Coin Turning Games By: Lauren Quattrocchi

What are Coin Turning Games? -Coin turning games are a class of combinatorial games in which there are a finite number of coins in a row, each coin either heads or tails. -A move consists of turning over a coin from heads to tails or from tails to heads in a way that abides by the rules of the specific game you are playing. - For all versions of one dimensional coin turning games the rightmost turned over coin must go from heads to tails, and the last player to move wins. This guarantees the game will end in finite time. -Another thing to keep in mind is the games are impartial.

Examples of Coin Turning Games ●Turning tables ●Ruler ●Mock Turtles

P-positions in Coin Turning Games For all coin turning games a position with k heads in positions x 1,..., x k is the disjuctive sum of k games, each with exactly one head, where for j = 1,...,k the head in game j is at x j Example: THHTTH = TH + TTH + TTTTTH g(THHTTH) = g(T H) g(TTH) g(TTTTTH)

Nim Multiplicatoin - 0 acts like 0 for multiplication: x 0=0 x = 0 for all x - 1 acts like unit for multiplication: x 1=1 x = x for all x - commutative laws hold: x y = y x for all x in y - associative law holds: x (y z)=(x y) (x z) for all x, y and z - every number other than 0 has a multiplicative inverse

Ex: 6 4 = 14 Ex: 11 5 = 12

Solving Nim Multiplication without table 1) the nim-product of a Fermat 2-power and any smaller number is their ordinary product 2) the nim-product of Fermat 2-power with itself is the Fermat 2-power times 3/2 in ordinary sense Example 1: 2 16 = 32 Example 2: = (3/2)16 = 24 Example 3: = (16 8) (16 1)= (16 16) (16 1) (8 16) (8 1) = = 128

Two Dimensional Coin Turning Games  is a two dimensional rectangular array  the conditions are as follows; coordinates of the array are numbered starting at (0,0) with coins at coordinates (x,y) with x ≥ 0 and y ≥ 0, the most southeast coin must go from heads to tails, and any other coins turned over must be in the rectangle {(a, b):0 ≤ a ≤ x, 0 ≤ b ≤ y}.

Examples of Two Dimensional Games Acrostic Twins Rugs (Turning Turtles) 2

Tartan Games Tartan Games are two dimensional coin turning games, G 1 xG 2. If turning coins at x 1, x 2, x 3,..., xm is a legal move in G 1, and turning coins at positions y 1, y 2, y 3,..., y n is a legal move in G 2, then turning coins at positions (x i, y j ), for all 1 ≤ i ≤m and 1 ≤ j ≤n is a legal move in the Tartan Game. Examples of Tartan Games include (Mock Turtles) 2 and Turning Corners Tartan Theorem: If g 1 (x) is the Sprague-Grundy function of some game G 1, and g 2 (y) is the Sprague Grundy function of some game G 2, then the Sprague Grundy function g(x,y) of G 1, G 2 is g(x,y) = g 1 g 2

Determining a Move Suppose you are at position (x,y) with Sprague Grundy value g 1 (x) g 2 (y) = v. Now, you want to replace Sprague Grundy Value v with a Sprauge Grundy value u. 1) Let v 1 = g 1 (x) and v 2 = g 2 (y), find a move in Turning Coins that takes (v 1, v 2 ) to a SG-value u. Let (u 1, u 2 ) denote the northwest corner of the move so ((u 1 u 2 ) (u 1 v 2 ) (v 1 u 2 )= u. 2) Find a move M 1 in the 1-D game G 1 that moves the SG-value g 1 (x) at x into a SG- value u 1. 3) Find a move M 2 in the 1-D game G 2 that moves the SG-value g 2 (y) at y into a SG- value u 2 The move M 1 x M 2 in G1xG 2 moves the SG-value to u, the desired SG-value to obtain a P-position.

Example of Method in Use Consider a (Mock Turtles)2 board is set up with Sprague Grundy values

Sources [1] T. Ferguson (1989), Game Theory, Coin Turning Games. I-29-I-39. [2] M. Lugo (2011), Stat 155, Proof of Tartan Theorem [3] A. Wang (2015), WikiAnswers, Intuitive Explanation of Nimbers