collisions. The time between the first two collisions, t$_{1}$ can be guaranteed to be positive because after 1 collides with 2, a collision with 3 is possible if 3 has the right orientation. But from relations (35) and (41), we see that for all the times to be positive, we need sin(( n − j + 1) θ ) / sin(( n − j ) θ ) < 0 and cos(( n − j + 1 / 2) θ ) / cos(( n − j − 1 / 2) θ ) < 0 for all 1 ≤ j ≤ n − 1 . The solution set for sin( kθ ) / sin(( k − 1) θ ) < 0 is and the solution set for cos(( k + 1 / 2) θ ) / cos(( k − 1 / 2) θ ) < 0 is In (42) and (43), k = n − j and j varies from 1 to n − 1 so k can take any value between 1 and n − 1. It is not hard to show that the intersection of the sets (42) having k between 1 and n − 1 is (( n − 2) π/ ( n − 1) , π ). The intersection of this interval with all the sets (43) having k between 1 and n − 1 is ((2 n − 3) π/ (2 n − 1) , π ). Thus a necessary condition for all positive time intervals is that θ be in the range ((2 n − 3) π/ (2 n − 1) , π ). We have seen that θ is real if β is. From the form of Eq. (14), the reality of α guarantees the reality of β. All the dynamical variables of the problem are found by multiplying the original velocity vector by the 2 × 2 matrices S$_{2}$ and S$_{3}$. So a process is classically allowed if α ∈ ℜ and (2 n − 3) π/ (2 n − 1) ≤ θ ≤ π. Eq. (26) may be rewritten as x = √ y exp( iφ ) , if cos φ = − (1 + y ) / (2 a √ y ) = − cos( β/ 2) /a. In this form, it is evident that if β ∈ ℜ , then α ∈ ℜ , iff cos $^{2}$( β/ 2) ≤ a $^{2}$. This inequality may be recast in terms of sines as: The lower bound on sin $^{2}$( θ/ 2) poses no constraint, it being guaranteed by Eq. (28), so we must solve that equation in the domain (2 $^{n}$− 3) π/ (2 $^{n}$− 1) ≤ θ ≤ π, subject to the constraint Let Q ( θ ) be defined by Q ( θ ) = cos $^{2}$( θ/ 2) / sin $^{2}$( nθ ). Then Eq. (28) can be written as 16 θ = Q − 1 ( a (1 + b ) a + b ) . (46) sin $^{2}$( θ/ 2) ≤ b/ (( a + b )( a + 1)) . (45) 1 − a 2 ≤ sin $^{2}$( β/ 2) ≤ 1 ⇔ b (1 − a ) ( a + b ) ≤ sin $^{2}$( θ/ 2) ≤ b ( a + b )( a + 1) . (44) k ⋃ l =1 ((2 l − 1) π/ (2 k + 1) , (2 l − 1) π/ (2 k − 1)) . (43) k − 1 ⋃ l =1 ( lπ/k, lπ/ ( k − 1)) (42)