Math/Physic/Economic/Statistic Problems

Define a game as follows: You begin with an urn
that contains a mixture of white and black balls,
and during the game you have access to as many
additional white and black balls as you might
need. In each move you remove two balls from
the urn without looking at their colors. If the balls
are the same color, you put in one black ball. If
the balls are different colors, you put the white
ball back into the urn and keep the black ball out.
Because each move reduces the number of balls
in the urn by one, the game will end with a single
ball in the urn. If you know how many white balls
and how many black balls are initially in the urn,
can you predict the color of the ball at the end of
the game?

1. Draw diagrams to map out all the possibilities for playing the game starting with two balls in the urn, then three balls, then four balls. For each case show all the possible configurations of black and white balls, and for each configuration all the possible first pick of two balls and resulting ball at the end of the game.

2. Looking at your answers to question 1, think about whether you can predict the colour of the final ball:

based on the initial number of black balls in the urn
based on the initial number of white balls in the urn
a) Make a conjecture about the colour of the final ball based on the initial number of black and white balls in the urn.

b) Translate that conjecture into a theorem in symbolic form using first order logic notation. You will need to invent some notation, including functions, to do so. Define your new notation and functions clearly.

3. Use mathematical induction to prove the formal conjecture you made in question 2.

Before you start, please identify the predicate function P(n) that you will be proving
In the inductive step of your proof, do not forget to clearly identify the Inductive Hypothesis (IH).