MCS Puzzle Contest Posted on April 27th, 2018 by

Hello All,


We have been sending out a weekly teaser problem in our announcements of the Pizza & Problem Solving sessions that the MCS Department has been hosting on Tuesday afternoons. We would like to make this a little more official and turn these into a problem solving contest. The plan is to post a problem (or two) maybe not every week, but every two weeks. We would like to encourage you to work on these and actually submit written up solutions to us. There will be prizes for the best solutions.  Here are the first two problems, one for math majors, the other for non-math majors. Whether you are a math major or not, you are welcome
to work on and submit a solution to either or both problems, but the problems intended for math majors are likely to be more challenging
and/or technical. Solutions are due by May 10 and can be submitted by e-mail or on paper (typed or handwritten, as you wish) to either Dr.
Siehler or to Dr Tuba.

Problem for math major types:

Let x and y be positive integers such that x+y=2018. Suppose that we have an x by y chessboard and each square has a pawn on it (the color of the pawn is irrelevant). When the clock strikes midnight, all of the pawns come alive and each one simultaneously moves to an adjacent square (but not diagonally) and then becomes a lifeless chess piece again. Upon the completion of all these moves, it is still true that each square of the board has exactly one pawn on it. For which values of the pair (x,y) is this possible?

Problem for non-math major types:

The four cards shown below have an interesting property:  the upper numbers run from one to four, and — this is the interesting part — each card’s lower number tells how many times the upper number occurs anywhere in the entire set of cards. `1′ occurs 3 times; `2′ occurs 1 time; `3′ occurs 3 times; `4′ occurs 1 time.   mazing! Your problem: Construct a set of ten cards with upper numbers 1–10 and the same “self-counting” property.