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Circle's slightly harder challenge

  1. Dec 23, 2006 #1
    Alright, well seeing as my last challenge lasted a whole 30 minutes before Jimmy Snyder solved all my brain teasers, I dug deep to find more challenging ones. :rolleyes: :smile:

    Please use and and the closing tag [/ color] to make your text unreadable to those who don't want to see the solution, thanks.

    If you recognize any of these puzzles from old threads please either ignore them or link to them and I may remove those puzzles from this thread to ensure that these are all new. Thanks.

    Good luck!!

    1.Three distinct points with integer coordinates lie in the plane on a circle of radius r > 0. Show that two of these points are separated by a distance of at least r^(1/3).

    2.What chemical compound is represented by the following: HIJKLMNO?

    3. On his way to work each day, a man living on the fifteenth floor of an apartment building takes the elevator to the first floor from the fifteenth floor. On his return, he is forced to take it to the seventh floor and walk the remaining eight floors to his apartment. Why?

    4.Three criminals just robbed a bank and go back to their hideout. They put the money behind a high tech security door. There are 3 locks on the door, each activated/deactivated by a button next to it. All locks are originally deactivated, and once a lock is activated it is impossible to tell whether it is activated or not. The three criminals want to work out a system so that any two of them can access the money but a single criminal cannot. The 2 criminals accessing the money must be assured that all locks are deactivated, otherwise an alarm will sound, and built-in lasers will shoot them. Also, each criminal may only give information about which locks he toggled to one other criminal. Figure out how their system will work.

    5.Punctuate the following so it makes sense: Alice while Matthew had had had had had had had had had had had a better effect on the teacher.













     
    Last edited: Dec 23, 2006
  2. jcsd
  3. Dec 23, 2006 #2
    2. H2O

    lol...
     
    Last edited: Dec 23, 2006
  4. Dec 23, 2006 #3
    edit: Yup, :tongue:
     
    Last edited: Dec 23, 2006
  5. Dec 23, 2006 #4
    3. A probable answer would be that the man has his office also on the first floor of the building, and the office extends from floors 1 to 7 (say) and the elevator is privately owned by the office for the employees . Note that the first statement of the question doesn't say from which floor the elevator is boarded, only the destination (1st floor). So the man may well have his residence on the 15th floor, walk upto the 7th floor and then take the elevator to the 1st floor.
    Who would construct such a building anyway ?

    5. Can we delete spaces, so that we may combine all but the last two hads into a very big surname for Matthew

    :biggrin:
     
  6. Dec 23, 2006 #5
    Lol, nice try on both :biggrin:.

    Highlight to see :


    I am afraid that you are not quite right on either, but your thinking on Q#3 is on the right path, it is something quirky like that.


    For Q#5, no you can't get rid of spaces, nice try :biggrin:
     
  7. Dec 23, 2006 #6
    Well, I only said that my answer to question 3 is only "possible", don't you agree. Maybe you could add a few more details so that you may contradict this case ?

    Cheers


    Arun
     
  8. Dec 23, 2006 #7
    True enough I suppose, I shouldn't have said you were incorrect. There is another answer though. heh. I'll post it in a few days.
     
    Last edited: Dec 24, 2006
  9. Dec 23, 2006 #8
    I also changed it so that he takes it from the 15th floor in the morning.
     
  10. Dec 23, 2006 #9
    He is too short to reach the buttons above the 7.


    Alice, while Matthew had had "had had" had had "had". "Had had" had had a better effect on the teacher.

    I showed this to my wife and she gave a similar answer to this except that I while she had had had had had had had had had had had had had had had had had had had had had had been in the answer I posted.
     
  11. Dec 23, 2006 #10
    the first problem is hard
     
  12. Dec 23, 2006 #11
    For Q#3, that is the "original answer" good job.

    Correct on Q#5 as well, that is the answer (pretty close to "original" answer but its right anyway)

    The "original answer" is::

    Alice, while Matthew had had "had," had had "had had." "Had had" had had a better effect on the teacher.
     
    Last edited: Dec 23, 2006
  13. Dec 24, 2006 #12

    daniel_i_l

    User Avatar
    Gold Member

    #4:
    the first two criminals activate or deactivate the first lock. then the first and third ones together either activte or deactivate the second one. then the second and third one either activate or deactivate the third lock. this way any given criminal knoes the position of only two locks.
    am i missing something?
     
  14. Dec 24, 2006 #13
    dontdisturbmycircles,
    Given the difference between my answer and the original answer, you should be able to punctuate the sentence at the bottom of my post.

    I showed this to my wife and she gave a similar answer to this except that I while she had had had had had had had had had had had had had had had had had had had had had had been in the answer I posted.
     
  15. Dec 24, 2006 #14
    "I showed this to my wife and she gave a similar answer to this except that I while she had had had had had had had had had had had had had had had had had had had had had had been in the answer I posted." - Jimmy Snyder

    I can't see any other way to correctly punctuate "except that I while she"

    :uhh:
     
    Last edited: Dec 24, 2006
  16. Dec 26, 2006 #15
    I showed this to my wife and she gave a similar answer to this except that I, while she had had 'had had "had" had had "had had"', had had 'had had "had had" had had "had"'. 'Had had "had"' had been in the answer I posted.

    By iterating in this fashion, any number of had's can be strung together and meaningfully punctuated.
     
  17. Dec 27, 2006 #16
    Yea definitely, it still takes some effort to do it though (although it is not a very creative process) I tried to select a few puzzles for different types of people. :smile: I already knew that any number of had's can be strung together in that fashion, it's not the kind of puzzle I enjoy doing but some may.

    Although of course you still realize that your answer to your question is not grammatically correct.
     
  18. Dec 27, 2006 #17
    It seems ok to me. What's the problem?
     
  19. Jan 2, 2007 #18
    This one's still bugging me. I've asked a few people I know, and so far everyone's stumped.

    Just so I'm clear, though, can the question can be re-written as the following?

    Three distinct points A, B, and C lie in the x,y plane, and each have exact integer coordinates. Each of the points also lies on a circle of radius R. Prove that at least one pair of these points is separated by a distance greater than or equal to the cube root of R.

    Any hints on the method of solving it?

    DaveE
     
  20. Jan 2, 2007 #19
    Had to ask a friend how to do this as I sorta forget how he showed me, use the points to form a triangle / circumcircle. If you need more help I have more hints. And yes I believe that your wording is fine.

    What is the equation for the Circumradius if the sides are a,b,c, and D = largest distance between two points, and A = area?
     
    Last edited: Jan 2, 2007
  21. Jan 5, 2007 #20
    Alright. Uncle!

    After trying this for a couple days, I asked a few of my more geeky friends, but nobody could figure it out. So I sent the problem to a math whiz at MIT, a college math professor, and someone who majored in math a long time ago. Nothing.

    So far, I think the best working theory someone had was that there were only a few quasi-rational coordinates on the unit circle. Not that the coordinates are actually *rational* per se, but that the radius could be adjusted to a particular value that would *make* them be rational. For example coordinates sqrt(3)/2, 1/2, which, while irrational, can be made to *be* rational with a radius that's a multiple of sqrt(2).

    Anyway, the assumption was that there's a fixed number of possible exact-integer coordinates on any given circle. And knowing that, you can calculate the circumradius of the points and compare it to the distance of the furthest points, and get an answer.

    But otherwise, I dunno. The sticky wicket in all cases is somehow establishing equasions wherein the coordinates of the three points are necessarily integers.

    Could you post (or PM me) the proof?

    DaveE
     
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