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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

2015.0. "Touche" by RUSURE::EDP (Always mount a scratch monkey.) Tue Nov 28 1995 10:14

    A little combinatorics exercise:
    
    The game Touche was found at a yard sale.  Under the playing surface,
    it contains five strips.  Each strip contains a row of five magnets,
    with the poles pointed up through the playing surface.  The magnets in
    each strip are inserted at the factory and cannot be altered.  The five
    strips can be placed under the playing surface to form a 5-by-5 array
    of magnets.  Each strip can be flipped over or rotated 180 degrees. 
    The playing surface is not symmetric under rotation (two of the sides
    are distinguished by the player they belong to).
    
    The box claims there are 245,760 games, where each game is a different
    configuration of the magnets.
    
    a) How did they get this number?
    
    b) Why is it wrong?
    
    c) What are the possible correct answers?
    
    
    				-- edp
    
    
Public key fingerprint:  8e ad 63 61 ba 0c 26 86  32 0a 7d 28 db e7 6f 75
To find PGP, read note 2688.4 in Humane::IBMPC_Shareware.

T.R	Title	User	Personal Name	Date	Lines
2015.1	I found five different strips	WIBBIN::NOYCE	EV5 issues 4 instructions per meter	`Tue Nov 28 1995 10:48`	33
	a) How did they get this number? 245760 = 5! * 4^5 * 2 There could be 5! ways to place the strips in their slots. Each strip could be flipped or not, and rotateded end-for-end or not. Perhaps the other factor of two represents players changing sides? Or can you choose to align strips parallel to either X or Y axis? (I assume not.) b) Why is it wrong? If the factor of two represents changing sides of players, that's already accounted for by the potential to rotate all the strips end-for-end. Here are five strips that are all different under flipping and rotation, so it seems to me that the 5! * 4^5 is achievable. straight Nssss NNsss NsNss NssNs NNssN rotated ssssN sssNN ssNsN sNssN NssNN flipped sNNNN ssNNN sNsNN sNNsN ssNNs flip & rot NNNNs NNNss NNsNs NsNNs sNNss c) What are the possible correct answers? 122880 is one possible correct answer. For each strip that has only two different orientations, reduce by a factor of 2. (Each strip is different from itself under flipping, since the middle magnet must change.) If n strips are identical, reduce by a factor of n!. In the limit, all strips are identical and symmetric (NNNNN or NNsNN or NsssN or NsNsN), and there are only 2^5 different configurations.
2015.2		RUSURE::EDP	Always mount a scratch monkey.	`Tue Nov 28 1995 13:12`	19
	Re .1: Yes, the alignment of the strips parallel to either the x or the y axis accounts for the other factor of two. In addition to the strips you listed that are different in all four positions, there is sNsss. The four other strips are symmetric under rotation, so they only have two positions each. Given any five strips, how many different boards can be constructed by positioning the five strips? Two boards are different iff their 25 magnet orientations are not all the same. -- edp Public key fingerprint: 8e ad 63 61 ba 0c 26 86 32 0a 7d 28 db e7 6f 75 To find PGP, read note 2688.4 in Humane::IBMPC_Shareware.