: Problem Set H

Problem Set H

Let P_1,\, P_2,\, \ldots P_n be distinct points in a plane. Connect the points with line segments P_1P_2,\,P_2P_3,\, \ldots P_{n-1}P_{n},\,P_{n}P_1,. Explore when it is possible to draw a line L that passes through the interior of every one of the line segments. Make a conjecture and prove your result supplying all details (Hint: See if odd or even n makes a difference.)
A bubble chamber contains three type of particles: 10 of type X, 11 of type Y, and 111 of type Z. Where ever particles meet (\oplus operation) the following happen:
- X \oplus Y \rightarrow 2Z
- X \oplus Z \rightarrow 2Y
- Y \oplus Z \rightarrow 2X
where \rightarrow means &dlquo;they transforms to.&drquo; With random meetings of particles can the system ever evolve to a point where there is only one type of particle? Prove you answer suppling all details. (Hint: look for invariants!)