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 13:16:51 -0500

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.

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).

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.

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.

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.

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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