Question

NEED HELP WITH DISCRETE MATH:

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Consider the following game. Alice and Bob have a an infinite quarter chessboard in front of them. The chessboard has a left edge and a bottom edge. There is one checker on some square the chessboard. The player whose turn it is can move the checker down any positive number of squares, or can move the check one column to the left, but anywhere in that column. The game ends when a player cannot move. We could consider the positions on the board as being the elements of NX N. From a position (x,y), a player can move to any cell (x,z) with 0 <2<y, z N, or, in the case that x > 0, to any cell (x - 1,w), for any w E N. For example, from cell (3,4), the player could move to any of (3,3), (3,2), (3,1), (3,0), (2.0), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), etc. The players alternate turns with Alice going first. Question (i). Prove that the game must end. Question (ii). If the game starts at position (x,y), how many moves will be needed before the game ends?

Answer 1

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Show that how many moves wilt be Now ae have hu dea before e game end to the lept are hudedtov th 2-Steps Column th - Co-oTdinates the checker to veach Equar to Where te game starts at Cai4 In this Column move to Cl ) directly. t Can be ih one Step Hence Steps it rs possible fof ave hoesed and tha end Ezactiy th Steps game to Note: A-t cue position thoir atleast onemove to change its position apart rom that point Whave there Can be no moves made to chang e its position because e this tho game end Whan tha checkay Peaches (,t).