College of Arts and Sciences

Department of Mathematics

Problem of the Fortnight

Current Problem


One hundred green-eyed logicians have been imprisoned on an island by a mad dictator. Their only hope for freedom lies in the answer to one famously difficult logic puzzle. Can you solve it? Alex Gendler walks us through this green-eyed riddle.



Previous Problems


Suppose I offer to give you one of three prizes - Prize A, B, or C. Prize A is the best, B in the middle, and C is the least best. You are to make a statement; if the statement is true, then I promise to award you either Prize A or Prize B, but if your statement is false, then you get Prize C. 

Of Course it is easy for you to be sure to win either Prize A or Prize B; all you need say is "Two plus two is four." But suppose you have your heart set on Prize A - what statement could you make which would force me to give you Prize A?


December 3rd/January 14 & 28: The Garden Winner: Alden Bradford

There exists a garden with flowers in it. There is at least one red flower, one blue flower, and one yellow flower. If you pick three flowers from the garden, then you will always pick a red and blue flower. If you pick three flowers, then what is the probability that one of them is yellow?


November 19th: Interval (1,12) Winners: Noah Austin and Sharif Ibrahim 

Let  d_1, d_2, ..., d_12  be real numbers in the open interval  (1,12).  Show that there exist distinct indices  i, j, k  such that  d_i, d_j, d_k  are the side lengths of an acute triangle.


November 5th: Bridge 17 minutes problem Winner: Noah Austin

There is a bridge, four friends, it's dark and they have one torch. You need to devise a plan to get all friends across but they can only move when they have the torch and only two friends can go across the bridge at a time. One friend takes 10 minutes, another 5, the other 2 and the last takes 1 minute to go across. Everyone has to get to the other side by 17 minutes. 


October 22nd:n! Winner: Noah Austin 

Prove that n! is not a divisor of n^n whenever n is an integer greater than 2.

n:     1  2   3    4     5  

n!:    1  2   6    24  120

n^n: 1  4  27  256  3125


October 8th: The ring, the lockbox, and the thief Winner: Michael Newsham

You and I each have one lock and a matching key. I want to mail you a box with an expensive ring in it, but any box that is not locked will be emptied before it reaches its recipient. Note that we each have keys to our own lock but not the other lock. How can I safely send you the ring?

Answers are due by meeting time on October 8th, and can be submitted in writing to the Math Club mailbox on the first floor of Neill or brought to the meeting. Of those who submit a correct answer, one lucky winner will receive a $25 gift certificate for Bruised Books.


March 7: Natural Numbers and Their Digits

Show that for all natural numbers   n, at least one of two numbers,  n  or   n +1, can be represented in the following form:  k+ S(k ) for some integer   k, where   S(k) is the sum of all digits in   k. For example,  21= 15+ (1+5). Answers are due by 5:00 PM on Friday, March 7. Please show all of your work. Good Luck!

February 14: Finding the Center of My Heart

A heart shape is formed from a square with sides 2 units long and two half-circles of radius 1 unit (see figure). Where should the heart be placed in the Cartesian plane so that the origin is at the center of the heart, i.e. so there is the same amount of area in each of the four quadrants? Answers are due by 5:00 PM on Friday, February 14. Please show all of your work. Happy Valintines Day!










Next Meeting

The next meeting is scheduled for September 6th, 5:10 pm in Neill 216. Yes, there will be pizza!