Re: There exists a Nim version that is a "draw" OS

From: Archimedes Plutonium (a_plutonium_at_iw.net)
Date: 07/10/04 

Date: Sat, 10 Jul 2004 15:24:42 -0500

Sat, 10 Jul 2004 13:16:51 -0500 Archimedes Plutonium wrote:

> 10 Jul 2004 00:25:53 -0700 Jan Kristian Haugland wrote:
>
> > A combinatorial game such as Nim can not be a draw.
> > Here is a distant relative that can be a draw:
> >
> > http://home.no.net/zamunda/split.htm
>
> I beg to differ.
>
> Yesterday I was working on a game of Nim, a morph of Nim where there are
> no draws in the game itself but where either player can win in the OS
> and not automatically that one player always wins the OS. Call it a
> pseudodraw.

The minimax theorem says a singular point. Thus a pseudodraw is
nonexistent.

Unless there is a draw within the game itself can the OS be a draw.

>
>
> Secondly, I was looking for another Nim morph where it actually has a
> draw within the game itself and the OS is a draw.
>
> Thirdly I was looking for a Tictactoe morph that was _not_ a draw in the
> OS and where either X or O can win in the OS. Call it a pseudodraw.
>
> Here is what I come up with:
>
> Nim-morph with pseudodraw OS: Let me call the person with first move as
> white and let me call the person with second move as black. The first
> move in this game is not the removal of any matchsticks but is the
> actual layout of the number of rows and the number of matchsticks within
> each row. Black then proceeds as in normal nim. I contend, thence, that
> this nim morph will end up as a win for one of the players but not
> automatically the black player (provided regular nim is considered a
> loss for the one who is forced to pick up the last matchstick).

This is a erroneous claim. Even if I added the rule that only one or two
matchsticks can be removed per move.

>
>
> Nim-morph with a Draw in the game itself: This is where white with first
> move determines the number of rows of matchsticks and the number of
> matchsticks in each row. And finally, determines that at least one row
> is a "Draw row" so that if this row or any of its matchsticks is picked
> up last then the entire game is a draw.

This is possible. It perhaps needs the rule of only one or two matchsticks
removed per move.

>
>
> TicTacToe-morph with pseudodraw OS: this one was a tough one to work out
> last night. I would have thought that Nim was going to be the tougher
> challenge. We have several rule changes to normal tictactoe. Call the
> first mover as X and the second mover as O. In this morph, O gets two
> first moves so that at the end of the game there will be five O on the
> board to four X. And the other change in rule is that if there are no
> three-in-a-row for a outright win then the win goes to the person who
> has the most two-in-a-row. Now I have not fully played out all the
> consequences. But I suspect, not sure of this suspection, that the OS of
> this morph tictactoe is a win for either X or O or a pseudodraw. And
> that every game played of this morph will produce a winner whether it be
> X or O.
>

Trouble with whether "end row middles" would count as 2-in-a-row rather
than having only "shortened 3-in-a-rows" count as 2-in-a-row. When X makes
first move with placing an X in center square then X has the most
2-in-a-row unless we count end-row-middles as 2 in a row for O.

Here again, the concept of Pseudodraw is erroneous, and that unless a draw
exists in the game itself can the OS be a draw. And the minimax theorem
says as much.

>
> Now, the most important aspect of the above, if true, implies that there
> exists a Pseudodraw for the games of checkers and chess, but more
> importantly, that those games OS is a draw with their current and
> present rules.

But the above is not all lost and wasted. I can salvage the idea that to
make Nim a draw is to add the rule that the player with first move decides
the arrangement of how many rows and number of matchsticks per row and
which row is the "Draw" row.

The implications for chess and checkers still remain. That if a game has a
draw possibility, then the OS of that game ends up into that draw play.

Nim OS is a win for one of the players always, well, because there is no
draw possibility while playing the game.

I never played Go. I suspect it has a draw possibility. If it does, then
that is its OS-- a draw. Chess has a draw possibility, thus chess OS is a
draw.
This claim can be made into a assertion and then a theorem.

Devise a game that is a VonNeumann game which has a draw possibility but
has a nonDraw OS. Nim has a nondraw OS but nim has no draw within the game
itself. So when we inject a draw possibility into Nim then does the one
player always win the OS??????

Archimedes Plutonium
www.archimedesplutonium.com
www.iw.net/~a_plutonium
whole entire Universe is just one big atom where dots
of the electron-dot-cloud are galaxies

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