A Few More Puzzles
Jul. 8th, 2011 10:36 am![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
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queensthiefSo, since some people seem to like the logical puzzles I previously posted, then here are a few more. Two of these are actually straight out of my assignments. ^__^
I. Sophos was lounging in his apartments, ill with a lovebug bite, when he suddenly started to crave some cake. He sent for someone to bring him a fresh slice from the kitchens. When a knock came from the doorways, there were two attendants who wanted to bring him some cake: one with a strawberry flavour, the other with chocolate.
There were only two attendants currently on duty: Ion from Attolia, whom he trusts unconditionally, and Ion Nomenus, whom he’d rather have behind bars for his lies (how he got out, we shall never know T__T). Since Sophos was sick, he did not want to open the doors himself, and the thick doors muffled their voices so he could not tell which one is carrying which cake.
He knew that his idol Gen could determine a solution to his predicament just by asking ONE question. What question could Sophos ask to one of them so that he’d only have to allow the Ion carrying the chocolate cake to enter?
(Hint: Since I wasn't able to solve this problem by myself, I figure I'd be mean not to give a clue. Try forming the question like this, "Does ____ carry ____?")
II. In the court of Attolia, friendship is mutual. The queen receives a strange message from Relius that warns, “If there are at least two people during dinner that has the same number of friends, then there will be poison in your husband’s wine tonight.”
Eugenides complains to his wife that he has no friends in the wretched place. Should Attolia send an order to the kitchen maids not to serve wine that night?
(This one is pretty easy. But I think the actual proof is a bit tediuos.)
III. The king of Attolia is entertaining a couple of his guards with a magic trick. He has a deck of regular cards on a table, shuffling them over and over again.
Gen tells Aris and Costis, “If the top 26 cards have more red cards than the bottom 26 cards have black, then I guarantee you that I have placed three cards in a row of the same colour somewhere in this deck.”
Aris calls the king a liar, "There can't be three cards in a row."
But Costis says Eugenides is telling the truth. Which guard was right?
I. Sophos was lounging in his apartments, ill with a lovebug bite, when he suddenly started to crave some cake. He sent for someone to bring him a fresh slice from the kitchens. When a knock came from the doorways, there were two attendants who wanted to bring him some cake: one with a strawberry flavour, the other with chocolate.
There were only two attendants currently on duty: Ion from Attolia, whom he trusts unconditionally, and Ion Nomenus, whom he’d rather have behind bars for his lies (how he got out, we shall never know T__T). Since Sophos was sick, he did not want to open the doors himself, and the thick doors muffled their voices so he could not tell which one is carrying which cake.
He knew that his idol Gen could determine a solution to his predicament just by asking ONE question. What question could Sophos ask to one of them so that he’d only have to allow the Ion carrying the chocolate cake to enter?
(Hint: Since I wasn't able to solve this problem by myself, I figure I'd be mean not to give a clue. Try forming the question like this, "Does ____ carry ____?")
II. In the court of Attolia, friendship is mutual. The queen receives a strange message from Relius that warns, “If there are at least two people during dinner that has the same number of friends, then there will be poison in your husband’s wine tonight.”
Eugenides complains to his wife that he has no friends in the wretched place. Should Attolia send an order to the kitchen maids not to serve wine that night?
(This one is pretty easy. But I think the actual proof is a bit tediuos.)
III. The king of Attolia is entertaining a couple of his guards with a magic trick. He has a deck of regular cards on a table, shuffling them over and over again.
Gen tells Aris and Costis, “If the top 26 cards have more red cards than the bottom 26 cards have black, then I guarantee you that I have placed three cards in a row of the same colour somewhere in this deck.”
Aris calls the king a liar, "There can't be three cards in a row."
But Costis says Eugenides is telling the truth. Which guard was right?
no subject
Date: 7/8/11 09:55 pm (UTC)(except I'd probably say yes for #2. :s )
no subject
Date: 7/8/11 10:06 pm (UTC)Yep, that's right. I think I might need a bit more explaining to do on #2, so if nobody still understands how to do it, I'll reword the problem a bit. Let's see. =)
For #3, I highly suggest that if you have a deck of cards lying around, you try the "trick" Gen is doing.
Yay! more puzzles :)
Date: 7/8/11 11:21 pm (UTC)I seem to remember that the answer to this question involves asking a question that forces them to give the same answer. Like, "Who would the other one of you say is carrying the chocolate cake?" in which case if Ion is carrying it, he'll be truthful and answer that the other guy would lie and say Nomenus has it. But Nomenus, knowing that Ion would be truthful and say he's carrying the cake, will have to lie and say Ion would also claim Nomenus is carrying it. So if Sophos gets the answer "The other one would say 'Nomenus'," he'll know it's actually not Nomenus, because both of them had to take a lie into account in one of their answers. Meanwhile, same thing if Nomenus actually does have the cake: both of them will answer that the other one would claim Ion has it.
Hey, I talked myself into figuring it out! I have to admit, I've seen the episode of Yu-Gi-Oh where they meet the Para Dox brothers and have to answer this question.
II: No wine! No matter how many people there are at dinner (provided there are a plural number of them and Gen isn't eating by himself, of course), if friendship is mutual, then every instance of "friends" involves two people, so there will always be at least two with the same number (even if that number is zero, or they both have the same number of friendships with other people but aren't friends with each other). I'd be interested to see the proof, if you have it, because I have the intuition that this will always hold, but I can't say for sure.
III: There are equal numbers of red and black cards in a deck, so however many red cards one half of the deck has, the other part of that half must be black; the other half of the deck will have the reverse proportion. I can't seem to divide them up so that the proportions are uneven, so that, for instance, one half of the deck will have more of a color than the other half has of the opposite color.
(I started with the idea that each set of 26 was entirely red or entirely black, and then substituted cards from one to the other---but in order for the halves to always be 26, I always had to switch them evenly, and the minority colors in each half were always equal. Does it count if you include the jokers??? haha) Okay, I give up.
Re: Yay! more puzzles :)
Date: 7/8/11 11:51 pm (UTC)I will now dissect your explanation. O__O
I'll wait for more people to try Question # 2, before I'll show the proof. I think yours is basically the english version of it though, haha... this one is more mathematical with 'n' variables and such. ^__^ It's called the Pigeonhole Principle, if you're interested! But yeah, you basically have it down!
***Spoilers for Q3****
For number three, you're actually right. Kekekeke... this is one of those trick questions in logic classes, called "vacuously true". As in since the "if" statement will never happen, we have no way to disprove the "then" statement, and therefore the whole thing is true! It's a hard concept to wrap your head around... but I guess it's one of those things where most people would ask "Why?", when we should all be asking, "Why not??" Hehehe. =)
Re: Yay! more puzzles :)
Date: 7/9/11 12:00 am (UTC)I used the question, "Does Ion carry the chocolate cake?"
It turns out whoever says "yes" will be the one who carries it.
If you use Nomenus's name, then whoever says "no" will be the one carrying it.
And you know that the Ions will give an answer that is opposite to the other one, so you'll be able to figure out what the other answers just by asking one of them.
So there are many variations to the question (if you vary the flavour too), but as long as Sophos manages to pinpoint which one is carrying the chocolate, then the question works. ^__^
But I like yours. ^__^
Re: Yay! more puzzles :)
Date: 7/10/11 03:44 am (UTC)no subject
Date: 7/10/11 03:24 am (UTC)If Sophos was trying to be Gen he could sneak through a secret passage and steal both pieces of cake ;)
I imagine that this would be a bit like "Who's on First?"
"Ion has the chocolate cake!"
"No Ion has the chocolate cake!"
XD
II. I just didn't understand. Probably safer for Gen to just avoid wine?
III. I probably learned this, but I forgot it after finals ;)
But SEA has helped me with advanced logic, yay!
no subject
Date: 7/10/11 09:38 pm (UTC)II. Oh, well the problem is that Attolia needs to figure out whether at least 2 people in her court have the same number of friends. Because if the was true then according to Relius's warning, Gen would be poisoned that night. If it doesn't, she doesn't need to worry about it. But you're right, lol... if Attolia ever receives a warning like that, she'd probably just disallow wine. ^__^
But as it turns out, it is possible for some people to have the same number of friends, so she should definitely have wine not served.
III. Ah yeah, the impossible card trick. As I explained a couple of posts above, Gen's statement was true, so Costis was right. ^^
Hehe, I'm glad you had fun!
no subject
Date: 7/10/11 12:39 pm (UTC)no subject
Date: 7/10/11 09:46 pm (UTC)Well, how I solved it was by using the question, "Does Ion have the chocolate cake?"
Suppose this was our situation.
1) Ion has chocolate.
2) Nomenus has strawberry.
In this case, Ion will say yes, Nomenus will say no.
Now suppose the situation was reversed.
1) Ion has strawberry.
2) Nomenus has chocolate.
Then Ion will say no, Nomenus will say yes.
In either case, the attendant who says yes is the one who has the chocolate cake. ^__^ And Sophos can let only the attendant who says 'yes' come into his room.
So the question can be structured like that, and by checking what each one answers, we can determine which of them has the chocolate cake. Also, he only needs to ask one of the attendants, not both, since they will always have an answer that is opposite to each other. ^__^ You can also check drashizu's method above. It's pretty interesting too.
I hope that helps you!
no subject
Date: 7/12/11 11:06 am (UTC)no subject
Date: 7/12/11 01:57 pm (UTC)But it's just not as fun!