Challenge of the Week: A Million Tokens

By Department of Mathematics

Deadline: 11:59 PM on Sunday, Nov. 17

The Rules: 

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 or put it in Bela’s departmental mailbox by the above deadline. Consistently successful participants will receive the Paul Mugabi Mathematics Problem Solving Award.

The Problem: 

Consider a sequence of tokens numbered 1,2,3,…,1000000. Each token has one side colored orange and the other side colored blue. Originally, each token has its orange side facing upward. At time t = 1, you turn each token upside down (and thus each token will have its blue side facing upward). Then, at time t = 2, you turn every second token upside down (and therefore, tokens numbered 2, 4, 6, etc. will have their orange side facing up) . At time t = 3, you turn every third token, and so on, until time t = 1000000, when you turn every 1000000th (that is, the last) token upside down. How many tokens will have their orange side facing up at the end?

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Author: Gettysburgian Staff

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