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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: The hexad game. | The hexad game. Joseph Kahane Dept of Mathematics Queens College CUNY Flushing NY 11367 e-mail jokqc@qcunix1.qc.edu and Alexander J. Ryba Dept of Mathematics Queens College CUNY Flushing NY 11367 e-mail ryba@qcunix1.qc.edu Submitted February 25 2000 Accepted May 18 2000. Abstract. We give a non-enumerative proof that the P-positions of the hexad game give the Steiner system S 5 6 12 . We show that the distributions of nim sums and Welter values of these P-positions have simple and surprising regularities. 1. Introduction. The hexad game also known as mathematical blackjack is played on a linear board ruled into twelve squares labelled 0 1 2 . 10 11. Six coins are initially placed on distinct squares whose labels have a sum of at least 21. Each player at his turn selects a coin and moves it to a smaller unoccupied square possibly skipping over several coins with the restriction that the sum of the 6 occupied squares must remain greater than or equal to 21. The first player unable to move when the sum of the occupied squares has just reached 21 is the loser. The hexad game was first investigated by Conway and Ryba their result that the P-positions form the hexads of the shuffle numbered Steiner system S 5 6 12 is reported in CS . We shall give a new proof of this result our proof is the only known argument that does not simply calculate the Grundy values of all 905 legal positions in the hexad game. Our proof makes use of certain properties of the Steiner system we will review these in Section 2. There is a close similarity between the hexad game and Welter s game We1 We2 . Indeed if we strike the clauses that mention the number 21 from our earlier description of the hexad game we obtain a description of a case of Welter s game. This similarity leads us to investigate the Welter values and nim sums of the Steiner hexads. We discover two surprising regularities in the distributions of these values. These regularities seem as striking as the known regularity in the .
