Pass 300
Let be a positive integer. Dan and Sam play a game in which the first to start says the number and the one who's next must say a multiple of the previous number, that is between the previous number and its square. They cannot repeat a number even if the number was said by the other. Also, the said number cannot be greater than 300.
For example, Dan begins saying , then Sam can reply 6 or 9, but not 3, because Dan said it before.
The winner is the one who cannot say a number in his turn. If Dan begins, and both players play optimally, for how many numbers does Dan win?
This is the nineteenth problem of the set Winning Strategies .
The answer is 80.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Easy to see that If k from 151 to 200, Sam wins
Therefore, if k is from 76 to 150, Sam is forced to say a number from 152 to 300, thus Dan wins.
If k from 9 to 75, Sam can say a number from 76 to 150 and wins the game
If k = 5 to 8, Sam is forced to say a number from 10 to 64 and loses the game
If k = 3 to 4, Sam can say 6 or 8 and force Dan to say from 12 to 64 and wins the game.
If k=1, Sam wins
k=2, Dan wins
Total cases that Dan wins is 80 case