Cricket Betting Site Betway
Page 56 of 58 FirstFirst ... 6465455565758 LastLast
Results 826 to 840 of 862
Like Tree110Likes

Thread: The Maths Thread

  1. #826
    International Coach ankitj's Avatar
    Join Date
    Jul 2010
    Location
    India
    Posts
    12,360
    Quote Originally Posted by vcs View Post
    1.) There is a string consisting of 2m black beads and 2n white beads, arranged in some random order. Prove that it is possible to make at most two cuts to the string such that two people can be given m black beads and n white beads.
    IIRC there is some theorem that states that for a continuous curve F(x) bounded between 2 points say X1 and X2 on x-axis, you can always find a segment of the curve with points x1 and x2 such that (x1, F(x1)] and [x2,F(x2)] can be connected with a line that has same slope as the slope line connecting [X1,F(X1)] and [X2,F(X2)] for any given length (x2-x1). Alternatively, if a car has average speed of v during a journey of duration, T you can always find a period t1 to t2 where average speed was exactly v, for any duration t2-t1<T.

    Same principal can apply here. Define a function F(k) in period 0 to 2m+2n such that F(k) is difference of number of black beads and white beads in first k beads on the string for integer k. The overall slope of line connecting end points of this curve is (2m-2n)/(2m+2n) = (m-n)/(m+n). For non-integer k, F(k) will fall on a straight line connecting points at 2 nearest integers.

    For this problem you have to find 2 points k1 and k2 that have exactly m black and n white beads between them i.e. k2-k1 = m+n and F(k2) - F(k1) = m-n. Line connecting end points of this segment will have a slope of exactly (m-n)/(m+n). By that theorem above it is always possible to find such points.

    This is obviously dirty version of the proof because I haven't named the theorem (probably it doesn't have a name, it's a common sense conclusion) and I have not proven how this will always be possible at integer k1 and k2.
    Last edited by ankitj; 20-03-2020 at 09:19 AM.
    vcs likes this.
    Allow me time to fix typos before liking my post, zorax

  2. #827
    rather mad Norwegian Magrat Garlick's Avatar
    Defend Your Castle Champion! Monkey Diving Champion!
    Join Date
    Aug 2003
    Location
    East Holland
    Posts
    27,021
    Quote Originally Posted by ankitj View Post
    IIRC there is some theorem that states that for a continuous curve F(x) bounded between 2 points say X1 and X2 on x-axis, you can always find a segment of the curve with points x1 and x2 such that (x1, F(x1)] and [x2,F(x2)] can be connected with a line that has same slope as the slope line connecting [X1,F(X1)] and [X2,F(X2)] for any given length (x2-x1). Alternatively, if a car has average speed of v during a journey of duration, T you can always find a period t1 to t2 where average speed was exactly v, for any duration t2-t1<T.
    isnt that just a corollary of the intermediate value theorem?
    vcs and ankitj like this.
    A follower of the schools of Machiavelli, Bentham, Locke, Hobbes, Sutcliffe, Bradman, Lindwall, Miller, Hassett, Benaud and Thunberg

    I had a glimpse of Eureka
    Chewing me up, chewing me up and spitting me out

    -jected, I was rejected
    I wasn't good enough to be elected
    Lava, it was like lava
    Moving through my body like fire

  3. #828
    vcs
    vcs is online now
    Cricket Web: All-Time Legend vcs's Avatar
    Join Date
    Apr 2009
    Location
    India
    Posts
    24,663
    Correct. Good work, you guys. A more general proof in this video.

    https://youtu.be/yuVqxCSsE7c
    Quote Originally Posted by benchmark00 View Post
    Chix love a man with a checkered posting history.

  4. #829
    Global Moderator Spark's Avatar
    Join Date
    Feb 2010
    Location
    A Blood Rainbow
    Posts
    56,221
    that theorem ankit mentioned is the mean value theorem
    Magrat Garlick and ankitj like this.
    citoyens, vouliez-vous une révolution sans révolution?

    those nights were on fire
    we couldn't get higher
    we didn't know that we had it all
    but no one warns you before the fall


  5. #830
    International Coach ankitj's Avatar
    Join Date
    Jul 2010
    Location
    India
    Posts
    12,360
    It should be possible to solve that problem using induction too. Not able to do it myself.

  6. #831
    vcs
    vcs is online now
    Cricket Web: All-Time Legend vcs's Avatar
    Join Date
    Apr 2009
    Location
    India
    Posts
    24,663
    I used a similar idea but not exactly induction.

    I defined a function F(k) = (number of black beads seen in the last m+n beads) - m, where k varies between m+n to 2m+2n.

    If F(m+n) == 0, we are done, since we can cut the necklace at the midpoint.

    If not, let us assume that F(m+n) > 0. In which case, F(2m+2n) must necessarily be < 0, since the second half of the necklace must have less than m black beads. Now we have a function whose value goes from positive to negative, and at every step the window shifts by exactly one bead. This means that there must exist a point p somewhere between m+n and 2m+2n, where F(p) = 0. We can guarantee the existence of such a point because, at every intermediate point, we drop one bead from the window at the left, and add one bead to the window at the right, as we keep moving the window from m+n to 2m+2n.

    A similar argument can be applied to the case F(m+n) < 0.
    Last edited by vcs; 21-03-2020 at 09:01 AM.

  7. #832
    International Coach ankitj's Avatar
    Join Date
    Jul 2010
    Location
    India
    Posts
    12,360
    There is a channel called MindYourDecisions on Youtube that posts Math problems. They range from fairly simple to very tricky. Ones on geometry are generally quite good. Take a look at this one for example: https://www.youtube.com/watch?v=dF67AJH9mjM
    vcs likes this.

  8. #833
    vcs
    vcs is online now
    Cricket Web: All-Time Legend vcs's Avatar
    Join Date
    Apr 2009
    Location
    India
    Posts
    24,663
    Quote Originally Posted by ankitj View Post
    There is a channel called MindYourDecisions on Youtube that posts Math problems. They range from fairly simple to very tricky. Ones on geometry are generally quite good. Take a look at this one for example: https://www.youtube.com/watch?v=dF67AJH9mjM
    Ha, that is excellent. I tried solving it but couldn't come up with the answer so looked at the solution.

    This one is tangentially connected in terms of the idea and approach.

    https://www.youtube.com/watch?v=wWXsajE-L_o

    http://jwilson.coe.uga.edu/EMAT6680F.../AS06/WU06.htm
    ankitj likes this.

  9. #834
    vcs
    vcs is online now
    Cricket Web: All-Time Legend vcs's Avatar
    Join Date
    Apr 2009
    Location
    India
    Posts
    24,663
    From that link :

    This is a special collection of problems that were given to select applicants during oral entrance exams to the math department of Moscow State University. These problems were designed to prevent Jews and other undesirables from getting a passing grade. Among problems that were used by the department to blackball unwanted candidate students, these problems are distinguished by having a simple solution that is difficult to find. Using problems with a simple solution protected the administration from extra complaints and appeals. This collection therefore has mathematical as well as historical value.
    WTF. How can otherwise intelligent people be such morons.

  10. #835
    Global Moderator Spark's Avatar
    Join Date
    Feb 2010
    Location
    A Blood Rainbow
    Posts
    56,221
    Quote Originally Posted by vcs View Post
    From that link :



    WTF. How can otherwise intelligent people be such morons.
    Serious 1880s areas here.

  11. #836
    Global Moderator Prince EWS's Avatar
    Join Date
    Jul 2003
    Location
    Cribbertopia
    Posts
    58,138
    Not gonna lie, when I saw this thread was bumped I thought it was gonna be Athlai trolling me about Terence Tao.
    Chewie likes this.
    Rejecting 'analysis by checklist' and 'skill absolutism' since Dec '09
    'Stats' is not a synonym for 'Career Test Averages'


    Quote Originally Posted by Jeffrey Tucker
    Someone asked me the other day if I believe in conspiracies. Well, sure. Here's one. It is called the political system. It is nothing if not a giant conspiracy to rob, trick and subjugate the population.
    Before replying to TJB, always remember:
    Quote Originally Posted by TheJediBrah View Post
    Next week I'll probably be arguing the opposite

    Code:
    Pixie Caramels won by an Innings and 258 runs.

  12. #837
    International Coach ankitj's Avatar
    Join Date
    Jul 2010
    Location
    India
    Posts
    12,360
    I googled cosine law and then wrote a complicated systems of equations. Got nowhere. Then saw the solution and was quite impressed by how elegant it was.
    vcs likes this.

  13. #838
    vcs
    vcs is online now
    Cricket Web: All-Time Legend vcs's Avatar
    Join Date
    Apr 2009
    Location
    India
    Posts
    24,663
    Slow day at work so tried another one out. This was fairly easy.

    https://www.youtube.com/watch?v=OuJQaxZvlYs
    SillyCowCorner1 likes this.

  14. #839
    U19 Cricketer
    Join Date
    Mar 2019
    Location
    india
    Posts
    428
    Quote Originally Posted by vcs View Post
    Slow day at work so tried another one out. This was fairly easy.

    https://www.youtube.com/watch?v=OuJQaxZvlYs
    I was quite impressed with myself for solving it under 5 minutes till I saw that some 5th graders did it under 1 minute.
    SillyCowCorner1 and vcs like this.

  15. #840
    International Coach SillyCowCorner1's Avatar
    Join Date
    Mar 2017
    Location
    On Earth
    Posts
    11,346
    Quote Originally Posted by vcs View Post
    Slow day at work so tried another one out. This was fairly easy.

    https://www.youtube.com/watch?v=OuJQaxZvlYs
    Thanks for sharing.

    Quote Originally Posted by Malcolm View Post
    I was quite impressed with myself for solving it under 5 minutes till I saw that some 5th graders did it under 1 minute.
    It was worth the 9 minutes. I had a good refresher in geometry and arithmetic. We take these things for granted when we are young.
    Symmetry is beauty, beauty is truth. Breaking symmetry means breaking truth.



Thread Information

Users Browsing this Thread

There are currently 1 users browsing this thread. (0 members and 1 guests)

Similar Threads

  1. Tech Geeks Thread
    By SupaFreak2005 in forum Off Topic
    Replies: 233
    Last Post: 03-11-2012, 03:15 PM
  2. *Nikhil Hearing* Details Thread
    By lord_of_darkness in forum Cricket Web XI
    Replies: 329
    Last Post: 17-12-2004, 03:21 PM
  3. **Official** South Africa in India Thread
    By Neil Pickup in forum Cricket Chat
    Replies: 453
    Last Post: 08-12-2004, 07:26 AM
  4. *Official* India in Australia Thread
    By Craig in forum Cricket Chat
    Replies: 2652
    Last Post: 08-01-2004, 07:00 PM
  5. *Official* West Indies in Zimbabwe Thread
    By Mr Mxyzptlk in forum Cricket Chat
    Replies: 554
    Last Post: 01-12-2003, 03:11 AM

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •