Practice Midterm: Statistics 155 - due Oct 18, 2004
Instructor: Yuval Peres Duration: 75 minutes.
Instructions: Please write your name on every page. This examination contains three
problems with weight 34 points each. Write each answer very clearly below the corresponding
question. (Use back of page if needed). No books, notebooks or other written materials are
allowed.
Good Luck!
1. Define precisely the Sprague-Grundy function gfor a progressively finite impartial game.
Consider the game which is played with piles of chips like nim, but with the additional
move allowed of breaking one pile of size k > 0 into two nonempty piles of sizes i > 0
and k−i > 0. Show that the Sprague-Grundy function gfor this game, when evaluated
at positions with a single pile, satisfies g(3) = 4. Find g(1000), that is, gevaluated at a
position with a single pile of size 1000.
Given a position consisting of piles of sizes 13, 24 and 17, how would you play?
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