You and your friend are playing a game.
There is a round table and unlimited quarters. You and your friend take turns to place a quarter on the table. The quarter should be placed flat on the table and should not overlap any other quarter. You cannot move the coins already present on the table. Whoever places the last quarter wins.
If you get to choose who goes first, whom will you choose and what will be your winning strategy?
Solution:
You choose to play first and place the coin in the center of the table.
When your opponent puts a coin, you put your coin in the spot directly opposite to his coin. E.g. your opponent has put the coin at a point which is distance x from the center, consider a diameter of the circle passing through this point. Then put your coin on this diameter at a distance x from the center but in the opposite direction.
With this strategy, wherever your opponent puts his coin, you will always have the described spot empty and you will get to put the last coin.
Winner: None :(
Suppose the table radius is "R" and coin radius is "X". The number of coins which can be placed on table are
ReplyDelete(pi * R^2)/(pi * X^2) = N (Ignoring the decimal part if any.
If N is even then I will decide to play second otherwise first.
What about the table space wasted between coins?
ReplyDeleteAlso, if, in the end, space for only two adjacent coins is left and your opponent places a coin in the middle, you cannot play even thought there is space left according to your calculations.
Assume that you cannot move the coins already present on the table. Sorry for omitting this minor but crucial detail.
Hmm did not think about it that way. Will need to think more.
ReplyDeletedo we know the diameter of the table and the coin?
ReplyDelete