Nhan Bao Ho

The Sprague-Grundy function of the real game Euclid

The game Euclid, introduced and named by Cole and Davie, is played with a pair of nonnegative integers. The two players move alternately, each subtracting a positive integer multiple of one of the integers from the other integer without making the result negative. The player who reduces one of the integers to zero wins. Unfortunately, the name Euclid has also been used for a subtle variation of this game due to Grossman in which the game stops when the two entries are equal. For that game, Straffin showed that the losing positions (a,b) with a

Min, a combinatorial game having a connection with prime numbers.

Abstract We introduce a two person game played with a pair of nonnegative integers; a move consists of subtracting from the larger integer, a positive integer no greater than the smaller integer. The player who reduces one of the integers to zero wins. The game is curious in several respects: in particular, its Sprague–Grundy values have an interesting connection with prime numbers.

A RESTRICTION OF EUCLID

Euclid is a well known two-player impartial combinatorial game. A position in Euclid is a pair of positive integers and the players move alternately by subtracting a positive integer multiple of one of the integers from the other integer without making the result negative. The player who makes the last move wins. There is a variation of Euclid due to Grossman in which the game stops when the two entrees are equal. We examine a further variation that we called M-Euclid in which the game stops when one of the entrees is a positive integer multiple of the other. We solve the Sprague-Grundy function for M-Euclid and compare the Sprague-Grundy functions of the three games. 10.1017/S0004972712000391

Variations of the Game 3-Euclid

We present two variations of the game 3-Euclid. The games involve a triplet of positive integers. Two players move alternately. In the first game, each move is to subtract a positive integer multiple of the smallest integer from one of the other integers as long as the result remains positive. In the second game, each move is to subtract a positive integer multiple of the smallest integer from the largest integer as long as the result remains positive. The player who makes the last move wins. We show that the two games have the same 𝒫-positions and positions of Sprague-Grundy value 1. We present three theorems on the periodicity of 𝒫-positions and positions of Sprague-Grundy value 1. We also obtain a theorem on the partition of Sprague-Grundy values for each game. In addition, we examine the misere versions of the two games and show that the Sprague-Grundy functions of each game and its misere version differ slightly.

The Max-Welter game

On a semi-infinite strip of squares rightward numbered 0,1,2,... with at most one coin in each square, in Welters game, two players alternately move a coin to an empty square on its left. Jumping over other coins is legal. The player who first cannot move loses. We examine a variant of Welters game, that we call Max-Welter, in which players are allowed to move only the coin furthest to the right. We solve the winning strategy and describe the positions of Sprague-Grundy value 1. We propose two theorems classifying some special cases where calculating the Sprague-Grundy value of a position of size k becomes easier by considering another position of size k-1. We establish two results on the periodicity of the Sprague-Grundy values. We then show that the Max-Welter game is classified in a proper subclass of tame games that Gurvich calls strongly miserable.

AMENABILITY FOR REAL C * -ALGEBRAS

It is shown that the complexification of a positive linear map on a real C -algebra need not be positive whereas the complexification of a completely positive linear map is completely positive. It is further shown that a real C -algebra is amenable if and only if its complexification is amenable and that a completely positive linear map is amenable if and only if its complexification is. Finally, a real version of the Choi‐Effros lifting theorem is established. 2000 Mathematics subject classification: 46L05.

On the Sprague-Grundyfunction of Exact k- Nim

Moores generalization of the game of {\sc Nim} is played as follows. Let

Some Remarks on End-Nim

n

Extended Complementary Nim

and

Utimately bipartite subtraction games

k