Student Question

In a room of 144 people who are joined by n other people who are each carrying k coins, when these coins are shared among all n + 144 people, each person has 2 of these coins. What is the minimum possible value of 2n + k?

Expert Answers

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This question asks the minimum number of new people (denoted by n in the problem) that solves the problem to generate a revised number of people once n new people are added to the original 144. To complicate matters, each new (or n) person carries a coin or multiple coins and the problem requires that once the n new people are added, each person in the room gets a presumably equal distribution of the coins so that the final tally of people in the room each has two coins. We can construct a spreadsheet to determine the minimum number of people (n) that are introduced into the room with the coins.

However, from there, there are some qualitative, rather than quantitative, assumptions to be made because we are not told the maximum number of coins each n incremental coins could carry. For instance, we can solve the problem by assuming n is 1. This means that when the new person enters the room, he or she brings the total number of people in the room to 145 (the original 144 plus 1). If the number of people in the room is 145 and each person has two coins, that implies that n—the one additional person who entered the room—brought 290 coins. While this is certainly possible, it seems unlikely. Therefore, n is probably greater than 1 (one). We can also discard any answer where the number of coins per n results in a fraction (as illustrated when n equals 5 or 7), as we know that people do not carry fractions of coins. An iterative process can be used to find the minimum possible value of 2n + k.

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