2015 Newsletter Puzzle and 2014 Solution

2015 Problem

Let’s call a positive integer quintish if it is divisible by some perfect 5th power (bigger than one). Do there exist 2015 consecutive positive integers each of which is quintish? Why or why not?

We invite you to submit your solution by email to sherman1@calpoly.edu. We will recognize those submitting correct solutions to the newsletter puzzle in next year’s newsletter.

2014 Problem

An elderly king with no heir wanted his kingdom to go to the smartest man in the land. He rounded up the three knights with the highest grades in Real Analysis with Applications to Jousting and sat them down at a table. “I need to find out which of you is the most intelligent, said the king, "so I will paint a dot on each of your foreheads. It will either be a black dot or a white dot, but you will not know which. You will have to figure it out using logic. The only hint I will give you is that I will not give all three of you a white dot. As soon as you have figured out what color dot you have, you should come to me and explain your reasoning to claim your prize.”

At this point, the king painted a black dot on each of their foreheads in such a way that none knew what color they wore, and then left to wait for the winner in the throne room. The three men sat for some time, each looking at the other two (and in particular, at the dots painted on the foreheads of the other two), without communicating in any way. Eventually, one cried, "Eureka! I must have a black dot!" He ran out to tell the king and, sure enough, claimed his prize.

Solution

Let’s call the knights Arthur, Ben and Camelot and say Arthur was the one who deduced his spot was black. Arthur reasoned as follows: suppose his dot were white. Then, say, Ben would see one white dot and one black dot. But Ben could then say that his dot would have to be black, since otherwise Camelot would see two white dots and be able to deduce immediately that his own dot is black. Since Camelot has kept quiet this can’t be the case. But then since Ben has also kept quiet, it must be that Arthur does not have a white dot to begin with.

Solvers

Don Gibson (B.S., 1980), Kyle Griffin (student), Tom Kremen (B.S., 1969), Craig Nelson (B.S., 1989), Gere Sibbach (B.S., 1973; M.S., 1975), Lumin Sperling (B.S., 2014), Hal Sudborough (B.S., 1965)