[Search for users] [Overall Top Noters] [List of all Conferences] [Download this site]

Conference rusure::math

Title:	Mathematics at DEC

Moderator:	RUSURE::EDP

Created:	Mon Feb 03 1986
Last Modified:	Fri Jun 06 1997
Last Successful Update:	Fri Jun 06 1997
Number of topics:	2083
Total number of notes:	14613

1461.0. "Mu-Torere, aka Squeeze Play" by JARETH::EDP (Always mount a scratch monkey.) Wed Jun 19 1991 19:43

    I just discovered a game of Maori origin.  It was marketed under the
    name "Squeeze Play" but the original name is apparently "Mu-Torere".
    
    The game is played on a board with a center space surrounded by eight
    other spaces which are essentially circularly oriented, so that each of
    the eight spaces has one neighbor on either side of it.  Players each
    have four identical pieces, and each player's pieces are placed in a
    group of four neighboring spaces, excluding the center, e.g.:
    
    	A A A
    	A o B
    	B B B
    
    Players alternate moves.  Moves are made according to one of three
    rules:
    
    	A piece may move from an outer point to the vacant center space
    	only if an opponent's marker occupies at least one neighboring
    	space.
    
    	A piece may move from the center to any vacant space.
    
    	A piece may move to any vacant neighboring space.
    
    Suprisingly, the first three moves are forced, but then players have
    choices.  What is the outcome if both players play optimally?
    
    
    				-- edp

T.R	Title	User	Personal Name	Date	Lines
1461.1	How does one win this game?	CIVAGE::LYNN	Lynn Yarbrough @WNP DTN 427-5663	`Wed Jun 19 1991 19:46`	4
	> What is the outcome if both players play optimally? What are the possible outcomes? What is the object of the game? Did I miss something?
1461.2		GUESS::DERAMO	duly noted	`Wed Jun 19 1991 22:00`	4
	I would guess it is like checkers ... if it is your turn to play and you don't have a legal move, you lose. Dan
1461.3		JARETH::EDP	Always mount a scratch monkey.	`Thu Jun 20 1991 10:25`	5
	Ah, yes, the object of the game is to leave your opponent without a legal move. -- edp
1461.4	It's a draw	HERON::BUCHANAN	object occidented	`Sat Jun 22 1991 22:04`	208
	Summary: Out of 46 possible positions (43 accessible from the starting position), I believe 8 are wins, 5 are losses, and 33 (including the starting position) are draws. I worked it through by hand, using symmetries to minimize the chance of making an untrappable error in my calculations. Details: Let's see how far we can get just by thinking a little. There are clearly three kinds of position: E] Where the centre space is empty [o] Where the centre space contains counter o (o not to move) [X] Where the centre space contains counter X (X to move) From each position of form E], X (the player to move) can go to 1,2,3 or 4 possible positions of form [o]. From each position of form [o], X can go to 0,1 or 2 possible positions of form [X]. From each position of form [X], X can go to a position of form E], and to 0,1 or 2 possible positions of form [o]. There's also a duality between the positions of form [o] [X]. If we label the positions o01, o02,..., X01, X02, we can pick the labelling such that om -> Xn iff on -> Xm. Similarly, if Xm -> on, then Xn -> om. Most of the E positions are there own duals, but E3 and E4 are duals of one another. If Em' denotes the dual to Em, then we can say that: if Em -> on, then Xn -> Em'. The converse of the second does not apply, since Em -> on is prohibited if it involves the movement of a buried counter. However, for checking purposes we can verify that if Xn -> Em, then either Em' -> on, or Em' cannot move to on due to buried counter. What are the positions, modulo all rotations and reflections? ------------------------------------------------------------------------------- E01 E]XXXXoooo o01 [o]ooXXXXo X01 [X]XXooooX E02 E]XXXoXooo o02 [o]oXoXXXo X02 [X]XoXoooX E03 E]XXXooXoo o03 [o]oXXoXXo X03 [X]XooXooX E04 E]XXoXXooo E05 E]XXooXXoo o04 [o]oooXXXX X04 [X]XXXoooo E06 E]XXoXoXoo o05 [o]ooXoXXX X05 [X]XXoXooo E07 E]XXoXooXo o06 [o]ooXXoXX X06 [X]XXooXoo E08 E]XoXoXoXo o07 [o]ooXXXoX X07 [X]XXoooXo o08 [o]oXooXXX X08 [X]XoXXooo o09 [o]oXoXoXX X09 [X]XoXoXoo o10 [o]oXoXXoX X10 [X]XoXooXo o11 [o]oXXooXX X11 [X]XooXXoo o12 [o]oXXoXoX X12 [X]XooXoXo o13 [o]oXXXooX X13 [X]XoooXXo o14 [o]XoooXXX X14 [X]oXXXooo o15 [o]XooXoXX X15 [X]oXXoXoo o16 [o]XooXXoX X16 [X]oXXooXo o17 [o]XoXooXX X17 [X]oXoXXoo o18 [o]XoXoXoX X18 [X]oXoXoXo o19 [o]XXoooXX X19 [X]ooXXXoo ------------------------------------------------------------------------------- o01-o03 are the lost positions, since there are no ways out of them. Dually, X01-X03 are Eden positions, and cannot be reached from any other position. o01-o03 can only be reached from X positions since moves from E positions would be with buried counters. From X01-X03, there are three legal moves, 2 to o positions, and 1 to a P position. o04-o13 have 1 legal move, and derive from 1 legal E position and one legal X position each. X04-X13 admit one move to an E position, and one to an o position, and each have a unique predecessor. o14-o19 have 2 legal moves to X positions, and derive from a unique E position each. X14-X19 admit one move, to an E position, and can derive from 2 o positions. Examining the E positions, we can say that for all except E3 & E4, the number of ways of moving, is the same as the number of ways of moving out. In practice, the number of links may be reduced by symmetries, but the duality will be preserved. This is about as much as we can say without actually getting down to the hard work of solving the game. ------------------------------------------------------------------------------- Position Comes from Goes to ------------------------------------------------------------------------------- E01 X01, X04 o04 E02 X02, X05, X07, X14 o05, o07, o14 E03 X08, X13 o06, o19 E04 X03, X06, X19 o08, o13 E05 X11 o11 E06 X09, X12, X16, X17 o09, o12, o16, o17 E07 X10, X15 o10, o15 E08 X18 o18 o01 X01, X04 - o02 X05, X07 - 003 X06 - X01 - E01, o01, o04 X02 - E02, o05, o07 X03 - E04, o06 o04 E01, X01 X14 o05 E02, X02 X15 o06 E03, X03 X16 o07 E02, X02 X13 o08 E04, X13 X17 o09 E06, X12 X18 o10 E07, X10 X12 o11 E05, X11 X16 o12 E06, X09 X10 o13 E04, X08 X07 X04 o14 E01, o01 X05 o15 E02, o02 X06 o16 E04, o03 X07 o13 E02, o02 X08 o17 E03, o13 X09 o18 E06, o12 X10 o12 E07, o10 X11 o16 E05, o11 X12 o10 E06, o09 X13 o07 E03, o08 o14 E02 X04, X19 o15 E07 X05, X17 o16 E06 X06, X11 o17 E06 X08, X15 o18 E08 X09 o19 E03 X14 X14 o04, o19 E02 X15 o05, o17 E07 X16 o06, o11 E06 X17 o08, o15 E06 X18 o09 E08 X19 o14 E04 ------------------------------------------------------------------------------- Building backwards from the lost positions, I don't think that many of the positions are decidable. Below, Ln denotes that the position loses in n moves, Wn denotes a win in n moves. /pos denotes that move loses. >pos denotes that move wins. Only thirteen positions are decidable, the other thirty-three, including E01, which is the starting position, are draws. We know this becuase from none of the 33 is any player ever forced to play to one of the 8 known winning positions. However, I haven't computed the drawing cycles that the players could chose. E01 X01, X04 o04 E02 X02, X05, X07, X14 o05, o07, /o14 E03 X08, X13 o06, o19 E04 W2 X03, X06, X19 o08, >o13 E05 X11 o11 E06 X09, X12, X16, X17 o09, o12, o16, o17 E07 X10, X15 o10, o15 E08 X18 o18 o01 L0 X01, X04 - o02 L0 X05, X07 - 003 L0 X06 - X01 W1 - E01, >o01, o04 X02 - E02, o05, o07 X03 - /E04, o06 o04 E01, X01 X14 o05 E02, X02 X15 o06 E03, X03 X16 o07 E02, X02 X13 o08 E04, X13 X17 o09 E06, X12 X18 o10 E07, X10 X12 o11 E05, X11 X16 o12 E06, X09 X10 o13 L1 E04, X08 /X07 X04 W1 o14 E01, >o01 X05 W1 o15 E02, >o02 X06 W1 o16 /E04, >o03 X07 W1 o13 E02, >o02 X08 W2 o17 E03, >o13 X09 o18 E06, o12 X10 o12 E07, o10 X11 o16 E05, o11 X12 o10 E06, o09 X13 o07 E03, o08 o14 W3 E02 /X04, >X19 o15 E07 /X05, X17 o16 E06 /X06, X11 o17 E06 /X08, X15 o18 E08 X09 o19 E03 X14 X14 o04, o19 E02 X15 o05, o17 E07 X16 o06, o11 E06 X17 o08, o15 E06 X18 o09 E08 X19 L2 o14 /E04 Andrew
1461.5	oho! :-)	GUESS::DERAMO	duly noted	`Sun Jun 23 1991 02:31`	22
	re 1457.0 by edp ... >> Consider arrangements of objects in m positions. There are n0 >> identical objects of one type, n1 of another type, n2 of a third, and >> so on up to nk. The sum of the n's is m. There are >> >> m!/(n0!n1!n2!...nk!) >> >> such arrangements. Call this number N. >>[...] >> The specific case I am considering at the moment is 9 positions with 4, >> 4, and 1 objects. re 1461.0 by edp ... >> I just discovered a game of Maori origin. It was marketed under the >> name "Squeeze Play" but the original name is apparently "Mu-Torere". "Just discovered"? Ha! It looks like you knew about it way back in topic 1457. :-) :-) Dan
1461.6	Burnside again	HERON::BUCHANAN	object occidented	`Mon Jun 24 1991 10:19`	54
1461.7	Burnside again	HERON::BUCHANAN	object occidented	`Mon Jun 24 1991 10:20`	54
1461.8		JARETH::EDP	Always mount a scratch monkey.	`Mon Jun 24 1991 16:37`	9
	Re .4, .6: Yes, I figured it was a draw as well. I also checked for 4, 6, 10, 12, 14, and 16 spaces around the center; they are all draws. The game is sort of interesting; it would be nice to find a variation or extension that resulted in challenging play. -- edp
1461.9		JARETH::EDP	Always mount a scratch monkey.	`Wed Jul 10 1991 10:44`	16
	Re .7: > btw, I wonder who wins the game if you play on a decagon instead of > an octogon, with 5 counters each. Or what happens if on the octogon, > one player has 5 and the other 3 counters? It's not clear a priori > whether having more or less counters is an advantage. I tried those variations; they are all draws. With 2 and 6 counters, it is a win for the player with more counters, regardless of who starts. Maybe there are interesting variations with more complicated boards (e.g., more than one center space). -- edp