Abstract
Previous authors have shown geometrically that the first \({N}\) digits of \({\pi}\) can be generated by counting the energy conserving, alternate ball-ball and ball-wall collisions in which the mass ratio of the balls is \({100^{N+1}}\). Here, after explicitly demonstrating the equality of separation and approach speeds in a ball-ball collision, the sequence of ball speeds is determined from a pair of linear difference equations. The asserted result is then readily deduced without consideration of the ball displacements and time intervals between collisions.
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References
Galperin, G. A., “Playing Pool with p (the Number p \({\pi}\) from a Billiard Point of View).” Regular & Chaotic Dynamics 8 (2003) 375–394.
Weidman, P. D., “On the Digits of p.” The Mathematical Intelligencer 35(4) (2013) 43–50.
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Davis, A.M.J. Digits of Pi. Math Intelligencer 37, 1–3 (2015). https://doi.org/10.1007/s00283-014-9502-0
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DOI: https://doi.org/10.1007/s00283-014-9502-0