# SOLN_MIDTERM (1) - Solutions to Midterm Problems Stat 155...

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Solutions to Midterm ProblemsStat 155: Game TheorySoumendu Sundar MukherjeeNote:If you have comments or questions about the solutions, feel free to emailme at[email protected].Problem 1Consider the take-away (subtraction) game with subtraction setS={2,4,7}. That is, this is a two player game, withnpiles ofxnchips each.At any player’s turn, they may take away 2, 4, or 7 chips from exactly oneof the piles.(a) What is the Sprague-Grundy function for a pile ofxchips, for allnon-negative integersx? Justify your answer.(b) What isg(91)?(c) Is the position (3,15,22) anN-position or aP-position? If it is anN-position, find all the winning moves.Solution:(a) Starting with the terminal positions, it is easy to calculate thefirst few values of the SG function; see Table1. From this it is easy to guessx012345678910111213141516171819g(x)00112203102102102102Table 1: First few values of the SG function.
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that the SG function is given, fork>3, byg(x) =1,ifx= 3k-1,0,ifx= 3k,2,ifx= 3k+ 1,whileg(0) throughg(7) can be read off from Table1. To prove this rigorously,we use induction onk.First of all, note that this can be easily verified fork= 3,4,5, as is done in Table1. Now assume that this has been verified fork= 3,4, . . . , m5. We shall show that this remains true fork=m+ 1.Suppose thatx= 3(m+ 1)-1 = 3
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