Mathematical Reasoning: Class Materials

Introductory problems #

Here are some unusual problems. In addition to solving them, we will think about how to write down our answers. I don’t want to hear “I know how to do it” - I want to see something written as well. A good problem solver is organized in their thinking!!

  1. Prime numbers? Is \(n^2 + n + 41\) a prime number for all natural numbers \(n\)? Make sure to give a reason for your answer!

  2. Philosophers’ hats. Each of three philosophers, whom we will call A, B and C, is given the opportunity to obtain their freedom from a common jail cell by solving the following problem.

The jailer presents a collection of three red hats and two white hats, announcing the assortment out loud. He then turns out the lights and puts one hat on each philosopher’s head, discarding the left over hats. The lights are turned on, and the three philosophers are permitted to look around. They are told that anyone able to deduce the color of his own hat will be allowed to go free.

Philosopher A looks around, thinks, and says out loud that he cannot determine the color of his own hat. Philosopher B looks around again, thinks, and says that he too cannot determine the color of his hat. Philosopher C, who is blind, thinks about what he has heard, and says, “Aha, I know the color of my hat.”

What is the color of philosopher C’s hat?

  1. Weighing marbles. Eight marbles all have the same color, size and shape. Only one marble differs in weight from the others. Using a balance scale (no weights available), find the heavy marble using no more than 2 weighings. Now write down a solution to this problem. Feel free to use words!

  2. Physicists and Mathematicians. Three physicists and three mathematicians are gathered together on one side of a river, and wish to get to the other side. They have a row boat which holds at most two people. When the boat reaches the shore, everyone in the boat gets out. How can they all get across if at no time more mathematicians than physicists can be left together anywhere? Now write down a solution to this problem.

  3. Physicists and Mathematicians, part 2. Three physicists and three mathematicians are gathered together on one side of a river, and wish to get to the other side. At no time can more mathematicians than physicists be on one side of the river. They have a row boat which holds at most two people. How can they all get across if all three physicists but only one mathematicians are able to row the boat?

  4. Weighted coins. Suppose you have twelve coins, all of which look exactly alike. One of the coins, however, is counterfeit, weighing either slightly more or slightly less than the others. Using only a balance scale (no weights available), how could you identify the counterfeit coin and indicate whether it is lighter or heavier than the rest, in no more than three weighings? As a warm up, see if you could do this in at most three weighings, if you knew that the counterfeit coin was heavier than the other coins.