## The Devil's Quarter Game

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posiduck
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### The Devil's Quarter Game

You die, and are brought before the devil to determine whether you will go to heaven or hell.

The Devil says that if you beat him at quarter placement game you can go to heaven. The rules of the game are this:

There is a large circular table. You and the devil each have an unlimited number of quarters. You take turns placing the quarters on the table. The quarters must be laid flat on the table, they cannot hang off the edges, they cannot overlap, but they may touch. A player loses when he cannot make a legal play (i.e. when it is a player's turn, and there is not enough room on the table for them to play another quarter).

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

What was the devil's first play, and what is his unbeatable strategy?

Edit: Oh god, I forgot to put some important rules, let me fix that.

MOD EDIT: Only one coin is played each turn.
Last edited by posiduck on Fri Aug 11, 2006 11:16 am UTC, edited 2 times in total.
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kira
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I might be the only one with this problem but... what on Earth is the "quarter placement game"?

DaveFP
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I don't know what it is either... A fuller explanation of the winning/losing conditions would help a lot. I think I know what they are, but I'm going to wait until it's explained properly.

xkcd
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I don't know either.

I suspect it's something like this game I've heard of with hexagons, but only because I've heard about it in the context of interesting proofs of the existence of perfect strategies.

posiduck
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I am foolish and forgot to include the victory condition. I fixed that above.
Last edited by posiduck on Fri Aug 11, 2006 12:00 pm UTC, edited 1 time in total.
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DaveFP
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Does the Devil have a quarter that is exactly the same size as the table? While I doubt that this is the answer, it is certainly amusing. Also (as a side note), posiduck's sig looks suspiciously like a Star Trek:TNG quote. I'm not sure what I should worry about most: The fact that he put it there, or the fact that I recognise it.

posiduck
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No, the solution isn't a trick like that. All the quarters are the same size (normal quarter size) and the table is fairly large (precise size is not important for the solution, provided it isn't too small for more than one move)
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Sitnaltax
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In Go terms, the devil plays his first move at tengen.
I will not succumb to evil, unless she's cute.

posiduck
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What is that in terms for people totally unfamiliar with Go?
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DaveFP
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According to Wikipedia, tengen is the central position on the Go board. I would have to say I probably agree, as the only other 'interesting' position on the table that it would be a good idea to occupy would be one where the quarter touches the edge of the table.

*suddenly remembers something Randy said earlier about hexagons*

Wait, I've got it! That's a good one, I like it. I'll pm you with the solution so that I don't spoil it for everyone else

ulnevets
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the devil plays it in the center of the table. after that, all he has to do is to mirror your move (reflect it across the diameter of the table that is perpendicular to the line drawn from the placed quarter to the middle quarter).

however, you cannot know that you will lose because a) the devil may not know of this strategy and b)it is difficult to exactly mirror a quarter placements. playing teh first quarter in the middle is a natural thing to do and does not indicate that he knows the strategy. you may still win by small fudgings and inaccuracies.

Sitnaltax
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When I go up against the Devil, I am definitely not counting on his lack of knowledge or skill as a cornerstone of my strategy. Just sayin'.
I will not succumb to evil, unless she's cute.

posiduck
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yeah, I guess I didn't specify that the devil won't make a mistake, and knows how to win, but since he's suggesting the game, I wouldn't hedge my bets on that happening.
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jgf
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I like this puzzle. As my math teacher always said, "Symmetry is math for sexy."

gotang
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This is one of my all-time favourite problems. It might interest you all to know that this was one of the regular interview questions for engineering at Balliol College in Oxford (it's been dropped now though). I know this from one of the old tutors himself, who used the question each year.

Heirtopendragon
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Whoa Whoa Whoa

if you stand before the Devil... aren't you already in hell?

Oort
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Does it matter what shape the table is? (ie, rectangualr, square, or circular?)

GreedyAlgorithm
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Oort wrote:Does it matter what shape the table is? (ie, rectangualr, square, or circular?)

If the table is too small for any quarters, you win. More generally, if the table consists of 2N disconnected areas each large enough for exactly one quarter, you win. Or they don't have to be disconnected, they could be connected with narrow smaller-than-a-quarter strips that are long enough.
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FluffyCubed
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Heh, not being american, my first thought was of a circular table, and big pieces of wood shaped like a quarter of a circle...

I'm assuming they are in fact 25 cent coins?

posiduck
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Oort wrote:Does it matter what shape the table is? (ie, rectangualr, square, or circular?)

The shape can matter.
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Erasmus
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The devil starts in the middle and then matches your placement with a quarter at the same radius but 180Â° away. After the devil's turn, any line through the table's centre is symmetric about that centre, so if you can play a quarter at any point, the devil is guaranteed to be able to play his strategy. Since the table is finite, eventually somebody has to lose, so it must be you.

jwwells
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Note that the rules say that the quarters may not overlap or hang off the edges, but they may touch. A possible way of dealing with the Devil:

If he is a perfect player, he'll have put his coin at the EXACT center. But if there is any leeway at all, you can beat him. So when you put down your next coin, use it to gently push his quarter away just a bit.

posiduck
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jwwells wrote:Note that the rules say that the quarters may not overlap or hang off the edges, but they may touch. A possible way of dealing with the Devil:

If he is a perfect player, he'll have put his coin at the EXACT center. But if there is any leeway at all, you can beat him. So when you put down your next coin, use it to gently push his quarter away just a bit.

Good point about the devil needing to play perfectly. Let's both of you have the capacity to put the coin precisely where you want it, and further that you would get caught if you push the devil's coin around, and the price of cheating is that you lose.
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ulnevets
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posiduck wrote:
jwwells wrote:Note that the rules say that the quarters may not overlap or hang off the edges, but they may touch. A possible way of dealing with the Devil:

If he is a perfect player, he'll have put his coin at the EXACT center. But if there is any leeway at all, you can beat him. So when you put down your next coin, use it to gently push his quarter away just a bit.

Good point about the devil needing to play perfectly. Let's both of you have the capacity to put the coin precisely where you want it, and further that you would get caught if you push the devil's coin around, and the price of cheating is that you lose.

no, the price is your soul

Heirtopendragon
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Raise your hand if you deserve to go to hell no matter if you win or lose

Jack Mac
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More restrictive question; raise your hand if you won't. Assume that it exists, no one's getting off on mere technicalities like religious beliefs!
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ulnevets
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i hear they cut off your hands in hell
Last edited by ulnevets on Sun Oct 01, 2006 5:54 pm UTC, edited 1 time in total.

Heirtopendragon
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Jack Mac wrote:More restrictive question; raise your hand if you won't. Assume that it exists, no one's getting off on mere technicalities like religious beliefs!

Apparently you haven't been listening to the evangelicals.

Jack Mac
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ulnevets wrote:i hear they cut off your hands in hell[/i]

That's okay; you need to wiggle your toes under the severed hand (try writing your name on your hands before you die, because hell has lots of residents and sorting out the hands could take a while) and sort of kick upwards.
Also re evangelicals.....no, no I haven't.
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Chariot
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to continue the go comparisons...
just play mirror go =P

Charodei
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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.

Verysillyman
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You can still lose if you play perfectly, or so i'm told in the movie 'A Beautiful Mind'. I also heard that the guy then went on to create a version of the game that wasn't flawed.

Regardless of that, it depends on the size and shape of the table. It could have some number of coin sized lumps on the edge. Then who wins would depend on how many lumps there were.

Chariot
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Verysillyman wrote:You can still lose if you play perfectly, or so i'm told in the movie 'A Beautiful Mind'. I also heard that the guy then went on to create a version of the game that wasn't flawed.

Regardless of that, it depends on the size and shape of the table. It could have some number of coin sized lumps on the edge. Then who wins would depend on how many lumps there were.

Nash didn't lose that game of go, btw, the "winning move" in that game was actually self-atari (I believe that's what it was, if not it was something equally stupid)

Just, you know...

Ah shit, I'm a fanboy

Cavendish
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What is the incentive to optimal packing?

There are lots of ineffecient packing schemes that still
prevent the addition of further coins (just create larger empty spaces).

This sort of thing hurts the symmetry argument.

posiduck
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Verysillyman wrote:You can still lose if you play perfectly, or so i'm told in the movie 'A Beautiful Mind'. I also heard that the guy then went on to create a version of the game that wasn't flawed.

Regardless of that, it depends on the size and shape of the table. It could have some number of coin sized lumps on the edge. Then who wins would depend on how many lumps there were.

in the original post I specified "large" and "circular". It is safe to assume I would have mentioned if the table also featured weird lumps or what have you.
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fjafjan
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Heirtopendragon wrote:Whoa Whoa Whoa

if you stand before the Devil... aren't you already in hell?

or the desert, but then that sucks pretty bad too so no big diff

What is the incentive to optimal packing?

There are lots of ineffecient packing schemes that still
prevent the addition of further coins (just create larger empty spaces).

This sort of thing hurts the symmetry argument.

But it doesnt have to be optimally packed, they just have to be identically packed, ultimatly you will have a symetrical table with no space left, be that with 50, 70 or 100 quarters[/quote]

EDIT: also i think it is fair to say the devil is alot lamer than death when it comes to these things
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fjafjan wrote:But it doesnt have to be optimally packed, they just have to be identically packed, ultimatly you will have a symetrical table with no space left, be that with 50, 70 or 100 quarters

Yeah, if the devil gets to go first you'd better be a mighty fine fiddle player.
Hooks to my brain are well in.

ikefalcon
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How about a nice game of chess to decide your fate? Anyone know the solution then?

FrizB
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I would punch the devil in the face.
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### Re: The Devil's Quarter Game

is the table made of spots that a quarter fits on
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