: it can be used to play Y-
like games on almost any board! One question, though: does it also
apply on star-shaped boards (i. e. boards with convex corners)? I'm
quite sure it does, but you're clearly the expert one here...
Bill Taylor
> I understand your point, though your 'blobs' wording, not being myself
> a very good English speaker, somewhat confuses me.
Actually, your English is marvellously good!
Here is a board with 7 randomish edge blobs...
| . . , , . . .
| , . . . . . . ,
| . . . . . . . . .
| , . . , . , . .
The commas show where the adjacent edges overlap.
For the newsreaderly-challenged....
| . . , , . . .
| , . . . . . . ,
| . . . . . . . . .
| , . . , . , . .
Calling the edges A B C D E F G cyclically,
the winner is whoever Y-connects ABE or ACE or any of their
rotational equivalents. So, there are 14 possible Y's which are
all mutually exclusive. It is as if there were just 7 regularly
circularly placed cells, ABCDEF, and whoever connects three
whose convex hull surrounds the centre of the circle, wins.
It is a mathematical miracle that this is equivalent to other
obvious definitions, and is in fact win/loss complementary.
Only for an ODD number of edges, of course!
Call this the "MIRACLE OF PARITY",
to help assuage the "CURSE OF PARITY".
> Anyway, it is a wonderful generalization: it can be used to play
> Y-like games on almost any board!
Absolutely.
> One question, though: does it also apply on star-shaped boards
Yes.
> but you're clearly the expert one here...
You are very kind, Luigi; please keep up your own good work!
-- Winding-number William
** Why do "PRO-LIFE" people usually support
** killing people by warfare and executions?