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(2) = 2. A pile of three chips can be moved to a pile of two or one chips,(SG-value 1 and 2) or to two piles of one chip each (SG-value 0), sog(3) = 3. Continuingin this way, we find more of the Sprague-Grundy values in Table 4.1.y\z012345678910110012314321426124127143214672441285472186736412314721827484128147214276041281472186772412814721827Table 4.1. The SG-values for Kayles. Entries for the Table are forg(y+z)whereyis on the left side andzis at the top.Fromx= 72 on, the SG-values are periodic with period 12, with the values in thelast line repeating forever. The 14 exceptions to the sequence of the last line are displayedin bold face type. The last exception is atx= 70. Since we know all the SG-values, wemay consider the game as solved. This solution is due to Guy and Smith (1956).3.Dawson’s Chess.One of T. R. Dawson’s fanciful problems inCaissa’s Wild Roses(1935), republished inFive Classics of Fairy Chessby Dover (1973), is give-away chessplayed with pawns. “Given two equal lines of opposing Pawns, White on 3rd rank, Blackon 5th,nadjacent files, White to play at losing game, what is the result?” (Captures mustbe made, last player to move loses.) We treat this game here under the normal play rule,that the last player to move wins.Those unfamiliar with the movement of the pawn in chess might prefer a different wayof describing the game as a kind of mis´ere tic-tac-toe on a line ofnsquares, with bothplayers using X as the mark. A player may place an X on any empty square provided itis not next to an X already placed. (The player who is forced to move next to another Xloses.)This game may also be described as a game of removing chips from a pile and possiblysplitting a pile into two piles. Ifn= 1 there is only one move ton= 0, ending the game.Forn >1, a move of placing an X at the end of a line removes the two squares at that endof the line from the game. This corresponds to removing two chips from the pile. Similarly,placing an X one square from the end corresponds to removing three chips from the pile.Placing an X in one of the squares not at the end or next to it corresponds to removingthree chips from the pile and splitting the pile into to parts. Thus the rules of the gameare: (1) You may remove one chip if it is the whole pile, or (2) you may remove two chipsfrom any pile, or (3) you may remove three chips from any pile and if desired split thatpile into two parts.The Sprague-Grundy sequence for Dawson’s chess beginsx0123456789101112131415161718g(x)0112031103322405223I – 25
It is eventually periodic with period 34. There are only 7 exceptions and the last exceptionoccurs atn= 51. Can you find the winning move on a line of 18 squares? Armed with theabove Sprague-Grundy sequence, you may hone your skill at the game by playing againstthe computer at ˜tom/Games/dawson.html .