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        riddles >> medium >> FAUSTIAN ROUND TABLE COIN GAME
        
(Message started by: Daleks on Jul 24^th, 2002, 1:55am)

Title: FAUSTIAN ROUND TABLE COIN GAME
Post by Daleks on Jul 24^th, 2002, 1:55am

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?

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Viorel Canja on Jul 25^th, 2002, 5:03am

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.

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Json921 on Jul 25^th, 2002, 4:08pm

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.

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by FlatLine on Jul 25^th, 2002, 6:47pm

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.

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by dlau on Jul 26^th, 2002, 12:23am

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 :)
David

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by srowen on Jul 26^th, 2002, 7:18am

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!

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Kozo Morimoto on Jul 28^th, 2002, 6:54am

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.

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Gareth Pearce on Jul 28^th, 2002, 6:39pm

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 ;)

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by The wedge of wood on Jul 31^st, 2002, 8:58am

The table could only be the diameter of one quater.any one think of that ;D

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Salem on Aug 1^st, 2002, 12:23pm

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 :)

But I think the correct answer is symmetry.

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by mook on Aug 3^rd, 2002, 10:07am

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.

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Kozo Morimoto on Sep 13^th, 2002, 7:17am

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?

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Blading on Sep 14^th, 2002, 4:56am

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.

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Icarus on Oct 15^th, 2002, 8:53pm

on 07/28/02 at 18:39:34, 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!

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by the_domino on Nov 29^th, 2002, 5:35am

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"

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Kozo Morimoto on Nov 29^th, 2002, 6:07am

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?

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Chronos on Dec 1^st, 2002, 9:10pm

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.

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by BenderBot80 on Feb 3^rd, 2003, 5:15pm

on 07/31/02 at 08:58:49, The wedge of wood wrote:

The table could only be the diameter of one quater.any one think of that ;D

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 ... :-)

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Icarus on Feb 3^rd, 2003, 7:13pm

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! :o

This puzzle may assume the devil is a little on the thick side, but I doubt he is going to be that stupid! ;)

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by wolfgang on Feb 3^rd, 2003, 7:52pm

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.

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Gaz on Oct 15^th, 2003, 12:14am

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. ???

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by mdh on Oct 17^th, 2003, 8:48pm

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.

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by travis on Nov 14^th, 2003, 9:46am

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.

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by jewish cow on Jun 13^th, 2004, 6:21pm

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.

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Squigs on Oct 4^th, 2004, 5:51am

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.

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by platodog on Jul 24^th, 2005, 6:17pm

Wow! After reading your responses I fear that my answer is quite unintellectual, nonetheless, I propose as follows:

[hideb]Put the quarter on the table (or place the quarter on the ground), then flip the table upside down so the quarter is sandwiched between the ground and the table surface. Since the table is upside down no more quarters can be placed on its surface.[/hideb]

Any thoughts?

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by towr on Jul 25^th, 2005, 4:26am

What's to stop the devil from flipping the table back?

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by River_Phoenix on Jul 25^th, 2005, 7:21am

If the devil puts a coin levitated at a slant, just slide your coin under his and keep going. :-/

The trouble is if he puts a coin down which whips around the table like an electron at infinite speed, knocking all your quarters off before they hit the surface.

Just don't look back.. and under NO circumstances should you eat the pomegranate.

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Jayant Chauhan on Jul 25^th, 2005, 8:05pm

Well, with all the Lateral thinking you guys have done, I think its the fact that the packing of quarters is gonna tend to be hexagonal no matter what happens, the only deciding factor beingthe first move. The cases:
1) First player places the coin at the center, that makes the total no. of coins that can go onto the table as odd, hence the first person can manage to win,

2) First player places the coin newhere else, which results in an even number of coins, though it could still be even if the second player goofs up, who in our case is the DEVIL HIMSELF, so that ain't possible.

SO thats how it is done I suppose, no tilting and cheating

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Dr Irvine on Sep 6^th, 2005, 2:28pm

The symmetry solution contains an implicit assumption that you are not allowed to move coins that have already been placed on the table. Nowhere in the problem is it stated that you cannot move already placed coins. If the devil is allowed to move coins, then there certainly exist tables on which an even number of coins can be placed (e.g. a table whose radius is exactly the diameter of a quarter).

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Icarus on Sep 6^th, 2005, 5:26pm

Gareth Pierce suggested nudging the coins around on the first page, and I pointed out that there are tables which allow only an even number of coins (using the exact same example as yours).

But as I also pointed out in the same post, the devil is a shifty creature and will twist the rules to his advantage, so you need to be sure before the start that you know exactly what is allowed, and what isn't. Or else, attempt all twists first, so the devil must state the restrictions (and therefore live by them).

But alas, he is known as the "Prince of Lies", so even if you win, your fate is still surely bad. :(

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Citizen on Jul 24^th, 2006, 1:48pm

The rule set as put to you by the devil:
- take turns putting quarters down
- no overlapping allowed
- the quarters must rest on the table surface
- the first guy who can't put a quarter down loses

Possible twists -
1. [hide]Starting first, put down all of your coins; this guarantees you're not the first guy who can't put a quarter down[/hide]
2. [hide]Stretch the first quarter to a size such that it covers the entire table (remember, this is hell - fire and brimstone abound. you could get the metal hot enough)[/hide]
3. [hide]Nick a quarter from the devil - if he runs out first then he can't put a quarter down (assumes you start with equal piles and the table can accomodate all coins)[/hide]
4. [hide]Take the coins, buy a tiny table with room for just one coin and put your quarter there[/hide]
5. Give your quarters to the poor beggar so he can buy some food - God will be so moved by your generous sacrifice that he'll rescue you from the Devil and hell

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Icarus on Jul 24^th, 2006, 6:18pm

(There's really no point in hiding at this stage in a thread. Hiding is only useful if you are posting a legitimate solution near the beginning of the thread. This prevents other people from seeing the answer before they get the chance to think about the problem themselves. But anyone reading here is well passed that and is sure to read everything anyway.)

on 07/24/06 at 13:48:07, Citizen wrote:

1. Starting first, put down all of your coins; this guarantees you're not the first guy who can't put a quarter down

On Nick's turn, he places a single coin of his down, then says "your turn". :(

Quote:

2. Stretch the first quarter to a size such that it covers the entire table (remember, this is hell - fire and brimstone abound. you could get the metal hot enough)

You quickly recognize that Satan has simply failed to provide any foundry tools (inhospitable fellow that he is). Attempts to stretch the coin with your hands and feet quickly leave them too burned to manipulate anything. Satan looks at you and says "I guess that means you are unable to place a coin, now doesn't it? :-[quote]3. Nick a quarter from the devil - if he runs out first then he can't put a quarter down (assumes you start with equal piles and the table can accomodate all coins).[/quote]

You slyly drop your purloined coin into your sack, amazed at how easy it was to put one over on ol' beelzebub. But something nags at your brain, and after considerable thought, it finally hits you: there was no "clink" when you dropped in the coin. In dread you open the bag and look in. There is your purloined coin, all alone. Bub's bag is looking suspiciously full, as he gives you a large grin. Wasn't there something about him also being the Prince of thieves? ::)

Quote:

4. Take the coins, buy a tiny table with room for just one coin and put your quarter there.

"Very clever!", says Lucifer, "indeed, I never specified that this was the table we would both use. I must bow to such brilliance and allow it. So, that is your table, where all of your moves are made, and this is mine, where all of my moves are made.:P

Quote:

5. Give your quarters to the poor beggar so he can buy some food - God will be so moved by your generous sacrifice that he'll rescue you from the Devil and hell

The Devil starts laughing uproarously at your generous gesture. "Fool!", he says, "has it escaped your notice that this is Hell? Did you not know that everyone here was sent here to be punished, even the beggars? Indeed, your beggar friend was a greedy miser on earth who never shared with anyone, and his punishment here was to experience the other end. Did you expect God, who knows your motives, to be moved by this false mercy?" :'(

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Citizen on Jul 25^th, 2006, 10:00am

Icarus - thanks for the warm welcome to Wu Riddles. Nothing like having your ideas smashed by a moderator. :-[

j/k

Actually, I should have seen the inherent contradiction between my first and third solutions. On the other hand, I don't think the rules said anything about whether you could lift coins from the table and place them back down again. Would be a possible response after the devil's counter to my first solution - though that would lead to a game of infinite length (on the other hand, you'd be helping out a lot of poor souls to whom the devil might otherwise turn his attention 8)).

What I don't like about the symmetry solution (which seems most popular) is that, like my scenarios, it is just that: a scenario. Why would the devil play with a symmetrical table? Furthermore, assuming he did, why would he then let you start first? He's gotta know some math, afterall.

I'm looking for some sort of infallible response to this problem - just not sure what it is yet.

BTW - Just to introduce myself. I'm Tom from Canada. I've lurked in this forum for about a week and I gotta say, I like it.

edited for typos

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Icarus on Jul 25^th, 2006, 12:31pm

There is a very common theme in fantasy stories about someone who is offered something they very much want, only to discover (or at least the reader discovers) that the one offering twists the reward so that while it technically matches the desired thing, it actually is just opposite the person's real desires.

For example, two stories by Orson Scott Card that I remember: In the first, a woman at the end of an ideal day runs into a peddler who brings up fears as to what the future holds, and then offers to let her have ideal days like this one for the rest of her life. She agrees, and ends up reliving that same day over and over in her mind, regardless of what actually is happening around her. The result destroys the lives of her family, whose wife and mother no longer has any attachment to their real lives.

The second involved a woman who is granted 3 wishes by a magical dragon. She wishes that her farm would alway provide enough to feed her family. So the dragon flies off and devours her family, then returns and informs her that now no matter how much her farm produces, it will always be enough to feed her family.

This riddle is a similar set-up. No matter how you try to limit things, this is the Father-of-Lies you are dealing with. He will twist things to your disadvantage. At the least, he is a liar, so how likely do you think it would be that he will let you free even if you did win?

Anyway, I must thank you for the fun challenge of figuring out how all your alternatives could go wrong. And, to be even handed:

on 07/25/02 at 05:03:22, Viorel Canja wrote:

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.

You lean back after placing the first coin, confident that even though you can't spell "symmetrical", you at least know what it means, and how to apply it here for certain victory. You close your eyes for a moment to savor the upcoming victory only to hear a crunching sound. You open your eyes to see a quarter-sized hole in one edge of the table. The evil one smiles and says "these matches always make me a little hungry". Then he places his coin exactly opposite the hole. :o

on 08/01/02 at 12:23:36, Salem wrote:

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 :)

After careful work, you succeed in balancing a coin on the top of the table edge. You were a little surprised that the Prince of Darkness did not object when you turned the table on its side, but he just set back and smiled. (Oh, how you hate that smile!) Relieved that you have pulled it off, you lean back and .... notice the table foot sticking straight up, with just enough room to balance a second coin. His grin deepens... :-X

on 10/04/04 at 05:51:26, Squigs wrote:

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.

"Ah, a sneaky one!", says the Deceiver, "Indeed, I cannot move if you don't. But then, you cannot move without releasing me from this condition, and letting the game go on to its inevitable conclusion. Which means that you are first unable to move on your turn!" :-/

on 07/24/05 at 18:17:15, platodog wrote:

Put the quarter on the table (or place the quarter on the ground), then flip the table upside down so the quarter is sandwiched between the ground and the table surface. Since the table is upside down no more quarters can be placed on its surface.

The Father of Lies growls "I thought you were educated, yet you don't understand simple instructions. The coins, I said, are to rest on the table surface, not the table surface to rest on the coin! The penalty for misplay is forfeiture of the game!" >:(

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by towr on Jul 25^th, 2006, 1:02pm

I suppose, to have a chance to win, you ought to get a fair arbiter to judge the game. God would be my choice.

Anyway, the devil hasn't the power to let you into heaven, so just repent.

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by zbudde on May 27^th, 2013, 6:14am

Put your quarter on the table, then throw the table, quarter and all, into the nearest pit of fire and brimstone, or destroy it by some other means. The rules did not state that the quarter must STAY at rest on the table surface, so after placing it, destroy the table, no more next turn for the devil.

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by Iz on Jun 17^th, 2013, 3:28pm

My take:

[hide]You place a quarter on the table and tell the devil, "Ok, I'll let you know when my turn is over."

Since you have placed a quarter on the table and your turn is still going, you have won the game. The Devil cannot place a quarter until you are done with your turn, and you choose to make your turn last infinitely.[/hide]

Ask for ice cream on the way to heaven.

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by yevvi on Jun 21^st, 2014, 6:44pm

Not sure if this solution has been already posted, here my take on it:

1st move: put a coin straight at the center (centers of coin and table coincide)
After that:

Every time devil puts a coin, we put coin symmetrically opposite to that coin relative to the center of the table (center of our coin is same distance from the center of the table as center of devils coin and 3 of these centers form straight line).

This way, no matter where devil puts the coin, we know we have available space to put the coin symmetrically opposite because each time before devil's move, the configuration is always symmetrical.

If we didn't put the coin at the center on the first move this wouldn't work because for any coin devil puts that overlaps with the center of the table we can't put the symmetrically opposite coin.

Title: Re: FAUSTIAN ROUND TABLE COIN GAME
Post by wakiza33 on Sep 16^th, 2014, 8:54am

Put the quarters in the shape of a cross.