Mathematics Problem Of the Week

Spring 2006

POW #10

Curves and Dots

A two-player game called “Curves and Dots” is played as follows:

o       Start with n dots.

o       On a player’s turn, the player must draw a curve either connecting any two dots OR from one dot to itself.  A new dot is then placed at the curve’s midpoint.

o       The rules are that no curve can cross any other curve, and no dot can have more than three curve lines coming from it.

o       A player loses when the player cannot make a valid move.

EXAMPLE with n = 2: 

Start with 2 dots.

First move: (Player 1)

Second move: (Player 2)

 

Third move: (Player 1)

Fourth move: (Player 2)

NO VALID MOVES.

(Player 2 wins.)

For each case starting with n=1, n=2, n=3 and n=4 dots, answer the following questions: 

(a)  How many initial possible moves are there? 

(b)  What is the maximum number of moves for a winner to be determined? 

(c)  Does Player 1 or Player 2 win?  (Assume Player 1 makes the first move.)

then see if you can answer the questions for starting with an arbitrary n dots.

             Due Friday, April 14th Noon

Solutions should be submitted to Dr. R. Lock's mailbox in the Math office or sent via e-mail to rlock@stlawu.edu.