Chris Watts. Problem Solving. DEFENSE OF THE THESIS 2k8. Chris Watts. Acknowledgement. To Kathleen Lewis, Lynn Carlson, Magdalena Mosbo, and Mary Harrell for consistently listening to and encouraging me through my mathematical insecurities.
To Kathleen Lewis, Lynn Carlson, Magdalena Mosbo, and Mary Harrell for consistently listening to and encouraging me through my mathematical insecurities.
To the whole Oswego math department, for helping me mature into a growing mathematician.
The average (non-problem-solver) math student is like someone who goes to a gym three times a week to do lots of repetitions with low weights on various exercise machines. In contrast, the problem solver goes on a long, hard backpacking trip. Both people get stronger. The problem solver gets hot, cold, wet, tired, and hungry. The problem solver gets lost, and has to find his or her way. The problem solver gets blisters. The problem solver climbs to the top of mountains, sees hitherto undreamed of vistas. The problem solver arrives at places of amazing beauty, and experiences ecstasy that is amplified by the effort expended to get there. When the problem solver returns home, he or she is energized by the adventure, and cannot stop gushing about the wonderful experience. Meanwhile, the gym rat has gotten steadily stronger, but has not had much fun, and has little to share with others (page x).
1. Let k ≥ 1 be an integer. Show there are exactly 3k-1 integers n such that:
Austrian-Polish Mathematics Competition 1996
2. We call an integer m “retrievable” if for some integers x and y, m = 3x2 + 4y2. Show that if m is retrievable, then 13m is retrievable.
AMTNYS, Jan. 2007
3. At ABC University, the mascot does as many pushups after each ABCU score as the team has accumulated. The team always makes extra points after touchdowns, so it scores only in increments of 3 and 7. For each sequence a1, a2, …, an where each ak = 3 or 7, let P(a1, a2, …, an) denote the total number of pushups the mascot does for the scoring sequence a1, a2, …, an. For example, P(3,7,3) = 3 + (3 + 7) + (3 + 7 + 3) = 26. Call a positive integer k accessible if there is a scoring sequence a1, a2, …,an such that P(a1, a2, …,an) = k. Is there a number K such that for all t ≥ K, t is accessible? If not, prove it, and if so, find K.
Pi Mu Epsilon, Spring 2007
4. Players 1, 2, 3, …, n are seated around a table and each has a single penny. Player 1 passes a penny to Player 2, who then passes two pennies to Player 3. Player 3 then passes one penny to Player 4, who passes two pennies to Player 5, and so on, players alternately passing one penny or two to the next player who still has some pennies. A player who runs out of pennies drops out of the game and leaves the table. Find an infinite set of numbers n for which some player ends up with all n pennies.
5. What is the expected length of a standard NHL shootout where the probability of each shooter scoring a goal is 1/3?
AMTNYS: Jan ’07
As a student…
As a prospective teacher…