math344 hw2s - Mathematical Game theory Assignment 2 due Problem 1 Find the Nim-sum of all numbers from 1 to 2n \u2212 1 where n > 1 is a natural

math344 hw2s - Mathematical Game theory Assignment 2...

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Mathematical Game theory Assignment 2, due 2019-09-20 Problem 1.Find the Nim-sum of all numbers from 1 to 2n-1, wheren >1 is a natural number.More precisely, compute12⊕ · · · ⊕2n-1.Hint: Show that there are exactly 2n-1numbers with 1 as thekth digit of their binary expansionfor eachk= 1,2, . . . , n. Problem 2.The columns of an 8×8 chess board are denoted by letters a–h, and the rows bynumbers 1–8. A game is played with two rooks on a chess board, with the following rules: A validmove is to move one of the rooks either to the left within its row, or down in its column, but not tothe right or up. Unlike in chess, the rooks can occupy the same square, and one can also jum overthe other. For example, if the rooks are in c4 and c7, it is valid to move the c7 rook to c1.Which player wins this game if initially the rooks are at d8 and g6? If it is the first player, whatare the winning moves?Hint: This is NIM in disguise.

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