Updates from the past two sessions

Here are the lists from last two sessions:  Probs-2014-10-10, Probs-2014-10-17.

Some hints to these problems are below the fold…

• Want to prove that if three integers have no common divisors and , then is a square.An example of such a situation is . Note that while pairwise these integers are not relatively prime, the three together do not share any common divisors. This turns out to be a common feature of all such examples. Clearing denominators we rewrite the given relation as . Every prime divisor of divides either or , but not both… Following this line of thought, we can write , , and , where the pairs $latex (m,n)$, , are relatively prime. We also have the relation and need to prove that this is a square.
•  As a hint to the prisoners problem i can suggest a (not quite true :)) personal story. All my 4 kids do not like to close the cap on the toothpaste after they brush their teeth in the morning. Surprisingly, they also do not like to use the toothpaste if it is open. So each of us occasionally walk into the bathroom; whenever I see an open toothpaste, I close the cap; if one of the kids sees a closed toothpaste, they use it and leave open… When can I be finally sure that all 4 of them brushed their teeth?
• For the pirates and coins, one has to reason backwards. Let’s rename the pirates according to their age, where is the youngest and is the oldest. For example, if   and are the last standing, then can take the whole loot, because he alone has 50% of the votes. Thus at the previous stage, when   were alive, would accept any offer of one coin or more (why?) and guarantee that such a suggestion by is accepted. Continue…