SOLUTION: Peter and Paul are playing a game involving 60 plastic sticks that are on the table between them. On a turn, a player may remove up to from 1 to 7 sticks from said table. The playe

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Question 1131954: Peter and Paul are playing a game involving 60 plastic sticks that are on the table between them. On a turn, a player may remove up to from 1 to 7 sticks from said table. The player that removes the last stick wins. Peter can't guarantee that he'll win the game unless he goes first and removes n sticks. What is the value of n?
Answer by ikleyn(52906)   (Show Source): You can put this solution on YOUR website!
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Peter and Paul are playing a game involving 60 plastic sticks that are on the table between them.
On a turn, a player may remove up to from 1 to 7 sticks from said table. The player that removes the last stick wins.
Peter can't guarantee that he'll win the game unless he goes first and removes "k" sticks. The value of "k" is?
a) 1 b) 2 c) 4 d) 6 e) 7
~~~~~~~~~~~~~~~~~~ 


Let's analyze it first.


1) Which final state is winning for Peter ?


   The final state "8 sticks remained on the table and Paul should make his move" is winning for Peter.

   Indeed, Paul will take whichever number of sticks from 1 to 7;  then Peter will take the rest and wins.



2)  Which state before the final is winning for Peter ?


    The state before the final, when "16 sticks are on the table and Paul should make his move" is winning for Peter.

    Indeed, Paul will take whichever number of sticks from 1 to 7;  then Peter will take so many to reduce the number 
    of stickers on the table to 8.  By doing it, Peter will provide the winning final state for himself, 
    according to n.1.


3)  Making arguments and thinking in this way, we see that the following states are winning for Peter:


    - having 56 sticker on the table and having Paul's move, Peter can reduce the number of stickers to 48.

    - having 48 sticker on the table and having Paul's move, Peter can reduce the number of stickers to 40.

    - having 40 sticker on the table and having Paul's move, Peter can reduce the number of stickers to 32.

    - having 32 sticker on the table and having Paul's move, Peter can reduce the number of stickers to 24.

    - having 24 sticker on the table and having Paul's move, Peter can reduce the number of stickers to 16.


As we just saw it in n.2 of the analysis above, this state is winning for Peter.


From what we analyzed, we see, that taking 4 stickers of 60 at the first move PROVIDES the winning strategy for Peter.


If Peter will violate this strategy at any of his move, then Paul can intercept the initiative and win the game.



ANSWER.  To win, Peter should take 4 stickers at his first move and then continue further as explained.

         By violating this strategy, Peter gives away the win to Paul.

Solved.



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