of collisions is

(1.3)

[

^r—l

I arctan y 777,1/7712 I

Exercise 1.3. Extend the upper bound on the numb

to a wedge convex inside; see figure 1.4.

Figur e 1.4. A plane wedge, convex inside

Exercise 1.4. a) Interpret the system of two point-m

ment, subject to elastic collisions with each other and

points of the segment, as a billiard.

b) Show that the system of three point-masses 777,1,7

line, subject to elastic collisions with each other, is iso

billiard inside a wedge in three-dimensional space. P

dihedral angle of this wedge is equal to

(1.4) arctan [ 777,2 \ I •

V V 777,1777,2777,3 /

c) Choose the system of reference at the center of ma

the above system to the billiard inside a plane angle (

d) Investigate the system of three elastic point-masses o

1.1. Digression. Billiard computes TT. Formula

possible to compute the first decimal digits of TT. Wh

brief account of G. Galperin's article [39].

Consider two point-masses on the half-line and ass

100fc777,i.

Let the first point be at rest and give the se

the left. Denote by N(k) the total number of collisions

in this system, finite by the above discussion. The cla

N(k) = 3141592653589793238462643383..