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Science Forum Index » Mathematics Forum » Secret-Move Linear-Board Game
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| Leroy Quet |
 Posted: Fri Dec 26, 2003 9:56 pm |
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I have a game-theory question below game-description.
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(the game below involves math, but only greater-than, lesser-than, and
addition. So it might appeal to a wider group of potential players.)
2-person game:
You have a number-line "board" with (2m) positions.
Player 1 is odd-integers, player-2 is even integers.
So there are (almost*) m moves where the players move simulateously.
Players, on the k_th move,
place the integer (2k-1) {for odd-player} or (2k) {for even-player}
at any integer-position on the board which has yet to have an integer
placed upon it.
But... the players do this by picking the position secretly (and
writing down this choice so as to avoid cheating) before publically
placing the integer upon the board.
It is okay for two integers to be at the same position IF the
integers' positions were both chosen on the same move.
(*)Play continues until there is one or no empty position(s).
Player-1 (odd) gets a point for every time the highest of every pair
of two adjacent integers is on the right side of the pair.
Player-2 (even) gets a point for every time the highest of two
adjacent integers is on the left side of the pair.
If two integers are at the same position, these values are added for
the purpose of scoring (determining if we have an ascent or descent).
(Players can count integers on the opposite sides of an empty position
as being 'adjacent' or not, as they agree.)
Example board at finish:
(1,2) 10, 3, 4, (5,6), (7, , 9, *
Scored,
P1, P2, P1, P1, P1, P2
Player 1 gets 4 points, player 2 gets 2 points.
(Note, 1+2 = 3, 5+6 = 11, 7+8, = 15)
(the '*' is an empty position)
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Related game theory question:
I noted that this game seems to be much more fun with the "secret
move" rule that if the players simply took turns, mainly because
player-1 could otherwise almost always win.
Are there many games with secret-moves rules? Such a rule could
balance games which otherwise have inherent biases towards one player
or another.
For instance, what if we modified Chess so that the players decide
their moves in secret and then move simultaneously?...If 2 pieces are
moved onto the same square on a move, then we would consider them to
be "at truce" and non-capturing.
(Intended moves would be written down so as to prevent cheating...)
Again, such a rule-change would be to eliminate the bias towards
White, but I would guess it would greatly alter the game in
general....
thanks,
Leroy Quet |
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| Bob Harris |
| Posted: Fri Dec 26, 2003 10:58 pm |
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Leroy Quet wrote:
Quote: ...
Players, on the k_th move, place the integer (2k-1) {for odd-player} or (2k)
{for even-player} at any integer-position on the board which has yet to have
an integer placed upon it.
Could you expand on that? At first read, it seems to me that on turn one 1
and 2 are played, on turn two 3 and 4, and so on.
That doesn't seem like enough freedom for an interesting game, so my next
thought was that the three k's are all different-- that the odd player
(that's me) plays an odd number and the even player plays an even number.
But then this statement is no clear to me:
Quote: It is okay for two integers to be at the same position IF the
integers' positions were both chosen on the same move.
For example, if 13 and 4 are played, do we use up board spots 6 and 2, only
using up the same spot if for example 12 and 12 were played?
Or am I just not getting it at all?
Bob H |
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| Leroy Quet |
| Posted: Sun Dec 28, 2003 5:35 pm |
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Bob Harris <plasticnitlion@wrappermindspring.com> wrote in message news:<BC126CFF.383D8%plasticnitlion@wrappermindspring.com>...
Quote: Leroy Quet wrote:
...
Players, on the k_th move, place the integer (2k-1) {for odd-player} or (2k)
{for even-player} at any integer-position on the board which has yet to have
an integer placed upon it.
Could you expand on that? At first read, it seems to me that on turn one 1
and 2 are played, on turn two 3 and 4, and so on.
That doesn't seem like enough freedom for an interesting game, so my next
thought was that the three k's are all different-- that the odd player
(that's me) plays an odd number and the even player plays an even number.
My original rules have 1 and 2 played on first move, 3 and 4 played on
the 2nd move, as you apparently assumed here. So, there is no choice
what integer is played on a particular move by any one player.
But allowing any integer to be played as long as it:
1) is in the range 1 to n, for some n (do not conflict with rule-3);
2) is always even for a player, is always odd for the other;
3) has not been used yet;
might be more fun.
Hmmm... Perhaps the integers can even be picked at random using a deck
of cards.
(Perhaps we should allow integer re-use here, if standard deck is
used.)
Quote:
But then this statement is no clear to me:
It is okay for two integers to be at the same position IF the
integers' positions were both chosen on the same move.
For example, if 13 and 4 are played, do we use up board spots 6 and 2, only
using up the same spot if for example 12 and 12 were played?
Or am I just not getting it at all?
I do not believe I have made things clear here.
Let us use letters to represent the positions so as to help avoid
confusion.
If position-c, say, is already occupied when a player wants to place a
4 down on the board, the 4 cannot be put at c.
So, the player puts 4 at d, which is empty.
On the *same* move, the other player had independently chosen to put
the 3 at d as well.
So, this is fine.
Now, let us say that c has a 2 and e has 1, and the value of d is now
(3+4) =7.
So, with [...2, 7, 1,...]
both players get a point here because there is one descent and one
ascent among these 3 positions.
Note: I should also point out that it is still fine for a player to
play the 7,
despite that 7 = 3 + 4.
(So, it might be that some adjacent integers neither give a point to
one player nor to the other.)
thanks,
Leroy Quet |
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| Bob Harris |
| Posted: Sun Dec 28, 2003 9:26 pm |
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Leroy Quet wrote:
Quote: ...
Players, on the k_th move, place the integer (2k-1) {for odd-player} or (2k)
{for even-player} at any integer-position on the board which has yet to have
an integer placed upon it.
and I asked:
Quote: Could you expand on that? At first read, it seems to me that on turn one 1
and 2 are played, on turn two 3 and 4, and so on.
That doesn't seem like enough freedom for an interesting game, so my next
thought was that the three k's are all different-- that the odd player
(that's me) plays an odd number and the even player plays an even number.
to which Leroy replied:
Quote: My original rules have 1 and 2 played on first move, 3 and 4 played on
the 2nd move, as you apparently assumed here. So, there is no choice
what integer is played on a particular move by any one player.
Rereading all that, I see what I missed was that when you play a number, you
can play it on any empty spot. For some reason I assumed that pieces 2k-1
and 2k had to be played at board location k.
Quote: Let us use letters to represent the positions so as to help avoid
confusion. ...
And I think that was why I was originally confused-- the board was numbered
and the pieces were numbered. Letters made it a lot clearer to me what you
have in mind.
Unfortunately, I don't have anything profound to say about the game though.
Thanks,
Bob H |
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