qts@houxa.UUCP (J.RAMMING) (02/14/86)
Keywords: round table, pennies ----- All puzzles should have elegant solutions; the more elegant, the better the puzzle. This is an excellent puzzle. Imagine a two-player game, in which each of the players begins with an infinite number of pennies. There exists a round table, and each player in his turn places a penny on the table. (Turns are alternated). The game ends when there is no more room on the table for any pennies. The person who last put a penny on the table is declared the winner. Question: Given that one of these players has a winning strategy, which player (the first, or the second) can always win? Prove your answer by giving the strategy. J. Christopher Ramming UUCP: decvax!bellcore!houxa!qts HOME: (201) 542-2079 WORK: (201) 949-9531
dim@whuxlm.UUCP (McCooey David I) (02/18/86)
> Imagine a two-player game, in which each of the players begins > with an infinite number of pennies. There exists a round table, > and each player in his turn places a penny on the table. (Turns > are alternated). The game ends when there is no more room on the > table for any pennies. The person who last put a penny on the table > is declared the winner. > > Question: Given that one of these players has a winning strategy, > which player (the first, or the second) can always win? > Prove your answer by giving the strategy. > The second player can always win if he uses the following strategy: Considering the center of the table as the "origin", always place his penny at a spot reflected through the origin from where his opponent just placed his last penny. Using this strategy, the second player will always have a spot to place his penny because he is simply mirroring the actions of the first player. Dave McCooey AT&T Bell Labs, Whippany ihnp4!whuxlm!dim
ags@pucc-h (Dave Seaman) (02/19/86)
In article <904@whuxlm.UUCP> dim@whuxlm.UUCP (McCooey David I) writes: >> Imagine a two-player game, in which each of the players begins >> with an infinite number of pennies. There exists a round table, >> and each player in his turn places a penny on the table. (Turns >> are alternated). The game ends when there is no more room on the >> table for any pennies. The person who last put a penny on the table >> is declared the winner. >> >The second player can always win if he uses the following strategy: > > Considering the center of the table as the "origin", always > place his penny at a spot reflected through the origin from > where his opponent just placed his last penny. > Actually this is a description of the first player's winning strategy, beginning at the second move. His first move, of course, is to place a penny at the exact center. -- Dave Seaman pur-ee!pucc-h!ags
meister@faron.UUCP (Philip W. Servita) (02/19/86)
In article <953@houxa.UUCP> qts@houxa.UUCP (J.RAMMING) writes: >Imagine a two-player game, in which each of the players begins >with an infinite number of pennies. There exists a round table, >and each player in his turn places a penny on the table. (Turns >are alternated). The game ends when there is no more room on the >table for any pennies. The person who last put a penny on the table >is declared the winner. > >Question: Given that one of these players has a winning strategy, > which player (the first, or the second) can always win? > Prove your answer by giving the strategy. First player win. The general strategy is simple: label the center of the table the origin, and, assuming you are the second player, and the first player just made a move M to (x,y), just make your move to (-x,-y). in this fashion there will always be a spot for the second player to move. This strategy will work for ANY symmetric starting position. How, then, is the round table a FIRST player win? at the beginning of the game there is exactly 1 move which preserves the symmetry of the playfield; (0,0). First player moves there and then pretends to be the second player. Better Problem: (for this problem, assume the Penny Width (PW) to be the unit of distance) Two players are about to play this game on a round table of diameter N PW, for unknown N. Unfortunately, a Malicious Mathematician has Krazy Glued a penny to position (0,.5PW). Determine the set of real numbers N for which this game is a first player win. -- --------------------------------------------------------------- sit down someday and make a list of all things that don't any difference to you. when you are done, throw away the list. --------------------------------------------------------------- -the venn buddhist
dim@whuxlm.UUCP (McCooey David I) (02/19/86)
> > Imagine a two-player game, in which each of the players begins > > with an infinite number of pennies. There exists a round table, > > and each player in his turn places a penny on the table. (Turns > > are alternated). The game ends when there is no more room on the > > table for any pennies. The person who last put a penny on the table > > is declared the winner. > > > > Question: Given that one of these players has a winning strategy, > > which player (the first, or the second) can always win? > > Prove your answer by giving the strategy. > > > The second player can always win if he uses the following strategy: > > Considering the center of the table as the "origin", always > place his penny at a spot reflected through the origin from > where his opponent just placed his last penny. > > Using this strategy, the second player will always have a spot to place his > penny because he is simply mirroring the actions of the first player. > > Dave McCooey > AT&T Bell Labs, Whippany > ihnp4!whuxlm!dim Actually, the above strategy has a flaw: The first player starts off by placing his penny on the origin. Therefore, it is the FIRST player that has a winning strategy. He just uses the strategy given above AFTER his first move, which is at the origin. Dave McCooey AT&T Bell Labs, Whippany ihnp4!whuxlm!dim
cipher@mmm.UUCP (Andre Guirard) (02/19/86)
In article <904@whuxlm.UUCP> dim@whuxlm.UUCP (McCooey David I) writes: >> Imagine a two-player game, in which each of the players begins >> with an infinite number of pennies. There exists a round table, >> and each player in his turn places a penny on the table. (Turns >> are alternated). The game ends when there is no more room on the >> table for any pennies. The person who last put a penny on the table >> is declared the winner. >> >> Question: Given that one of these players has a winning strategy, >> which player (the first, or the second) can always win? >> Prove your answer by giving the strategy. >> >The second player can always win if he uses the following strategy: > > Considering the center of the table as the "origin", always > place his penny at a spot reflected through the origin from > where his opponent just placed his last penny. > >Using this strategy, the second player will always have a spot to place his >penny because he is simply mirroring the actions of the first player. This is a very elegant solution, but it is unfortunately wrong. This strategy will work, but it will only work for the first player, after s/he has started the game by playing to the exact center of the table (a move impossible to "mirror"). -- /''`\ Andre Guirard ([]-[]) High Weasel \ x / speak no evil ihnp4!mmm!cipher `-'
matt@oddjob.UUCP (Matt Crawford) (02/19/86)
dim@whuxlm.UUCP (McCooey David I) Sez: >The second player can always win if he uses the following strategy: > > Considering the center of the table as the "origin", always > place his penny at a spot reflected through the origin from > where his opponent just placed his last penny. Hey, David, wanna play the penny game for a $1000 stake? (Sorry, it's my mercenary streak again.) You can go second, and I'll place my first penny exactly at the center of the table. Now what do you do? _____________________________________________________ Matt University crawford@anl-mcs.arpa Crawford of Chicago ihnp4!oddjob!matt
ins_anmy@jhunix.UUCP (Norman M Yarvin) (02/20/86)
> > Imagine a two-player game, in which each of the players begins > > with an infinite number of pennies. There exists a round table, > > and each player in his turn places a penny on the table. (Turns > > are alternated). The game ends when there is no more room on the > > table for any pennies. The person who last put a penny on the table > > is declared the winner. > > > > Question: Given that one of these players has a winning strategy, > > which player (the first, or the second) can always win? > > Prove your answer by giving the strategy. > > > The second player can always win if he uses the following strategy: > > Considering the center of the table as the "origin", always > place his penny at a spot reflected through the origin from > where his opponent just placed his last penny. > > Using this strategy, the second player will always have a spot to place his > penny because he is simply mirroring the actions of the first player. > > Dave McCooey NO! The first player has the winning strategy! He uses the strategy detailed above, except that he places his first penny exactly in the center of the table. -- Norman Yarvin UUCP: seismo!umcp-cs \ ihnp4!whuxcc > !jhunix!ins_anmy allegra!hopkins / BITNET: INS_ANMY@JHUNIX ARPA: ins_anmy%jhunix.BITNET@wiscvm.WISC.EDU "By God, it's Uncle Irwin from the city sewers"
vsh@pixdoc.UUCP (Steve Harris) (02/21/86)
If there is a way one player can always win, it must be the first
player, since only s/he can win when the table is the same size as the
penny.
--
Steve Harris | {allegra|ihnp4|cbosgd|ima|genrad|amd|harvard}\
xePIX, Inc. | !wjh12!pixel!pixdoc!vsh
51 Lake Street |
Nashua, NH 03060 | +1 605 881 8791markb@sdcrdcf.UUCP (Mark Biggar) (02/21/86)
In article <953@houxa.UUCP> qts@houxa.UUCP (J.RAMMING) writes: >Imagine a two-player game, in which each of the players begins >with an infinite number of pennies. There exists a round table, >and each player in his turn places a penny on the table. (Turns >are alternated). The game ends when there is no more room on the >table for any pennies. The person who last put a penny on the table >is declared the winner. >Question: Given that one of these players has a winning strategy, > which player (the first, or the second) can always win? > Prove your answer by giving the strategy. Necessary Assumption: pennies must not overlap, but may touch. Winning strategy for the first player is as follows: 1. Make your first move to the exact center of the table. 2. On each following move place your penny exactly on the other side of of the center penny form your opponents move. If the other player places a penny 3 inches north of the center penny you place one 3 inches south of it. As there will always be a place for you to put a penny you will always place the last one. A much more difficult problem is a winning stratagy if the last person to be albe to place a penny loses. Mark Biggar {allegra,burdvax,cbosgd,hplabs,ihnp4,akgua,sdcsvax}!sdcrdcf!markb
jbs@mit-eddie.UUCP (Jeff Siegal) (02/21/86)
In article <904@whuxlm.UUCP> dim@whuxlm.UUCP (McCooey David I) writes: >> Imagine a two-player game, in which each of the players begins >> with an infinite number of pennies. There exists a round table, >> and each player in his turn places a penny on the table. (Turns >> are alternated). The game ends when there is no more room on the >> table for any pennies. The person who last put a penny on the table >> is declared the winner. >> >> Question: Given that one of these players has a winning strategy, >> which player (the first, or the second) can always win? >> Prove your answer by giving the strategy. >> >The second player can always win if he uses the following strategy: > > Considering the center of the table as the "origin", always > place his penny at a spot reflected through the origin from > where his opponent just placed his last penny. > >Using this strategy, the second player will always have a spot to place his >penny because he is simply mirroring the actions of the first player. Almost. The first player can place his penny _AT_ the center of the table. Then he can always mirror the actions of his opponent. Therefore, the first player can always win. Jeff Siegal - MIT EECS
ewa@sdcc3.UUCP (Eric Anderson) (02/21/86)
In article <904@whuxlm.UUCP> dim@whuxlm.UUCP (McCooey David I) writes: >> Imagine a two-player game... >The second player can always win if he uses the following strategy: > Considering the center of the table as the "origin", always > place his penny at a spot reflected through the origin from > where his opponent just placed his last penny. >Using this strategy, the second player will always have a spot to place his >penny because he is simply mirroring the actions of the first player. Close .. but actually the FIRST player will win. Suppose the second player mirrors the first players moves. Noting this, the first player aviods the origin. Then, when the rest of the table is full, the first player plays the origin, and wins. Winning strategy: Play the origin first, then mirror the other player. Eric Anderson, UC San Diego {elsewhere}!ihnp4!ucbvax!sdcsvax!sdcc3!ewa Home: (619)453-7315 Work: (619)586-1201 White House: (202)456-1414