Define a game as follow: you begin with an urn that contains a mixture of black and white balls, and during the game you have access to as many extra black and white balls as you need. In each move of the game, you remove two balls from the urn without being able to see what colour they are. Then you look at their colour and do the following: If the balls are the same colour, you keep them out of the urn and put a black ball in the urn. if the balls are different colours, you keep the black one out of the urn and put the white one back into the urn. Each move reduces the number of balls by one, and the game will end when only one ball is left in the urn. In this you will figure out how to predict the colour of the last ball in the urn and prove your answer using mathematical induction. Q1) 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. Q2) Looking at your answers to Q1, 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 Q3) Use mathematical induction to prove the formal conjecture you made in Q2. 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).
please do the last question(Q3) as well
Define a game as follow: you begin with an urn that contains a mixture of black and white balls, and during the game you have access to as many extra black and white balls as you need.
In each move of the game, you remove two balls from the urn without being able to see what colour they are. Then you look at their colour and do the following:
- If the balls are the same colour, you keep them out of the urn and put a black ball in the urn.
- if the balls are different colours, you keep the black one out of the urn and put the white one back into the urn.
Each move reduces the number of balls by one, and the game will end when only one ball is left in the urn.
In this you will figure out how to predict the colour of the last ball in the urn and prove your answer using mathematical induction.
Q1) 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.
Q2)
Looking at your answers to Q1, 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
Q3)
Use mathematical induction to prove the formal conjecture you made in Q2.
- 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).
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