Question 1188391: 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
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20054)   (Show Source):
You can put this solution on YOUR website!
Peter's object is to leave Paul with 8 sticks the last time. Because Paul
can't then pick them all up. So regardless of how many Paul removes, Peter
will be able to remove the rest, for there will be 7 or less.
So to leave Paul with 8 the last time Peter must have left him with 16
before that. For instance, if Pau1 removes 1, leaving Peter with 15, Peter will
then remove 7 and leave Paul with 8. Or, say, if Pau1 picks up 7, leaving Peter
with 9, Peter will then pick up 1, leaving Paul with 8.
So to win, Peter must leave Paul with a multiple of 8 each time. Peter can
always do that if he starts first. Beginning with 60 sticks, Peter must leave
Paul with 56, the largest multiple of 8 not exceeding 60. So Peter must begin
by first removing k=4 sticks, to leave Paul with 56.
[BTW, if Paul starts first, and doesn't know the trick, then if Paul at any time
leaves Peter with anything other than a multiple of 8, Peter can then win
because all he has to do is remove enough to leave Paul with a multiple of 8.]
Edwin
Answer by ikleyn(52776)  (Show Source):
You can put this solution on YOUR website! .
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
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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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