Results 1 to 2 of 2

Thread: Billiard balls and the number theory result

  1. #1
    Newbie
    Joined
    Dec 2009
    Posts
    2

    Exclamation Billiard balls and the number theory result

    Hello,

    "Given a rectangular billiard table with only corner pockets and sides of integer lengths m and n (with m and n relatively prime), a ball sent at a 45 degree angle from a corner will be pocketed in another corner after m+n-2 bounces ".

    This is the Steinhaus and Gardner result.

    Can anyone please tell were i can find (on the Internet = for free) the proof of this result.
    i could find the books published by this authors on ebay and similar websites but all i need a free version as i can not effort any of them.

    Thanks in advance,
    Pitagora's
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1
    Hello Pitagora's
    Quote Originally Posted by Pitagora's View Post
    Hello,

    "Given a rectangular billiard table with only corner pockets and sides of integer lengths m and n (with m and n relatively prime), a ball sent at a 45 degree angle from a corner will be pocketed in another corner after m+n-2 bounces ".

    This is the Steinhaus and Gardner result.

    Can anyone please tell were i can find (on the Internet = for free) the proof of this result.
    i could find the books published by this authors on ebay and similar websites but all i need a free version as i can not effort any of them.

    Thanks in advance,
    Pitagora's
    Since no-one has been able to find a proof of this on the internet, I have come up with this.

    Look at the attached diagram. It shows a grid made up of $\displaystyle m \times n$ rectangles, each one representing an image of the billiard table. The ball is projected from $\displaystyle A$ at $\displaystyle 45^o$ to the bottom edge $\displaystyle AP$. In this diagram, instead of being reflected when it hits a side of the table, the ball continues on its straight-line path. So the new path is, at each impact, the mirror-image of the actual path the real ball would take.

    This 'virtual ball' will clearly enter another 'pocket', then, as soon as it strikes two sides simultaneously - in other words, where the diagonal line in our diagram passes through the point where two of the grid-lines meet. Denote this point by $\displaystyle B$. Then, since $\displaystyle AB$ is at $\displaystyle 45^o$ to the bottom edge of the grid, the distances $\displaystyle AP$ and $\displaystyle BP$ are equal, and each is equal to a multiple of $\displaystyle m$ and a multiple of $\displaystyle n$ respectively. In other words:
    $\displaystyle AP = BP\Rightarrow pm = qn$, for some integers $\displaystyle p, q$.
    Now $\displaystyle m$ and $\displaystyle n$ are co-prime. Therefore $\displaystyle p$ is a multiple of $\displaystyle n$ and $\displaystyle q$ is a multiple of $\displaystyle m$. Clearly the smallest values that satisfy this equation are:
    $\displaystyle p=n$, and $\displaystyle q=m$

    $\displaystyle \Rightarrow AP=BP=mn$
    So there are $\displaystyle n$ rectangles placed horizontally side by side to make up the line $\displaystyle AP$, the line $\displaystyle AB$ intersecting the (vertical) right-hand edge of $\displaystyle (n-1)$ of these before entering the virtual pocket at B. Similarly AB intersects $\displaystyle (m-1)$ horizontal edges of the $\displaystyle m$ rectangles that make up the line $\displaystyle PB$.

    Each of these intersections represents an impact with an edge of the table. There are therefore $\displaystyle n-1+m-1=m+n-2$ impacts in total.

    Grandad
    Attached Thumbnails Attached Thumbnails Billiard balls and the number theory result-untitled.jpg  
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Jun 5th 2011, 04:55 PM
  2. "Never-returning billiard balls"
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: Oct 21st 2010, 09:24 AM
  3. Deduce this result from the Prime Number Theorem
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: Oct 8th 2010, 03:44 AM
  4. a basket contains 5 red balls, 3 blue balls, 1 green balls
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: May 28th 2010, 02:39 AM
  5. Replies: 6
    Last Post: May 26th 2009, 02:44 PM

Search Tags


/mathhelpforum @mathhelpforum