The Devil's Quarter Game

A forum for good logic/math puzzles.

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rath358
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Re: The Devil's Quarter Game

Postby rath358 » Thu Mar 12, 2009 1:09 am UTC

I wonder if the ridges on the edges could be utilized
Spoiler:
to make it impossible for him to perfectly match your placement...

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Re: The Devil's Quarter Game

Postby bittyx » Thu Mar 12, 2009 3:39 pm UTC

rath358 wrote:I wonder if the ridges on the edges could be utilized
Spoiler:
to make it impossible for him to perfectly match your placement...

Spoiler:
No, because he is always mirroring your exact move, so if there was enough room for you to place a coin, there will be enough room for him as well. Perhaps, if he he didn't orient the coins the same way you do, he could be in trouble, but it's easy for him to mirror that too.

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Re: The Devil's Quarter Game

Postby Puck » Thu Mar 12, 2009 10:18 pm UTC

Yes, but:
Spoiler:
there are 119 ridges on a US quarter. That's an odd number.

If we imagine them, for a moment, to be much larger ridges than they actually are, and then imagine that after the devil places his coin in the center of the table, you place yours so that it "interlocks" with the ridges in the devil's quarter, then he will be unable to exactly mirror your position.
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Re: The Devil's Quarter Game

Postby rat4000 » Fri Mar 13, 2009 9:54 pm UTC

To the post above:

Spoiler:
Do you seriously think the Devil would not have thought of that?

I mean, the Devil. If I were playing, I'd pick Hell's currency, which just hapens not to have ridges, and I'm pretty sure he could outsmart me...

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Re: The Devil's Quarter Game

Postby pancake bunny » Sat Mar 14, 2009 1:03 am UTC

he uses devil powers?
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Re: The Devil's Quarter Game

Postby crzftx » Tue Mar 17, 2009 5:34 am UTC

rat4000 wrote:To the post above:

Spoiler:
Do you seriously think the Devil would not have thought of that?

I mean, the Devil. If I were playing, I'd pick Hell's currency, which just hapens not to have ridges, and I'm pretty sure he could outsmart me...


Apparently the devil didn't think of it, because he picked the quarter, not the dime, with 118 ridges, or a smooth penny/nickel.
Spoiler:
But in all actuality, the devil just makes a 180 degree rotation of your move, instead of a symmetrical flip.
You can still always prevent the devil's strategy, however, if you place a quarter like this:
quarters.jpg
quarters.jpg (38.23 KiB) Viewed 11459 times
If you're lucky, you can find a way to make the devil's quarter not fit. It'll be close, though... the ridges are really tiny.
Last edited by crzftx on Sun Apr 05, 2009 9:34 pm UTC, edited 1 time in total.

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Re: The Devil's Quarter Game

Postby mrbaggins » Tue Mar 24, 2009 10:50 am UTC

Wouldn't any move by the devil that has no symetrical move cause him victory?

IE: It doesn't need to be exactly centered, just covering the exact center?
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Re: The Devil's Quarter Game

Postby douglasm » Tue Mar 24, 2009 1:09 pm UTC

mrbaggins wrote:Wouldn't any move by the devil that has no symetrical move cause him victory?

IE: It doesn't need to be exactly centered, just covering the exact center?

Spoiler:
No. The point of the devil's move is not to prevent you from having a symmetric move, but to guarantee that the devil will have a symmetric move when his turn comes up again. At the end of each and every turn the devil takes, the table must be completely symmetric. Anything else does not guarantee that any valid move you can make has a corresponding valid response from the devil. If symmetry is irrevocably broken, the game becomes vastly more difficult to analyze.

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Re: The Devil's Quarter Game

Postby lordlicorice » Sat Apr 04, 2009 6:01 am UTC

Time to rock the boat a little.

I think that (with a circular table) the devil will win even if he doesn't initially play in the center.
Spoiler:
This is a more general winning algorithm for the devil:

-Devil starts anywhere

LOOP {
-Player places somewhere

IF the Player's move is mirror-able
-Devil mirrors the move

IF the Player's move is not mirror-able (the Player just filled the center spot)
-Devil mirrors his starting move
}

:wink:

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Re: The Devil's Quarter Game

Postby HenryS » Sat Apr 04, 2009 3:48 pm UTC

What happens if the following are the first two moves (played with circles of radius 1):
Spoiler:
Devil: (-1,0)
Player: (1,0)

Then neither the players move nor the Devil's starting move is mirrorable.

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Re: The Devil's Quarter Game

Postby lordlicorice » Sat Apr 04, 2009 4:58 pm UTC

HenryS wrote:What happens if the following are the first two moves (played with circles of radius 1):
Spoiler:
Devil: (-1,0)
Player: (1,0)

Then neither the players move nor the Devil's starting move is mirrorable.


Yeah I forgot that case but it still works. Correction:

Spoiler:
-"Starting move" - Devil starts anywhere

LOOP {
-Player places somewhere

//Three exclusive cases
IF the Player's move is mirror-able
-Devil mirrors the move
IF the Player's move is in the center
-Devil mirrors the starting move
IF the Player's move mirrors the starting move
-Devil places in the center
}


It would certainly work on a grid but I'm not sure it always works with quarters on a table. If the devil places his first quarter too close to the center, then the player can block the mirror-move and fill the center.

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Re: The Devil's Quarter Game

Postby HenryS » Sat Apr 04, 2009 5:45 pm UTC

Spoiler:
Right, that's exactly the problem with the example I gave: since the quarters are radius 1, there is no room to put one in the center. If the game is played "on a grid", by which I assume you mean that there are a finite number of non-overlapping spots on which a quarter can be played, then the game isn't that interesting: who wins or loses just depends on if there are an even or odd number of spots.

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Re: The Devil's Quarter Game

Postby lordlicorice » Sat Apr 04, 2009 6:02 pm UTC

Spoiler:
So then all the devil has to do is not make a stupid first move that's too close to the center.

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Re: The Devil's Quarter Game

Postby HenryS » Sat Apr 04, 2009 9:53 pm UTC

Spoiler:
Ok then, again with quarters of radius 1:
Devil plays first at (-n,0) (n is some big number)
Player plays at (0.5,0) (which is not mirror-able and blocks the center position so:)
Devil plays at (n,0)
Player plays at (-1.5,0) (not mirror-able, center is already blocked, mirror of the starting move is already taken)
The Devil's next move is not determined by the loop.

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Re: The Devil's Quarter Game

Postby lordlicorice » Sun Apr 05, 2009 4:02 am UTC

Very interesting objection: the ".5" ruins this simple algorithm.
However, I think the idea still works; you just have to fudge it a little.
Spoiler:
In your example

D1 the devil played out to the far left
P1 the player played in the middle (unmirrorable) but offset a little to the right
D2 the devil played out to the far right
P2 the player adjoined his quarter just to the left of his first move

At this point the center is played, he can't mirror the player, and he can't mirror his starting move.. as you pointed out.

What he can do is adjoin his own quarter to the right of the center-play (2.5,0). At this point the strategy becomes
Don't be the first one to make a mirrorable move aka playing anywhere outside (x,0).

Unfortunately the devil is the one who runs out of moves first. He has to fit a two-unit circle into a one-unit space. But if the devil had simply shifted his D2 move right a bit in response to the player's offset middle play, he would have had space! Your objection only works if P1 places his circle [imath]-1<x<1[/imath]. If the player shifts his P1 right (away from D1) [imath]x[/imath] then the devil needs to shift D2 right [imath]x+.5[/imath] to make room for a collinear D3. If the player shifts left (toward D1) [imath]x[/imath] then placing D2 shifted left [imath]x-.5[/imath] leaves exactly enough room for the player to squeeze in a P2. So the devil can either constrict his D2 a tiny bit or expand it [imath]-x+1.5[/imath] to make room for a D3 to squeeze in. The problem with expanding it is that if there's enough room "outside" of D1 to expand D2 by as much as 2.5, then there's enough room for the player to leap-frog and place a quarter to the left of D1! And even if the devil shrinks his D2 then if there's enough room on the right of D2 for at least one quarter, then again it turns into
Don't be the first one to make a mirrorable move aka playing anywhere outside (x,0).

Unfortunately the devil is the one who runs out of moves first. Last time we second guessed our placement of D2. What is there left to second-guess? The placement of D1. If the devil makes his initial move on the edge of the table then there's not "enough room on the right of D2 for at least one quarter!" In other words, if the devil places D1 on the edge and corrects his D2 placement for an offset P1, then you're both limited to placing your moves between D1 and D2, which is a contest we've already seen that the devil can win.

Notes:
this process of who-will-fill-the-space-first also applies to the potentially troublesome case of P1=(-1,0).. the devil wins. This is actually simpler than the x=.5 case above where you had to deal with weird fractional offsets. Try visualizing this simpler case if you can't see where I got my conclusions.
the constants ".5" and "1.5" are dependent on the size of the table mod the size of a quarter, but they should be able to be generalized

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Re: The Devil's Quarter Game

Postby jestingrabbit » Sun Apr 05, 2009 7:07 am UTC

For me, whilst the question posed at the top of this page "what about the coin ridges?" was an interesting piece of lateral thinking, it wasn't about the logic puzzle.

The puzzle that I like would probably be best formulated in terms of the Australian 50 cent coin.

http://upload.wikimedia.org/wikipedia/e ... ts_Rev.jpg

Because satan would clearly use Australian currency.
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Re: The Devil's Quarter Game

Postby 6453893 » Mon Apr 06, 2009 3:34 am UTC

jestingrabbit wrote:For me, whilst the question posed at the top of this page "what about the coin ridges?" was an interesting piece of lateral thinking, it wasn't about the logic puzzle.

The puzzle that I like would probably be best formulated in terms of the Australian 50 cent coin.

http://upload.wikimedia.org/wikipedia/e ... ts_Rev.jpg

Because satan would clearly use Australian currency.


It's true. Before I moved here I never would have guessed hell could be situated in Brisbane.

If he really wanted to be a dick he would use australian 5 cent pieces. They are, as if by some voodoo magic, always, ALWAYS grimy.

lordlicorice: The devil runs out of moves unfortunately? Whose side are you on?

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Re: The Devil's Quarter Game

Postby ircmaxell » Fri Apr 10, 2009 7:06 pm UTC

Spoiler:
The table is non-horizontal. It's a spiral shapped table, standing on edge, perpendicular (or mostly, enough to break friction between coin and table, and still close enough that the top edge of the table touches the ground) to the floor. When he places his first coin, he stands it on the floor against the table in the exact spot the table is tangental to the floor. Technically, the coin is "on the table" and doesn't hang over. Now, so long as there isn't enough table ABOVE the devil's coin, at the sole point of contact of the table, you can't play your move...

Think of a spiral table, with the width of the spirals 1.5 times the width of the coin. you can't go above, because your coin would be "off the edge", so the only other moves available to you would in effect force a loss due to the tangental nature of the point of contact. You loose in the first move...

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Re: The Devil's Quarter Game

Postby mrbaggins » Mon Jun 08, 2009 5:45 am UTC

I had a thought about this problem just now.

Playing in the middle doesn't guarantee the devil anything.

Let the grid be isometric (As it is how circles tesselate best) with sides 1, coins have a radius of 0.5 (diameter of 1)

Devil plays in the middle. Player plays to the up and right of the center (coins touching). Devils plays directly below (for symetry). Player plays off grid, slightly to the above and left of the center. Devil has no symmetrical move, and so loses surety.
Image
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Re: The Devil's Quarter Game

Postby Nitrodon » Mon Jun 08, 2009 5:48 am UTC

The devil would place his second coin down and left of center, not down and right.

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Re: The Devil's Quarter Game

Postby Poohblah » Mon Jun 08, 2009 5:53 am UTC

Mirror. As in, reflected about the origin (center), not some arbitrary line.

You haven't a snowball's chance in hell if the devil is a perfect player.

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Re: The Devil's Quarter Game

Postby mrbaggins » Mon Jun 08, 2009 6:56 am UTC

Bah... yeah.... I realised that the first time I read this problem, but for some reason I discarded it whilst thinking earlier...
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Re: The Devil's Quarter Game

Postby helloearthling » Mon Jun 08, 2009 6:39 pm UTC

posiduck wrote:The devil plays first, and you realize that you will lose.


Well, yeah. He's the devil.

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Re: The Devil's Quarter Game

Postby lordlicorice » Mon Jun 08, 2009 10:58 pm UTC

Anyway he does have a "symmetric" move on the bottom left in your image. You didn't really think that through too well :roll:

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Re: The Devil's Quarter Game

Postby mrbaggins » Tue Jun 09, 2009 2:40 am UTC

lordlicorice wrote:Anyway he does have a "symmetric" move on the bottom left in your image. You didn't really think that through too well :roll:

Please show me where.
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Re: The Devil's Quarter Game

Postby lordlicorice » Wed Jun 10, 2009 1:13 am UTC

Image

I assume you meant the black move to be on the intersection of triangles like the others, since that seems to be why you used the triangle background. It's an obvious move, basically identical to the other one on the right.

Analysis:

Now wherever black goes red can mirror it on the opposite side. There's no move without a mirror on the other side. Well there may be (if the other moves were asymmetric) but then red can "always" make a complementary unmirrorable move that either puts them in a position where black is out of unmirrorable moves or where red can again complement an unmirrorable move. I put "always" in quotes because it works in normal euclidian 2d geometry but you might be able to construct a space where it doesn't work, like the tesselated triangles.

Actually I'm not sure if it does work in the triangles space. There's no 2-pairing of opposites so you might be working with 3s and the winner would be determined by the parity of how many quarters you can squeeze in around the middle quarter. On the other hand it's not a different space just a subspace and you kind of have opposites. Hm. This is much more complicated than I intended to make it. :? :? :?

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Re: The Devil's Quarter Game

Postby mrbaggins » Wed Jun 10, 2009 3:49 am UTC

Nah, the second black move is supposed to be off center, thus making that next move not possible.
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Re: The Devil's Quarter Game

Postby jestingrabbit » Wed Jun 10, 2009 4:37 am UTC

mrbaggins wrote:Nah, the second black move is supposed to be off center, thus making that next move not possible.


Fair enough, but the second red move would be where the red dot is in lordlicorice's post. Basically, if the centre of the table is the origin in the Cartesian plane, and we label moves by the vector that is the centre of the coin, then when the human plays v, the devil follows with -v. In your image you have v=(x,y) being followed by (x,-y), whereas the devil would move (-x,-y) instead, if the devil is playing the strategy that is known to be a winner (the devil might be more cunning still and use a different winning strategy that makes you think you can win, but which still ends with you losing).
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Re:

Postby Random832 » Wed Jun 10, 2009 5:07 am UTC

Charodei wrote:The problem is more complex on other table shapes. On an asymmetrical table, I don't think there is any clear winning strategy. A table with bilateral, but not rotational, symmetry (e.g. a semicircle) has some strategy. The winner is the one who can place the last coin on the line of symmetry. Once that line is covered, one player mirrors the moves of the other. Since the critical play is on a straight line, it may be possible to find a winning strategy. I think the first person can always win, but don't have a proof yet. Possibly induction on the length of the line, with the unit of measure the width of a coin.


No. Once a coin is placed that crosses the line but is not exactly centered on the line, neither player can cause the board to be restored to a state that has any symmetry. This then becomes equivalent to an asymmetrical table.

lordlicorice wrote:
HenryS wrote:What happens if the following are the first two moves (played with circles of radius 1):
Spoiler:
Devil: (-1,0)
Player: (1,0)

Then neither the players move nor the Devil's starting move is mirrorable.


Yeah I forgot that case but it still works. Correction:

Spoiler:
-"Starting move" - Devil starts anywhere

LOOP {
-Player places somewhere

//Three exclusive cases
IF the Player's move is mirror-able
-Devil mirrors the move
IF the Player's move is in the center
-Devil mirrors the starting move
IF the Player's move mirrors the starting move
-Devil places in the center
}


It would certainly work on a grid but I'm not sure it always works with quarters on a table. If the devil places his first quarter too close to the center, then the player can block the mirror-move and fill the center.


Devil's first move: (50, 0)
Player's first move: (-49.9, 0)
OR
Player's first move: (-0.1, 0)

Symmetry is irrevocably broken if a quarter is placed that blocks either the center or the mirror of the devil's first move without _being_ the move that is being blocked. The devil has to play the center first.

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Re: The Devil's Quarter Game

Postby lordlicorice » Wed Jun 10, 2009 11:34 pm UTC

Random832:
Player's first move: (-49.9, 0)

Then the devil plays the middle.
Player's first move: (-0.1, 0)

Then the devil plays -50.

Yeah exact symmetry can be broken but it still works out for the devil if D's initial move is right on the edge of the table. I think. See my spoilered post above.

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Re: The Devil's Quarter Game

Postby Random832 » Thu Jun 11, 2009 4:07 am UTC

lordlicorice wrote:Random832:
Player's first move: (-49.9, 0)

Then the devil plays the middle.
Player's first move: (-0.1, 0)

Then the devil plays -50.


That doesn't prove anything. Symmetry is still broken.

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Re: The Devil's Quarter Game

Postby ServantOfScience » Thu Aug 13, 2009 7:40 am UTC

Spoiler:
I would say that the devil places a quarter such that the quarter is centered on the center of the table. If he does this, then it is impossible for your to perform a move that he cannot mirror, by placing all of his quarters such that your last move, the center of the table and his quarter are colinear. By following this scheme, you cannot place a quarter he cannot mirror, since the space required to place a quarter he cannot mirror is already blocked by a quarter one of you placed.
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Re: The Devil's Quarter Game

Postby lordlicorice » Thu Aug 13, 2009 4:59 pm UTC

ServantOfScience wrote:
Spoiler:
I would say that the devil places a quarter such that the quarter is centered on the center of the table. If he does this, then it is impossible for your to perform a move that he cannot mirror, by placing all of his quarters such that your last move, the center of the table and his quarter are colinear. By following this scheme, you cannot place a quarter he cannot mirror, since the space required to place a quarter he cannot mirror is already blocked by a quarter one of you placed.

We established that more than 3 years ago. :?

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Re: The Devil's Quarter Game

Postby Random832 » Thu Aug 13, 2009 6:35 pm UTC

Right - the latest digression in the thread was based on explaining just why the center has to be his _first_ move (following an argument that he can still win if you take the center), because you could block the center without placing a quarter in the exact center.

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Re: The Devil's Quarter Game

Postby talvinen » Mon Aug 17, 2009 1:25 pm UTC

This reminds me of a game I came across as a kid. It's pretty easy to win actually:
Start with any number, say, 21. You and your opponent alternately substract anything from 1 to 5 from this number. The player who substracts the last digit (to or below zero), loses the game.
Spoiler:
We can easily see that we want to reduce the number to 1, your opponent HAS to substract at least 1 and loses automatically. Before 1, we try to reach 7: No matter what digit (1-5) your opponent substracts, you can bring the number down to 1. Same goes for 13->7, 19->13 etc.
As long as you begin, and the starting number is not 1+6x, you will always win.


I think the solution might be similiar here.
Spoiler:
The devil can (by a very small margin) choose how many legal moves to remove from the table - by either efficiently using space or wasting it. If he can control how many (theoretical) legal moves the table holds, he might be able to bring the game down to the situation that there are 3 spots left - either you cover 2 of them, he covers the last, or you cover one, he covers two. In any case, you lose.


okay, this thread was necro-ed :roll:

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Re: The Devil's Quarter Game

Postby Random832 » Mon Aug 17, 2009 5:25 pm UTC

talvinen wrote:I think the solution might be similiar here.
Spoiler:
The devil can (by a very small margin) choose how many legal moves to remove from the table - by either efficiently using space or wasting it. If he can control how many (theoretical) legal moves the table holds, he might be able to bring the game down to the situation that there are 3 spots left - either you cover 2 of them, he covers the last, or you cover one, he covers two. In any case, you lose.


Yeah, except until it gets very close to the endgame, the number of legal moves is: infinite, and he can reduce it to: infinite. Without a constructively proveable strategy like the
Spoiler:
"play the center, then copy the opposing player's moves rotated 180 degrees",
there's no way to guarantee that _he_, rather than _you_, will be the one to create this situation.

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Re: The Devil's Quarter Game

Postby keithbrunkala » Mon Aug 17, 2009 9:54 pm UTC

The first post doesn't say that the players are limited to one quarter per turn, so my thinking is that the Devil fills the entire table with his first move.

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Re: The Devil's Quarter Game

Postby jestingrabbit » Tue Aug 18, 2009 5:07 am UTC

keithbrunkala wrote:The first post doesn't say that the players are limited to one quarter per turn, so my thinking is that the Devil fills the entire table with his first move.


I think its there In the "take turns placing quarters" part. But let me just make it very clear that each player plays one coin per turn.

Also, spoiler solutions people.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

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Re: The Devil's Quarter Game

Postby Random832 » Tue Aug 18, 2009 1:55 pm UTC

jestingrabbit wrote:Also, spoiler solutions people.


I didn't think that applied to problems that had been solved years ago and which we'd long since moved on to analyzing the details of proposed alternate solutions.

Oh, and where you yourself made a clear unspoilered reference to the very same "strategy that is known to be a winner".
Last edited by Random832 on Tue Aug 18, 2009 1:58 pm UTC, edited 1 time in total.

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Re: The Devil's Quarter Game

Postby jestingrabbit » Tue Aug 18, 2009 1:57 pm UTC

Random832 wrote:
jestingrabbit wrote:Also, spoiler solutions people.


I didn't think that applied to problems that had been solved years ago and which we'd long since moved on to analyzing the details of proposed alternate solutions.


When there are users in the thread that are currently trying to solve it, I'd prefer that they were spoilered.
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