Essay Alhazen's Billiard Problem

5102 Words 21 Pages
Alexander Zouev 000051 - 060

Extended Essay – Mathematics Alhazen’s Billiard Problem
Antwerp International School May 2007

Word Count: 3017 -0-

Alexander Zouev 000051 - 060

Abstract The research question of this Mathematics Extended Essay is, “on a circular table there are two balls; at what point along the circumference must one be aimed at in order for it to strike the other after rebounding off the edge”. In investigating this question, I first used my own initial approach (which involved measuring various chord lengths), followed by looking at a number of special cases scenarios (for example when both balls are on the diameter, or equidistant from the center) and finally forming a general solution based on coordinate
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Investigating this problem gave me an opportunity to fill this void. That being said, the problem was in itself also very appealing to me as I personally enjoy playing billiards or pool and was eager to find out about the mathematics of the game. The problem appeared in the Daily Telegraph news in 1997 when Dr Peter Nueman, an Oxford don of Queen’s Collage, managed to provide a new solution to the problem. Inspired by early mathematician Descartes, Nueman cleverly translated the billiards table geometry simply into x and y coordinates on two axes. 6 This is a method I intend to use further into my extended essay. Please note that this essay (and the solution to the focus question) is narrowed down to emphasize the algebraic solution to Alhazen’s Problem - however in the conclusion, other methods are briefly discussed.

Pre-examination of the problem: The great difficulty with this investigation lies within two concepts. First of all, the balls in question are randomly scattered on the table with no specific locations – in other words our solution would need to be generalized for any set of billiard balls. Second of all, the balls need to be treated as fixed points. To begin this investigation one should first consider where and how many possibilities there can be on a circular pool table that would allow for a ball to strike once off the edge and then hit another ball. Moreover, what

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