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Math Help - Billiard balls and the number theory result

  1. #1
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    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
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  2. #2
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    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 m \times n rectangles, each one representing an image of the billiard table. The ball is projected from A at 45^o to the bottom edge 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 B. Then, since AB is at 45^o to the bottom edge of the grid, the distances AP and BP are equal, and each is equal to a multiple of m and a multiple of n respectively. In other words:
    AP = BP\Rightarrow pm = qn, for some integers p, q.
    Now m and n are co-prime. Therefore p is a multiple of n and q is a multiple of m. Clearly the smallest values that satisfy this equation are:
    p=n, and q=m

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

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

    Grandad
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