Dr. Anita Burris
Math 683 Some Interesting Math Problems
Fall 96
In Math 683 you will be introduced to the nature of theoretical mathematics. Topics
include logic, sets, concept formulation, and methods of proof. The following problems
(from Discrete Mathematics with Applications by H.F. Mattson, Jr.) are some
examples which at first may appear to be atypical mathematical problems. Following
completion of this course, you should be able to solve these types of problems as well as
many other types of problems requiring mathematical reasoning and/or proof. See section
20 of the course text (Doing Mathematics by Galovich) for more examples of the
types of problems that we will consider in this course.
(1) Assume items (i) - (iii). Is (iv) a valid conclusion?
(i) If Jones is transferred to North Africa, then Smith is going to a new post.
(ii) Brown will not be appointed undersecretary if Jones is transferred to North Africa
or Smith is going to a new post.
(iii) Smith is not going to a new post.
(iv) Therefore, Brown will be appointed undersecretary.
(2) In a Transylvania factory there are four types of workers: (i) sane humans; (ii) insane
humans; (iii) sane vampires; (iv) insane vampires. Whatever a sane human says is true;
whatever an insane human says is false; whatever a sane vampire says is false; and
whatever an insane vampire says is true.
(a) If a worker says "I am human or I am sane," what type is he/she?
(b) If a worker says "I am not a sane human," what type is he/she?
(c) If a worker says "I am an insane human," what type is he/she?
(3) (Russell's paradox) Define set A by A = {X | X is a set, X X}. Is A A?
(4) All people have the same eye color - a proof by mathematical induction.
Basis step: S(1) is true since one person has the same eye color as him/herself.
Inductive step: Suppose S(n) is true. Then every set of n people have the same eye color.
From a room of n+1 people, remove one person (Fred). By the induction hypothesis, the
remaining n have the same eye color. Now put Fred back in the room, and remove
someone else (Betty). Again by the induction hypothesis, the n people in the room have
the same eye color. Since Betty's eye color matched the others before, when she reenters
the room, her eye color will still match the others. Hence the n+1 people all have the
same eye color. Thus S(n) S(n+1), and by induction all people
have the same eye color.
Find the fallacy in this argument.
(5) Guess and prove a formula for 1 - 2 + 3 - 4 + ... + n(-1)n-1.
