First presenter Co-presenter(s) Name : Aleksandr Myllari * Name: E-mail: E-mail: Affiliation: University of Turku Name: Department: E-mail: City: Name: State/Province: E-mail: Country: Finland Name: TalkNumber: 06-09 E-mail: Session: 6- Applications of Math Software to Mathematical Research Schedule:  Room: Saturday, 10:30  B-2620 Related website: Title ofpresentation: Conway Matrices Related to a Non-transitive Head-or-Tail Game with a q-sided die Abstract: We consider the well-known problem of string overlapping in connection with the so-called Penney Ante game with a $q$-sided die. Let two players, (1) and (2), agree on some integer $n \ge 2.$ Then both of them select a $q$-ary $n$-word, say $w_1$ and $w_2,$ and begin rolling a die until either $w_1$ or $w_2$ appears as a block of $n$ consecutive outcomes. Player (1) wins if $w_1$ appears before $w_2$ does. The problem is to find the probability $P(w_1, w_2)$ that player (1) will win for the chosen $w_1$ and $w_2.$ A solution of the problem was proposed by J.H. Conway (but was not published), and the key tool of his solution is the so-called Conway matrix $C_n^{(q)}$ whose indices encode $n$ consecutive outcomes. We extend our previous results obtained for the case $q = 2$ ("coin") to the case of arbitrary $q >1.$ We propose a simple and effective algorithm for calculation and visualization of Conway matrices $C_n$ and the corresponding matrices $P_n$ (that give probabilities that player (1) will win for the chosen words $w_1$ and $w_2$) via the standard technique of the (inverse) multidimensional DFT. Computer Algebra system {\it Mathematica} (versions 5 and 6) was used for intermediate calculations, check of hypotheses and visualization of results. This is joint work with N. Gogin and T. Myllari.