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It's Math-Puzzle Time

Apothe0sisApothe0sis Registered User regular
edited April 2008 in Debate and/or Discourse
It's that time of year again (Ok, so I made that part up) which means that it's time to play the "let's post amusing mathematical puzzles, oddities or anomalies game". Why? Because we can, and who doesn't love maths?

And to begin, here is the first mathematical puzzle of the thread.

---

There are three train stations, A, B and C, B sitting equidistant between A and C. Next to stations A and C there are Klimpy's chain restaurants. Given that stations A and C are in commercial districts of the city and B sits in the only nearby residential district the restaurants at A and C are almost entirely reliant upon business from people who have caught the train from B.

Customers, knowing that the distance between their station and the stations either side do not plan which Klimpy's they will eat at, instead they arrive at the train station and simply catch the next train, it's it's going to station A, they eat at that restaurant, if it goes to B, they eat at that restaurant.

However, Klimpy's central management discovers that the Klimpy's restaurant at station A does about three times as much business as the restaurant at station C.

Assume that the same number of trains travel to each station each day, that the gap between any particular two trains traveling to the same station is 20 minutes and that the times at which the customers show up to the train station is random/evenly distributed throughout the day.

Explain.

Apothe0sis on
What I see sees me.
SODOMISE INTOLERANCE
Tide goes in. Tide goes out.
«13456

Posts

  • enlightenedbumenlightenedbum Registered User regular
    edited April 2008
    Doc wrote: »
    The solution to this is available on the 'net, so no cheating!

    A prison warden decides to play a game with the inmates. If they can win, they all go free. If they lose, they all get stuck in there for life. The prisoners can all talk to each other to come up with a strategy before the game starts, but not after. The game is as follows: a prisoner will be randomly selected daily. He will enter a room where he sees two switches. He must toggle exactly one of the two switches and then exit the room. To "win," a prisoner must go to the warden and say "All prisoners have visited the room." If he is correct, they win. If not, they lose.

    Again, after the game starts, the prisoners cannot communicate with each other. What's a good strategy for them to come up with before the game starts?

    Can prisoners be selected more than once?

    Lose: to suffer defeat, to misplace (Ex: "I hope I don't lose the match." "Did you lose your phone again?")
    Loose: about to slip, to release (Ex: "That knot is loose." "Loose arrows.")
  • DocDoc Registered User, ClubPA regular
    edited April 2008
    Linden wrote: »
    Are the switches known to be stable across days? Do they start in known positions? Can the number of days passed be counted by the prisoners?

    Let's say they start both down/off. The only people that mess with the switches are the prisoners, and the prisoners are free to count days passed. But a prisoner can be selected multiple times, possibly even twice in a row (as it's random every day).

  • AbsurdistAbsurdist Registered User
    edited April 2008
    Doc wrote: »
    The solution to this is available on the 'net, so no cheating!

    A prison warden decides to play a game with the inmates. If they can win, they all go free. If they lose, they all get stuck in there for life. The prisoners can all talk to each other to come up with a strategy before the game starts, but not after. The game is as follows: a prisoner will be randomly selected daily. He will enter a room where he sees two switches. He must toggle exactly one of the two switches and then exit the room. To "win," a prisoner must go to the warden and say "All prisoners have visited the room." If he is correct, they win. If not, they lose.

    Again, after the game starts, the prisoners cannot communicate with each other. What's a good strategy for them to come up with before the game starts?

    I've seen this puzzle before. For it to be solvable, the prisoners must be given enough information beforehand to be able to describe the different possible positions of the switches. That is, they must be able to make a list of instructions for themselves that refers to switches as either in position A or position B. On or Off, Left or Right, Up or Down - it doesn't matter what the two states are, but the prisoners must be able to walk into the room and know whether each switch is in position A or B. Example: what if the switches are not labeled, and do not turn anything "on" or "off" that the prisoners can see? What the first prisoner walks into the room on day one only to discover that switch 1 flips Left/Right while switch 2 flips Up/Down? What if the switches are buttons that get pressed and change colors from Red to Green with each push? etc.

    This is a great puzzle though. Number of prisoners doesn't matter. Solution is easier if no prisoner can be called more than once, but it can be solved even if any prisoner can be called any random number of times. It's also easier if you know the initial state of the switches (both down/off, as Doc posted here) but it can even be done without knowing the initial state. All in all, rockin hard puzzle!

    I'll leave the solution to somebody else and post one of my favorite tricky puzzles.

    [SIGPIC][/SIGPIC]
  • DocDoc Registered User, ClubPA regular
    edited April 2008
    Absurdist wrote: »
    Doc wrote: »
    The solution to this is available on the 'net, so no cheating!

    A prison warden decides to play a game with the inmates. If they can win, they all go free. If they lose, they all get stuck in there for life. The prisoners can all talk to each other to come up with a strategy before the game starts, but not after. The game is as follows: a prisoner will be randomly selected daily. He will enter a room where he sees two switches. He must toggle exactly one of the two switches and then exit the room. To "win," a prisoner must go to the warden and say "All prisoners have visited the room." If he is correct, they win. If not, they lose.

    Again, after the game starts, the prisoners cannot communicate with each other. What's a good strategy for them to come up with before the game starts?

    I've seen this puzzle before. For it to be solvable, the prisoners must be given enough information beforehand to be able to describe the different possible positions of the switches. That is, they must be able to make a list of instructions for themselves that refers to switches as either in position A or position B. On or Off, Left or Right, Up or Down - it doesn't matter what the two states are, but the prisoners must be able to walk into the room and know whether each switch is in position A or B. Example: what if the switches are not labeled, and do not turn anything "on" or "off" that the prisoners can see? What the first prisoner walks into the room on day one only to discover that switch 1 flips Left/Right while switch 2 flips Up/Down? What if the switches are buttons that get pressed and change colors from Red to Green with each push? etc.

    I'll leave the solution to somebody else and post one of my favorite tricky puzzles.

    Right, they are physical switches with an up and a down. This can be known ahead of time.

  • CantideCantide Registered User regular
    edited April 2008
    Doc wrote: »
    The solution to this is available on the 'net, so no cheating!

    A prison warden decides to play a game with the inmates. If they can win, they all go free. If they lose, they all get stuck in there for life. The prisoners can all talk to each other to come up with a strategy before the game starts, but not after. The game is as follows: a prisoner will be randomly selected daily. He will enter a room where he sees two switches. He must toggle exactly one of the two switches and then exit the room. To "win," a prisoner must go to the warden and say "All prisoners have visited the room." If he is correct, they win. If not, they lose.

    Again, after the game starts, the prisoners cannot communicate with each other. What's a good strategy for them to come up with before the game starts?

    I think I have a solution for this, although it may not be the fastest one.
    Spoiler:

    2v0ltm9.jpg
  • TheLawinatorTheLawinator Registered User regular
    edited April 2008
    That won't work if someone is called more than once.

    My SteamID Gamertag and PSN: TheLawinator
  • CantideCantide Registered User regular
    edited April 2008
    Yes it will, because
    Spoiler:

    2v0ltm9.jpg
  • enlightenedbumenlightenedbum Registered User regular
    edited April 2008
    That won't work if someone is called more than once.

    No, it's fine because if they have already switched the important switch they switch the other one and the control dude ignores it. Of course, the expected length of time for this whole method to work is some enormous times, surely longer than the shortest sentence any of the prisoners is serving, but I think it would work.

    Lose: to suffer defeat, to misplace (Ex: "I hope I don't lose the match." "Did you lose your phone again?")
    Loose: about to slip, to release (Ex: "That knot is loose." "Loose arrows.")
  • ProtoProto Registered User regular
    edited April 2008
    That won't work if someone is called more than once.
    Spoiler:

    and her knees up on the glove compartment
    took out her barrettes and her hair spilled out like rootbeer
  • AbsurdistAbsurdist Registered User
    edited April 2008
    The solution to like every math puzzle ever is on the internet at this point, so no cheating if you want a good challenge! Some people get this right away, but it was really hard for me. Like, keep-me-up-all-night-and-then-eureka-while-showering-the-next-morning kind of hard. PM me for a hint.

    What is the next number in this sequence?

    1, 4, 7, 12, 15, 18, 21, 24, 27, ?

    [SIGPIC][/SIGPIC]
  • CorlisCorlis Registered User regular
    edited April 2008
    Actually, I think that will work, but it will take a really long time.

    But I don't mind, as long as there's a bed beneath the stars that shine,
    I'll be fine, just give me a minute, a man's got a limit, I can't get a life if my heart's not in it.
  • CantideCantide Registered User regular
    edited April 2008
    Corlis wrote: »
    Actually, I think that will work, but it will take a really long time.

    That's why I said it's probably not the fastest, although that's partly due to the relatively large example value of X. If we were talking about 10 prisoners, my method would probably take only a few months.

    And if the prisoners don't have to flip any switches, there's an easy way to cut down the time:
    Spoiler:

    EDIT: Actually, that doesn't work because it would require flipping two switches on the same visit. Never mind.

    2v0ltm9.jpg
  • cyphrcyphr Registered User
    edited April 2008
    Absurdist wrote: »
    The solution to like every math puzzle ever is on the internet at this point, so no cheating if you want a good challenge! Some people get this right away, but it was really hard for me. Like, keep-me-up-all-night-and-then-eureka-while-showering-the-next-morning kind of hard. PM me for a hint.

    What is the next number in this sequence?

    1, 4, 7, 12, 15, 18, 21, 24, 27, ?

    I love ambiguous sequences!
    Spoiler:

    steam_sig.png
  • AbsurdistAbsurdist Registered User
    edited April 2008
    cyphr wrote: »
    Absurdist wrote: »
    The solution to like every math puzzle ever is on the internet at this point, so no cheating if you want a good challenge! Some people get this right away, but it was really hard for me. Like, keep-me-up-all-night-and-then-eureka-while-showering-the-next-morning kind of hard. PM me for a hint.

    What is the next number in this sequence?

    1, 4, 7, 12, 15, 18, 21, 24, 27, ?

    I love ambiguous sequences!
    Spoiler:

    Jeez, some people.

    Ok, to clarify, what is the next number in this sequence that does not require the use of multiple rules or "if...then" statements. Since, you know, you can obviously define any sequence via an infinite number of functions if you allow the use of compound functions.

    1, 4, 7, 12, 15, 18, 21, 24, 27, ?

    [SIGPIC][/SIGPIC]
  • cyphrcyphr Registered User
    edited April 2008
    Alright alright, point taken. But now I'm also going to be up for a while thinking about it. Bastard. :-p

    steam_sig.png
  • AbsurdistAbsurdist Registered User
    edited April 2008
    Here is another tricky one!

    What's the next number in this sequence?

    10, 0, 3, 20, 16, 51, 32, 67, 74, ?

    [SIGPIC][/SIGPIC]
  • cyphrcyphr Registered User
    edited April 2008
    I'll post my own while I ponder yours.

    1, 11, 21, 1211, 111221, 312211, 13112221, ?

    steam_sig.png
  • AdrienAdrien Registered User
    edited April 2008
    Cantide wrote: »
    Doc wrote: »
    The solution to this is available on the 'net, so no cheating!

    A prison warden decides to play a game with the inmates. If they can win, they all go free. If they lose, they all get stuck in there for life. The prisoners can all talk to each other to come up with a strategy before the game starts, but not after. The game is as follows: a prisoner will be randomly selected daily. He will enter a room where he sees two switches. He must toggle exactly one of the two switches and then exit the room. To "win," a prisoner must go to the warden and say "All prisoners have visited the room." If he is correct, they win. If not, they lose.

    Again, after the game starts, the prisoners cannot communicate with each other. What's a good strategy for them to come up with before the game starts?

    I think I have a solution for this, although it may not be the fastest one.
    Spoiler:

    Technically that's not a solution, as technically the problem isn't solvable— every prisoner is not necessarily going to be called into the room after any amount of time.

    That is a "good strategy", though :P

    tmkm.jpg
  • AbsurdistAbsurdist Registered User
    edited April 2008
    cyphr wrote: »
    I'll post my own while I ponder yours.

    1, 11, 21, 1211, 111221, 312211, 13112221, ?

    A classic! This sequence was the first of it's type that I had ever come across, and I never did get it. I had to ask for the answer. Since then, of course, I look for solutions of a similar type in any stumper puzzle.
    Spoiler:

    [SIGPIC][/SIGPIC]
  • AbsurdistAbsurdist Registered User
    edited April 2008
    Adrien wrote: »
    Technically that's not a solution, as technically the problem isn't solvable— every prisoner is not necessarily going to be called into the room after any amount of time.

    That is a "good strategy", though :P

    He's got you there, Doc. You need to stipulate that the prisoners and the universe they live in are immortal.

    hehe

    [SIGPIC][/SIGPIC]
  • DocDoc Registered User, ClubPA regular
    edited April 2008
    Cantide wrote: »
    Doc wrote: »
    The solution to this is available on the 'net, so no cheating!

    A prison warden decides to play a game with the inmates. If they can win, they all go free. If they lose, they all get stuck in there for life. The prisoners can all talk to each other to come up with a strategy before the game starts, but not after. The game is as follows: a prisoner will be randomly selected daily. He will enter a room where he sees two switches. He must toggle exactly one of the two switches and then exit the room. To "win," a prisoner must go to the warden and say "All prisoners have visited the room." If he is correct, they win. If not, they lose.

    Again, after the game starts, the prisoners cannot communicate with each other. What's a good strategy for them to come up with before the game starts?

    I think I have a solution for this, although it may not be the fastest one.
    Spoiler:

    That is a solution. It does take a really long time.

  • Rufus_ShinraRufus_Shinra Registered User
    edited April 2008
    Doc wrote: »
    Cantide wrote: »
    Doc wrote: »
    The solution to this is available on the 'net, so no cheating!

    A prison warden decides to play a game with the inmates. If they can win, they all go free. If they lose, they all get stuck in there for life. The prisoners can all talk to each other to come up with a strategy before the game starts, but not after. The game is as follows: a prisoner will be randomly selected daily. He will enter a room where he sees two switches. He must toggle exactly one of the two switches and then exit the room. To "win," a prisoner must go to the warden and say "All prisoners have visited the room." If he is correct, they win. If not, they lose.

    Again, after the game starts, the prisoners cannot communicate with each other. What's a good strategy for them to come up with before the game starts?

    I think I have a solution for this, although it may not be the fastest one.
    Spoiler:

    That is a solution. It does take a really long time.
    Does that mean there's a better solution? I must know.

  • jothkijothki Registered User regular
    edited April 2008
    Adrien wrote: »
    Cantide wrote: »
    Doc wrote: »
    The solution to this is available on the 'net, so no cheating!

    A prison warden decides to play a game with the inmates. If they can win, they all go free. If they lose, they all get stuck in there for life. The prisoners can all talk to each other to come up with a strategy before the game starts, but not after. The game is as follows: a prisoner will be randomly selected daily. He will enter a room where he sees two switches. He must toggle exactly one of the two switches and then exit the room. To "win," a prisoner must go to the warden and say "All prisoners have visited the room." If he is correct, they win. If not, they lose.

    Again, after the game starts, the prisoners cannot communicate with each other. What's a good strategy for them to come up with before the game starts?

    I think I have a solution for this, although it may not be the fastest one.
    Spoiler:

    Technically that's not a solution, as technically the problem isn't solvable— every prisoner is not necessarily going to be called into the room after any amount of time.

    That is a "good strategy", though :P

    If every prisoner isn't called in at some point, it's unwinnable anyway.

  • AdrienAdrien Registered User
    edited April 2008
    The game as described is not necessarily winnable, yes.

    tmkm.jpg
  • SavantSavant Registered User regular
    edited April 2008
    Doc wrote: »
    Cantide wrote: »
    Doc wrote: »
    The solution to this is available on the 'net, so no cheating!

    A prison warden decides to play a game with the inmates. If they can win, they all go free. If they lose, they all get stuck in there for life. The prisoners can all talk to each other to come up with a strategy before the game starts, but not after. The game is as follows: a prisoner will be randomly selected daily. He will enter a room where he sees two switches. He must toggle exactly one of the two switches and then exit the room. To "win," a prisoner must go to the warden and say "All prisoners have visited the room." If he is correct, they win. If not, they lose.

    Again, after the game starts, the prisoners cannot communicate with each other. What's a good strategy for them to come up with before the game starts?

    I think I have a solution for this, although it may not be the fastest one.
    Spoiler:

    That is a solution. It does take a really long time.
    Does that mean there's a better solution? I must know.

    I've come up with a solution, but it seems a little slow (but not as slow as the previous one). This assumes that they HAVE to flick a switch every day.
    Spoiler:

  • areaarea Registered User regular
    edited April 2008
    The traditional prisoner puzzle only has one switch, and a lot of work has been done on it:

    http://www.segerman.org/prisoners.pdf
    http://www.ocf.berkeley.edu/~wwu/papers/100prisonersLightBulb.pdf

    Edit: Obviously, these links can be considered massive spoilers for the prisoner problem. You've been warned.

    Most methods can be extended to two lightbulbs (switches) in some respect; the first paper has a small section on two switches, as well as a bunch of other variants.

    Interestingly, to my knowledge, no-one has been able to prove what the best solution is.

    minisig.jpg
  • AbsurdistAbsurdist Registered User
    edited April 2008
    Nobody can solve my puzzlers. Do I win something?

    10, 0, 3, 20, 16, 51, 32, 67, 74, ?

    1, 4, 7, 12, 15, 18, 21, 24, 27, ?

    [SIGPIC][/SIGPIC]
  • cyphrcyphr Registered User
    edited April 2008
    Spoiler:

    steam_sig.png
  • Apothe0sisApothe0sis Registered User regular
    edited April 2008
    This is most assuredly not my series, and math geeks will probably see why this is funny.

    2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837 ... ?

    What I see sees me.
    SODOMISE INTOLERANCE
    Tide goes in. Tide goes out.
  • AbsurdistAbsurdist Registered User
    edited April 2008
    Apothe0sis wrote: »
    This is most assuredly not my series, and math geeks will probably see why this is funny.

    2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837 ... ?

    Usually when they jump right from small numbers into enormous numbers it means that the series is a list of numbers with an unusual property.
    Spoiler:

    EDIT:
    Spoiler:

    [SIGPIC][/SIGPIC]
  • Apothe0sisApothe0sis Registered User regular
    edited April 2008
    Absurdist wrote: »
    Apothe0sis wrote: »
    This is most assuredly not my series, and math geeks will probably see why this is funny.

    2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837 ... ?

    Usually when they jump right from small numbers into enormous numbers it means that the series is a list of numbers with an unusual property.
    Spoiler:
    You probably have to recognise the series. It's concievably solvable, but you'd have to have an Erdos number of 0 or something to do so.

    REAL SPOILER
    Spoiler:

    What I see sees me.
    SODOMISE INTOLERANCE
    Tide goes in. Tide goes out.
  • AbsurdistAbsurdist Registered User
    edited April 2008
    Apothe0sis wrote: »
    Absurdist wrote: »
    Apothe0sis wrote: »
    This is most assuredly not my series, and math geeks will probably see why this is funny.

    2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837 ... ?

    Usually when they jump right from small numbers into enormous numbers it means that the series is a list of numbers with an unusual property.
    Spoiler:
    You probably have to recognise the series. It's concievably solvable, but you'd have to have an Erdos number of 0 or something to do so.

    REAL SPOILER
    Spoiler:

    I'm just happy that I know what an Erdos number is. :)

    [SIGPIC][/SIGPIC]
  • SuckafishSuckafish Registered User regular
    edited April 2008
    Absurdist wrote: »
    Nobody can solve my puzzlers. Do I win something?

    10, 0, 3, 20, 16, 51, 32, 67, 74, ?
    Spoiler:

    Absurdist wrote: »

    1, 4, 7, 12, 15, 18, 21, 24, 27, ?
    Spoiler:

  • SuckafishSuckafish Registered User regular
    edited April 2008
    Here is a fairly easy one.

    If my three were a four, and my one where a three, what I am would be nine less than half what I'd be.
    I am a three digit, whole number. What am I?

  • SavantSavant Registered User regular
    edited April 2008
    Suckafish wrote: »
    Here is a fairly easy one.

    If my three were a four, and my one where a three, what I am would be nine less than half what I'd be.
    I am a three digit, whole number. What am I?
    Spoiler:

  • ScooterScooter Registered User regular
    edited April 2008
    It has multiple answers, doesn't it?

  • SithDrummerSithDrummer Registered User regular
    edited April 2008
    Suckafish wrote: »
    Absurdist wrote: »
    Nobody can solve my puzzlers. Do I win something?

    10, 0, 3, 20, 16, 51, 32, 67, 74, ?
    Spoiler:

    Absurdist wrote: »

    1, 4, 7, 12, 15, 18, 21, 24, 27, ?
    Spoiler:
    And either way, it's not a math puzzle.

    It's an easy game to hate
  • FunkyWaltDoggFunkyWaltDogg Registered User regular
    edited April 2008
    Absurdist wrote: »
    Apothe0sis wrote: »
    Absurdist wrote: »
    Apothe0sis wrote: »
    This is most assuredly not my series, and math geeks will probably see why this is funny.

    2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837 ... ?

    Usually when they jump right from small numbers into enormous numbers it means that the series is a list of numbers with an unusual property.
    Spoiler:
    You probably have to recognise the series. It's concievably solvable, but you'd have to have an Erdos number of 0 or something to do so.

    REAL SPOILER
    Spoiler:

    I'm just happy that I know what an Erdos number is. :)

    Speaking of, wouldn't anyone with an Erdos number of 0 be dead, and thus unable to solve the problem?

    Burnage wrote:
    FWD is very good at this game.
  • SithDrummerSithDrummer Registered User regular
    edited April 2008
    Scooter wrote: »
    It has multiple answers, doesn't it?
    Not for 3 digits, as far as I can tell.

    More complex explanation:
    Spoiler:


    Also, Paul Erdös is the only person with an Erdös number of 0, and he is dead. There are a lot of people still alive with an Erdös number of 1, however. And Natalie Portman has an Erdös number of 5.

    It's an easy game to hate
  • ScooterScooter Registered User regular
    edited April 2008
    Spoiler:

«13456
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