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θ = π/ 3 and our analysis gives the allowed region described by: 3 ab − a − b − 1 ≥ 0 . To ascertain which θ is the physical one, we insist that the time duration between collisions are all positive. The formulas for the time ratios here are: 

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These must be nonnegative for j ≤ n − 1 , which requirement holds only for ( n − 1) π/n ≤ θ ≤ π. The only suitable θ from Eq. (98) is (2 $^{n}$− 1) π/ (2 n + 1) and so the allowed region for (31) n is given by ab cot $^{2}$((2 n − 1) π/ (2(2 n $^{+ 1)))}$− a − b − 1 ≥ 0 . That is, it lies to the right of the right branch of the hyperbola given by: 

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