Seven puzzles

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Antichrist_Online
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Seven puzzles

Postby Antichrist_Online » Fri Mar 02, 2007 9:26 pm

I got sent these by a maths lecturer so here you all go:

1)
The names of 100 prisoners are placed in 100 wooden boxes, one name to a box, and the boxes are lined up on a table in a room. One by one, the prisoners are led into the room; each may look in at most 50 boxes, but must leave the room exactly as he found it and is permitted no further
communication with the others.
The prisoners have a chance to plot their strategy in advance, and they are going to need it, because unless every single prisoner finds his own name all will subsequently be executed.
Find a strategy for them which which has probability of success exceeding 30%.

2)
At many train stations, post offices and currier services around the world, the cost of sending a rectangular box is determined by the sum of its dimensions; that is, length plus width plus height.
Prove that you can't "cheat" by packing a box into a cheaper box.

3)
Each resident of Dot-town carries a red or blue dot on his (or her) forehead, but if he ever figures out what color it is he kills himself. Each day the residents gather; one day a stranger comes and tells them something|anything|non-trivial about the number of blue dots. Prove that eventually every resident kills himself.

4)
Suppose you have an algebraic expression involving variables, addition, multiplication, and parentheses. You repeatedly attempt to expand it using the distributive law. How do you know that the expression doesn't continue to expand forever? "Non-trivial" means here that there is some number of blue dots for which the statement would not have been true. Thus we have a frighteningly general version of classical problems involving knowledge about knowledge.

5)
Jan and Maria have fallen in love (via the internet) and Jan wishes to mail her a ring. Unfortunately, they live in the country of Kleptopia where anything sent through the mail will be stolen unless it is enclosed in a padlocked box. Jan and Maria each have plenty of padlocks, but none to which the other has a key. How can Jan get the ring safely into Maria's hands? (Two solutions, but one is much more elegant and worth more points).

6)
As a result of temporary magical powers, you have made it to the Wimbledon finals and are playing Roger Federer for all the marbles. However, your powers cannot last the whole match. What score do you want it to be when they disappear, to maximize your chances of hanging on for a win?

7)
A logician is again visiting the South Seas, and is as before at a fork, wanting to know which of two roads leads to the village. This time, present are three willing natives, one each from a tribe of invariable truth-tellers, a tribe of invariable liars, and a tribe of random answerers. Of course the logician doesn't know which native is from which tribe. Moreover, he is permitted to ask only two yes-or-no questions, each question being directed to just one native. Can he get the information he needs?
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Antichrist_Online
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Postby Antichrist_Online » Wed Mar 14, 2007 7:40 pm

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joo
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Postby joo » Thu Dec 11, 2008 2:24 am

5)
  1. Jan sends Maria a padlocked box containing the ring, and keeps the key.
  2. Maria puts on another padlock beside Jan's and sends the box back.
  3. Jan removes his padlock and sends the box locked with Jan's padlock, containing the ring.
  4. Maria removes her lock and obtains the ring.

Do I win a prize? :roll:
Last edited by joo on Thu Dec 11, 2008 2:52 am, edited 1 time in total.
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Dudel
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Postby Dudel » Thu Dec 11, 2008 2:49 am

1) Open up a box and the two on eithor side of it. On each one you know who is inside so you leave a not that says something like

<----Name

----->Name
That will quickly get everything and has a greater then 30% change of success. If you open up 50. because you open them and know who's where etc.

Question: Do you take your own box?

Eithor way my brain says this will work.
Last edited by Dudel on Thu Dec 11, 2008 2:55 am, edited 2 times in total.
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joo
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Postby joo » Thu Dec 11, 2008 2:52 am

Dudel wrote:1) Open up a box and the two on eithor side of it. On each one you know who is inside so you leave a not that says something like

<Name>Name

That will quickly get everything and has a greater then 30% change of success. If you open up 50. because you open them and know who's where etc.

Question: Do you take your own box?

Eithor way my brain says this will work.

I don't think you're allowed to leave notes - that would be communication.
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Dudel
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Postby Dudel » Thu Dec 11, 2008 2:57 am

Okay its a form of cominication... give me a minute. I just craped that out. Never could play by the rules and always argued with teachers about problems not being plausable.

My answers were always okay but you broke the rules

Damn me and thinking outside the box... or is that smashing the box with a hammer?

Edit: I got a new idea... Kill the guard. Take his keys and just leave. You morons there are 100 of you in a room and none of you knows how to throw a punch?
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Rebma
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Postby Rebma » Thu Dec 11, 2008 7:01 am

3) This is simple. Say you have ten people in town. Dude comes and says there are 7 blue dots.

Well everyone is going to count those around them right? So I count, and I see 7 dots, I know right away I'm red. Death to me. If I only count 6, I know I'm blue. Death to me.

EDIT_ he posted more in number 4 lol. So instead of the number... he says 7 people with the blue dots have brown hair. I don't-I know im red. I do-I know I'm blue.

EDIT TO THE EDIT YO - Nevermind. I misunderstood.. and now get it. But don't feel like typing it out.
kronos wrote:like a nice trim is totally fine. short, neat. I don't want to be fighting through the forests of fangorn and expecting treebeard to come and show me the way in
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Tiamo
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Postby Tiamo » Thu Dec 11, 2008 7:51 am

6) is easy, because there are only 2 situations to consider: two sets up, 6-0 in the tiebreak of the third set.
The other would have been 2 sets up, 5-0, 40-0 up, but it is MUCH harder to win a game vs. Federer than to win a point!
I think ...
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Tiamo
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Postby Tiamo » Thu Dec 11, 2008 8:06 am

5) Jan sends the ring in a padlocked box. Maria destroys the box to retrieve the ring. :P
I think ...
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Rebma
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Postby Rebma » Thu Dec 11, 2008 8:08 am

Tiamo wrote:5) Jan sends the ring in a padlocked box. Maria destroys the box to retrieve the ring. :P


Tiamo ftw! (for the win)....(or f*** the world)lol.
kronos wrote:like a nice trim is totally fine. short, neat. I don't want to be fighting through the forests of fangorn and expecting treebeard to come and show me the way in
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joo
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Postby joo » Sat Dec 13, 2008 1:07 am

Tiamo wrote:5) Jan sends the ring in a padlocked box. Maria destroys the box to retrieve the ring. :P

I agree, much more elegant.

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