Within The Curve: Game
Leroy Quet
Each player has m (n-by-n) grids, where m is the number of players.
(So there are m^2 grids all together.)
I suggest an n of at least 12.
On one of their grids each player secretly draws a closed non-self-
intersecting curve. (The curve is bounded within the n-by-n grid.)
Each player's curve does not go through any intersections of the grid-
lines.
Next, on one of each of the other player's blank n-by-n grids each
player copies his/her curve over.
The copies of each curve must go through the same respective squares
of each grid as the original curve did.
So, there are m copies each of m curves, each player in possession of
one copy of each curve.
Next, secretly and simultaneously, each player fills in the squares
each curve goes through on any particular grid with 1,2,3,...., the
integers placed in order and next to each other along the curve. The
numbers can start anywhere along a curve, and can go either clockwise
or counterclockwise.
Next, each player secretly fills in the squares within each curve's
interior with 1,2,3,..., the numbers placed in order, each number
placed in any empty interior square such that all other numbers
(including possibly numbers along the curve) above, right of, left of,
or below the number are coprime to that number.
(Any pair of adjacent numbers that are both in squares a curve passes
through don't have to be coprime. Only interior numbers have to be
coprime to adjacent numbers along the curve, or to adjacent numbers
that are also on the curve's interior.)
Players continue to fill the interior of each curve with numbers until
the players can't fill in any more numbers under the rules.
When each player has filled in each curve as far as they can, the
score for each player is the sum of the top numbers in the interior
squares of each of the m curves the player filled (partially) in.
Highest score wins.
Players may check their opponents' grids after the game is over to
make sure that all applicable pairs of adjacent numbers are actually
coprime. If a player made a mistake, that player automatically loses
the game.
Thanks,
Leroy Quet
Mensanator
> Here is a game for any number of players.
>
> Each player has m (n-by-n) grids, where m is the number of players.
> (So there are m^2 grids all together.)
> I suggest an n of at least 12.
>
> On one of their grids each player secretly draws a closed non-self-
> intersecting curve. (The curve is bounded within the n-by-n grid.)
> Each player's curve does not go through any intersections of the grid-
> lines.
>
> Next, on one of each of the other player's blank n-by-n grids each
> player copies his/her curve over.
> The copies of each curve must go through the same respective squares
> of each grid as the original curve did.
Yeah, right. That'll happen without error. Better use a copy machine.
> So, there are m copies each of m curves, each player in possession of
> one copy of each curve.
No, each player is in possesion of m copies of his own curve.
Are you trying to imply each player should get a copy from
each other player? Don't you think it's important to state this?
>
> Next, secretly and simultaneously, each player fills in the squares
> each curve goes through on any particular grid with 1,2,3,...., the
> integers placed in order and next to each other along the curve. The
> numbers can start anywhere along a curve, and can go either clockwise
> or counterclockwise.
What do you mean "next to the curve"? These get written into a square
the curve passes through? Or squares along the curve that the curve
doesn't touch?
>
> Next, each player secretly fills in the squares within each curve's
> interior with 1,2,3,..., the numbers placed in order,
I assume that means each number is only used once.
And this implies that a number can't be placed unless it's adjacent to
another number (either a curve number or an interior number).
> each number
> placed in any empty interior square such that all other numbers
> (including possibly numbers along the curve) above, right of, left of,
> or below the number are coprime to that number.
So, some interior squares may not get a number placed in them?
And if you get stuck trying to place 4, you might be able to
if you re-arrange the previous placed numbers? Assuming the
re-arrangement still follows the rules.
>
> (Any pair of adjacent numbers that are both in squares a curve passes
> through don't have to be coprime. Only interior numbers have to be
> coprime to adjacent numbers along the curve, or to adjacent numbers
> that are also on the curve's interior.)
>
> Players continue to fill the interior of each curve with numbers until
> the players can't fill in any more numbers under the rules.
>
> When each player has filled in each curve as far as they can, the
> score for each player is the sum of the top numbers in the interior
> squares of each of the m curves the player filled (partially) in.
>
> Highest score wins.
>
> Players may check their opponents' grids after the game is over to
> make sure that all applicable pairs of adjacent numbers are actually
> coprime. If a player made a mistake, that player automatically loses
> the game.
You probably could automate this using Excel.
>
> Thanks,
> Leroy Quet
Mensanator
What about this as an example? Does it pass all your rules?
Obviously, I can't post an Excel file, so here's a text approximation.
Empty squares are shown as "____" (for spacing in text files).
The curve consists of bracketed numbers [ 1] through [30].
The interior is filled with unbracketed numbers 1 through 11.
All interior cells are filled giving this curve a score of 11, right?
I built the curve such that only primes touch interior cells.
That's legal, right? By that I mean how [13][14][15][16][17]
loops around without creating any interior cells.
____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____
____ ____ ____ ____ [ 9] [10] ____ ____ ____ ____ ____ ____
____ ____ ____ ____ [ 8] [11] [12] ____ ____ ____ ____ ____
____ ____ ____ [ 6] [ 7] 1 [13] [14] [15] ____ ____ ____
____ ____ [ 4] [ 5] 2 5 3 [17] [16] ____ ____ ____
____ [ 2] [ 3] 8 7 4 [19] [18] ____ ____ ____ ____
____ [ 1] 10 11 6 [21] [20] ____ ____ ____ ____ ____
____ [30] [29] 9 [23] [22] ____ ____ ____ ____ ____ ____
____ ____ [28] [27] [24] ____ ____ ____ ____ ____ ____ ____
____ ____ ____ [26] [25] ____ ____ ____ ____ ____ ____ ____
____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____
____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____
I assume I could make a curve with only composites touvhing
the interior cells. Or a curve that has 0 interior cells forcing
a score of 0.
Have I got this all correct?
Leroy Quet
Mensanator wrote:
> On Oct 27, 4:49�pm, Leroy Quet <qqq...@mindspring.com> wrote:
> > Here is a game for any number of players.
> >
> > Each player has m (n-by-n) grids, where m is the number of players.
> > (So there are m^2 grids all together.)
> > I suggest an n of at least 12.
> >
> > On one of their grids each player secretly draws a closed non-self-
> > intersecting curve. (The curve is bounded within the n-by-n grid.)
> > Each player's curve does not go through any intersections of the grid-
> > lines.
> >
> > Next, on one of each of the other player's blank n-by-n grids each
> > player copies his/her curve over.
> > The copies of each curve must go through the same respective squares
> > of each grid as the original curve did.
>
> Yeah, right. That'll happen without error. Better use a copy machine.
>
> > So, there are m copies each of m curves, each player in possession of
> > one copy of each curve.
>
> No, each player is in possesion of m copies of his own curve.
>
> Are you trying to imply each player should get a copy from
> each other player? Don't you think it's important to state this?
Did I not state this. Sorry. Yes, each player is in possession of one
copy of each curve.
> >
> > Next, secretly and simultaneously, each player fills in the squares
> > each curve goes through on any particular grid with 1,2,3,...., the
> > integers placed in order and next to each other along the curve. The
> > numbers can start anywhere along a curve, and can go either clockwise
> > or counterclockwise.
>
> What do you mean "next to the curve"? These get written into a square
> the curve passes through? Or squares along the curve that the curve
> doesn't touch?
The numbers get written in the squares the curve passes through.
> >
> > Next, each player secretly fills in the squares within each curve's
> > interior with 1,2,3,..., the numbers placed in order,
>
> I assume that means each number is only used once.
Yes.
> And this implies that a number can't be placed unless it's adjacent to
> another number (either a curve number or an interior number).
>
The curve numbers are placed first before the interior numbers. And
each curve number is next to another number. (Each number k is between
k-1 and k+1, except 1, which is between 2 and the largest curve
number.)
> > each number
> > placed in any empty interior square such that all other numbers
> > (including possibly numbers along the curve) above, right of, left of,
> > or below the number are coprime to that number.
>
> So, some interior squares may not get a number placed in them?
>
Yes.
> And if you get stuck trying to place 4, you might be able to
> if you re-arrange the previous placed numbers? Assuming the
> re-arrangement still follows the rules.
>
We could make a rule that the players use pens and aren't allowed to
erase. Or players could decide to allow erasing.
It is up to the players.
Leroy Quet
Mensanator wrote:
I think you understand the game correctly, except that you placed a 6
(interior number) next to a 21 (curve number). You also have a 9
(interior) above a 27 (curve).
You lose by default. :)
You could force a 0 score. But all players play using the same curves.
So making a zero-score curve would not benifit anybody, including you.
Thanks,
Leroy Quet
Mensanator
Duh! That's what I get for not writing the Excel verification macro.
I assumed my curve had only primes touching the interior and
only verified the interior numbers for co-primeness.
And about losing. That's a BAD rule because it destroys possible
strategy. Keep in mind that this is a GAME, not a mathematical
research thesis (an Excel verification macro would eliminate bluffing,
so maybe you want hand verification so that you could try to
deliberately make an error hoping your opponents don't catch it).
But the penalty shouldn't be automatic loss, which would discourage
anyone from trying it. In Scrabble, we always add a rule that
says a challenger loses his turn if his challenge fails. That way
you can place a word knowing it would fail a challenge and then
grab the OSPD and wave it under your opponent's nose and shout
"Go ahead! Challenge my word! I DARE you!" Congress deliberately
passes unconstitutional laws knowing they'll stand until challenged
(if ever).
This works best if you have a history of beating challenges
that make your opponents actually fear to make a challenge.
And do the OSPD waving once or twice on words that won't
fail a challenge so your opponent can't tell whether or not
you're bluffing.
Also, your rules state the interior numbers are to be placed in
order, but there's no way to verify that it was done that way,
so this rule is unnecessary.
I'm working on another curve using the strategy of placing all the
composites first so that as the space gets tight, I only have
primes to place, which precludes placing them in order. This,
of course, requires that ALL interior squares get filled (I'm
trying to make a curve that's worth a big score).
You should modify the scoring rule such that verification is done
in ascending order from 1 with the score being the last number
correctly placed. I screwed up my first attempt at a big score
by drawing a bad curve and started at 52 and worked down to 1
(and if all YOU saw was the final grid, you would have no idea
I did it that way). But I got stuck with only 10 interior numbers
to place, so I would have ended up with a score of 10 or less,
rather than the 52 I was shooting for. With the automatic lose
rule, there's no point in even trying that strategy (making the
game a LOT less fun).
>
> You could force a 0 score. But all players play using the same curves.
> So making a zero-score curve would not benifit anybody, including you.
Right. "You'll get no help from me." This might be best if you
think your opponents might be better than you. Wouldn't want to
hand them a potential high score you can't achieve yourself.
If you KNOW you can fill them all, then a big interior space
is a better strategy hoping your opponents will not figure out
how to fill all cells.
>
> Thanks,
> Leroy Quet
Leroy Quet
Mensanator wrote:
>...
> You should modify the scoring rule such that verification is done
> in ascending order from 1 with the score being the last number
> correctly placed.
>....
Agreed.