Problem Set 3 : Issues of Parity

Due 02/17/04

1.  The numbers 1 through 10 are written in a row. Can the signs "+" and "-" be placed between them in such a way that the resulting expression adds up to 0?                           



2.   a) We have a container that contains 9 quarts and another that contains 4 quarts. We fill these containers by immersing them in the river. How can you put exactly exactly 6 quarts of water into the large container?
     b) Now suppose that you have a 6 quart container and a 4 quart container.How can we use them to fill one of the containers with 3 quarts of water?



3.   Below is a picture of the town of Konigsberg, Prussia (now Kaliningrad, Russia). There are seven bridges connecting the two islands in the middle of the river and the sides of the river. The Swiss mathematician Leonhard Euler (1707-1783) traveled there and wondered if it would be possible for a person to walk around the city and cross each of the seven bridges exactly once.
    
   a)  Is this possible?  b) Suppose that the top left bridge in the picture is closed.  Now is it possible?



4.   If n is a positive integer such that 2n+1 is a perfect square show that n+1 is the sum of two successive perfect squares.



5.   Twenty-five men and twenty-five women are seated at a round table. Show that there must be at least one person at the table who is seated between two men.



6.   Remove the lower left corner square and the upper right corner square from an ordinary 8-by-8 chessboard. Can the resulting board be cover by 31 dominoes? Assume each domino will cover exactly two adjacent squares of the board.