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   FAUSTIAN ROUND TABLE COIN GAME
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   Author  Topic: FAUSTIAN ROUND TABLE COIN GAME  (Read 22256 times)
Daleks
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FAUSTIAN ROUND TABLE COIN GAME  
« on: Jul 24th, 2002, 1:55am »
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I say there will always be an odd number of coins placable on the table, since it is round. The first coin goes in the center, and from there 6 coins can fit around it. All subsequent layers have an even number of coins since they have to be symmetric along vertical and horizontal axes. So if you go first, you always force him to make the coin count even, and he eventually loses. Anyone agree?
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Viorel Canja
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #1 on: Jul 25th, 2002, 5:03am »
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This explanation has one flaw : the devil can place a coin overlaping 2 layers. The coin doesn't have to be "layer aligned"
.
 
The right explanation is that by placing a coin in the center the resulting free table surface has the property that any valid coin position has a coresponding valid position that is simetrical to the center of the table. If the devil can place a coin then you can place a coin in the associated simetrical position.
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Json921
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #2 on: Jul 25th, 2002, 4:08pm »
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The devil says, "ok, we'll take turns putting quarters down ... " So, if you go first, lay all your quarters down, covering the whole table. Nowhere does it say that you have to put only one quarter down at a time.
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FlatLine
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #3 on: Jul 25th, 2002, 6:47pm »
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The round table only has a finite amount of space on which to place coins. If you and the devil each have an equal set of coins (the riddle doesn't explicitly mention this though, but that would make the game make sense... should i not be able to assume this?) then the first person that places down his coin will always be one coin ahead of the other one, and therefore will place all of his coins down first.
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dlau
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #4 on: Jul 26th, 2002, 12:23am »
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hehe, wu asked me this and i was stumped for a bit until i realized one thing: the table is symmetrical. One point anywhere matches to exactly one other point on the other side of the table... hope this helps Smiley
David
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S. Owen
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #5 on: Jul 26th, 2002, 7:18am »
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I agree with Viorel's solution - well stated.
 
However I think you can assume that you cannot stack quarters, layer on layer... if you could, then the game would go on endlessly (just keep piling up the quarters). Agreed,  even with this assumption you will never lose, but neither will the devil!
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Kozo Morimoto
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #6 on: Jul 28th, 2002, 6:54am »
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I think when Viorel said over lapping two layers he meant:
Coin   Coin
    Coin   Coin  (in 2D, flat on the table, not vertically)
 
as opposed to
coin coin
coin coin
 
sort of like staggered (like bricks on the side of my house) and not like 2 CD stacks side by side.  (my examples are bad because they are vertical but you get the picture in flat dimension, right?)
 
The riddle doesn't also mention that you can't place coins on its side, so the symmetry answer won't hold.  And it doens tell you how many coin/s you can place in your turn.  And it also doesn't say what the shape/s of the coins are - like are they all the same shape and size?  Are they symmetrical coins?
 
So I have to go with the Json921 answer of laying down as many coins to cover the whole table surface in one go.  This solution works without making any assumptions and only relies on the information give in the riddle.
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Gareth Pearce
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #7 on: Jul 28th, 2002, 6:39pm »
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Hmm I thought the question forbid stacking,  
however I would love to see a proof that hexagonal close pack is always odd for any given circle.
 
the 'quaters' answer is probably right though...
The 'symetry' answer to me assumes that when placing a quater you cant nudge the pile about at all... which I think is too much to ask from the devil Wink
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The wedge of wood
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #8 on: Jul 31st, 2002, 8:58am »
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The table could only be the diameter of one quater.any one think of that   Grin
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Salem
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #9 on: Aug 1st, 2002, 12:23pm »
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When I first thought of this I came up with:
lay the table on it's side, place your quarter on the new top of the table (which was the side of the table). This is the only place a quarter will
stay Smiley
 
But I think the correct answer is symmetry.
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mook
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #10 on: Aug 3rd, 2002, 10:07am »
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symmetry is the answer. since the table can be divided into two equal halves, whoever goes second can place their quarter exactly opposite of where the first person places theirs, unless the first person lays their first coin down flat in the exact center of the table.  In that case the first person will always have a space opposite of where the second person places his coin.
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Kozo Morimoto
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #11 on: Sep 13th, 2002, 7:17am »
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Is symmetry guaranteed?
 
If you place coins just far enough apart that the space in between are just too small to put another coin in, and the spaces are uneven, is it possible to end up with the 2nd player winning?
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Blading
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #12 on: Sep 14th, 2002, 4:56am »
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Kozo:
It's only possible for the second player to win if the first player doesn't play perfectly.  Symmetry is guaranteed in a perfect game, because the playing area is 360 degrees symmetrical.  Once player 1 places his coin in the center, anywhere player 2 can place a coin has an opposing point free for p1 to play, reflected in his center coin.
 
If p2 spaced his coins out, and p1 played for space and packed his in closely (each on separate halves), it's possible for p2 to win.  Imagine stacking coins symmetrically around the center hole on an old CD.  Once the area is full, you remove one coin, and shift one of the remaining coins so that there's 'no space' on the CD.
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Icarus
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #13 on: Oct 15th, 2002, 8:53pm »
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on Jul 28th, 2002, 6:39pm, Gareth Pearce wrote:
however I would love to see a proof that hexagonal close pack is always odd for any given circle.

 
Two and a half months late, but here is answer to this question:
 
Daleks argument that any round table allows an odd number of quarters only depends on 1) the first quarter being placed in the center - you control this, so no problem, and 2) that the remaining quarters are placed neatly around it in expanding horizontal layers, which is not specified anywhere.
 
It is easy to choose tables that allow a maximum even number of quarters. For instance, a table with diameter twice that of a quarter allows a max of 2 quarters on it. By using symmetry in your placement as has been described, it is always possible to force the required (but not max) odd number needed to beat the devil. And if the devil cheats, as you suggest, NO strategy is going to be successful!  
 
I would definitely start with Json921's strategy, not because it would work (the devil's setting the rules here), but because that will force him to set the rules so that the symmetry strategy can be played. If you start with a single coin in the middle, he will fill the table on his turn, and fat chance you have of arguing then that he wouldn't of allowed you to do it!
« Last Edit: Feb 3rd, 2003, 7:07pm by Icarus » IP Logged

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the_domino
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #14 on: Nov 29th, 2002, 5:35am »
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the size nor shape of the table matter in any way.
 
the rules of the game state that you lay down 1 coin per turn on the table (not ontop of any other coins) and it must not over lap any other coins.
 
there is no rule that governs how the coins should be laid out. close packed etc. you just lay them down in any free space available.
 
this is where going first ensures victory
 
by taking the first turn you alway have total control over the free space availble. no matter how long the game goes on it will always boil down the situation where there is either space for only 2 or 1 coins on the table.
 
with one space you simply fill it with a coin and win..
 
with 2 spaces normal rules would dictate that you lose and you put your coin down followd by the devil leaving you no space for your next turn.
 
however what is stopping you from playing dirty and placing your coin in the middle of the space, effectively filling 2 spaces, and denying the devil the space to put down a coin on his next turn.
 
simply by going first ensures that you control what space, if any, is left on the table after each round.
 
goes back to day at schooll playing games like "squares"
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Kozo Morimoto
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #15 on: Nov 29th, 2002, 6:07am »
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What happens if the devil stands the coin on its edge.  It follows all the rules - 1 coin per turn, no overlap etc.  Would that ruin your symmetry?
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Chronos
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #16 on: Dec 1st, 2002, 9:10pm »
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A coin on edge doesn't present a problem, provided you're as skilled as the devil (which seems a risky assumption, given the stakes, but it's the best we've got).  Whenever he puts down a coin on its edge, you put down one on edge, too, in the symmetrical position.  Of course, if he does something like setting down at coin slanted at 45 degrees, with the other end levitated (hey, this is a supernatural being, after all, and he can't be expected to play fair), then you're in pretty bad shape.
 
There are some shapes, by the way, that guarantee player 2 the victory, such as a table shaped like an annulus with the central hole larger than a quarter.  The same strategy applies, except that player 1 can't play in the center any more, so it's player 2 who can mirror the moves.  When it comes down to 2 positions left, they'll be on opposite sides of the table, so you won't be able to steal both at once.
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BenderBot80
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #17 on: Feb 3rd, 2003, 5:15pm »
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on Jul 31st, 2002, 8:58am, The wedge of wood wrote:
The table could only be the diameter of one quater.any one think of that Grin

Yeah that was my thought as well. This puzzle is 'devilish' in it's simplicity.  You are offered to play this game. You see the setup, and immediately you realize that you can win if you go first.  How? Well if the table is as big as a quarter ... Smiley
 
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Icarus
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #18 on: Feb 3rd, 2003, 7:13pm »
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Yeah...Right - the devil is going to challenge you to a game where the first player automatically wins on his first move, and then GIVE YOU the choice of going first! Shocked
 
This puzzle may assume the devil is a little on the thick side, but I doubt he is going to be that stupid! Wink
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wolfgang
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #19 on: Feb 3rd, 2003, 7:52pm »
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He's not going to just let you go first unless he knows something you don't. Expect him to cheat. The rules just say, "The first person who can't put a quarter down loses." So I suspect something like, all your quarters have superglue on them so you can pick them up but you can't put them down. Or they'll be red hot so you can't pick one up. Or he'll move the table where you can't reach it.  Or he'll break a quarter in half with his bare hands and put the pieces in two different places to ruin the symmetry. Or he'll turn the lights out and use a table big enough so that you can't tell just by feel where the exact center is.  
If he says, Go ahead, sure you can go first, you know you're in trouble.
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Gaz
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #20 on: Oct 15th, 2003, 12:14am »
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Surely who ever goes fisrt with equal number of coins will lose.
If you have no coins left you cant put one down and under the rules you lose. Huh
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mdh
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #21 on: Oct 17th, 2003, 8:48pm »
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Gaz,
 
I think the table is necessarily quite small, becuase we immediately figure out the symmetrical solution, and to be sure of it (as the question says we are) we have to also be immediately confident that we won't run out of coins before the game ends.
 
I agree with Icarus though, and go a step further: the devil doesn't offer you games that are as arbitrary as this one - he's got something more up his sleeve!  I like idea of "turning the table" on him and putting it on its side.
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travis
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #22 on: Nov 14th, 2003, 9:46am »
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no where in the riddle does it say that you must put ONLY 1 quarter down at a time, therefore you go first, and FILL THE TABLE UP WITH QUARTERS, then tell him it's his turn.
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jewish cow
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #23 on: Jun 13th, 2004, 6:21pm »
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The trick to this riddle is'nt in beating the devil.  The trick is finding the optimal number of quarters that you can sneak out of hell with.  I suggest that everyone wear cargo pants on the day you are ganna die.  I also suggest Old Navy,  They have the best prices.
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Squigs
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Re: FAUSTIAN ROUND TABLE COIN GAME  
« Reply #24 on: Oct 4th, 2004, 5:51am »
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If you want to cheat, simply refuse to put a coin down.  Since he has to wait until you put yours down, the devil will be unable to put his down.
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