By Department of Mathematics
Deadline: 11:59 PM on Sunday, Sept. 15
The contest is open to everyone. Individuals or teams of at most three members may submit solutions. Do not discuss the problem with anyone other than members of your team. You may use any source, written or electronic, but all sources must be properly cited. You may use any computational tools. Your solution will be graded on a 0–4 point scale. All your assertions must be completely and fully justiﬁed. At the same time, you should aim to be as concise as possible; avoid overly lengthy arguments and unnecessary components. Your grade will be based on both mathematical accuracy and clarity of presentation. Either send your solution to firstname.lastname@example.org or put it in Bela’s departmental mailbox by the above deadline. Consistently successful participants will receive the Paul Mugabi Mathematics Problem Solving Award.
At a college event, 28 chairs, with 14 of them orange and another 14 blue, are arranged in a row in such a way that neighboring chairs are of diﬀerent color. Initially, n students are seated on the ﬁrst n chairs, and 28−n professors are seated on the last 28−n chairs (n is a positive integer less than 28). Your goal is to reverse the ordering so that the 28−n professors are seated on the ﬁrst 28−n chairs and the n students are seated on the last n chairs. You are allowed to ask a professor and a student to switch places, but only if they are seated on chairs of the same color. Find, with proof, all values of n for which this is possible.