Professor Simonson History of Math Ingenuity

Spring 1998 MA 149

Final Examination - 100 points - Take Home

1. (10 points) Mathematics

In reviewing the book "Fermat's Enigma", a more complete written version of the film you saw, the reviewer concludes: "I strongly recommend this book to anyone wishing to catch a glimpse of what is one of the most important and ill-understood, but oldest, cultural activities of humanity." (New York Times Book Review, Sunday November 30, 1997)
Please comment on this quote with resepct to your own experience with mathematics as it has changed through this course, and in contrast to the thoughts you wrote in the very first assignment.

2. (10 points) Nim Once More

Find a kernel and explain how to always win the following Nim game when you are allowed the choice of whether to move first or second. The game starts with an odd number of sticks. You may take 1, 2 or 4 sticks each turn and the goal is take the last stick.

3. (15 points) Pythagrean Triples

a. Prove or disprove: A Pythagorean triple (x, y, z) is either primitive (none of the 3 numbers have a common factor) or else they all have a common factor.

b. Prove or disprove: In any primitive Pythagorean triple (x, y, z), where x<y<z, either x or y must be divisible by 3.

c. Prove or disprove: In any primitive Pythagorean triple (x, y, z), where x<y<z, either y or z must be divisible by 5.

4. (10 points) Number Systems

a. In the 13th century in western Europe, it was common practice to use a mixed positional number system. Base 10 was used for the whole number part, and base 60 for the part after the decimal point. Show how they would write the binary number 110001.0101

b. In the Old Testament Genesis 41:49, Joseph is in charge of the agriculture in Egypt. It says that at the end of the famine, he "stopped counting all the grain because there was too much to count". What do you think this means in terms of the Egyptian number system?

5. (10 points) Euclid and Proofs

Prove that if each pair of opposite sides of a 4-sided figure are equal, then the pairs of opposite angles are also equal, and the adjacent angles sum to 180 degrees.

6. (8 points) Sums of Squares - Conjectures

Although every number can be written as a sum of 4 squares, NOT every number can be written as a sum of two squares. By experimenting, find a conjecture that describes exactly which numbers can and which cannot be written as a sum of two squares. Hint: Look at numbers in groups of 4.

7. (7 points) Pentagonal Numbers

Does the infinite series of fractions with 1 in the numerator and the pentagonal numbers in the denominator converge or diverge? Prove your answer. (Hint: Compare it to another series).

8. (10 points) Pythagorean Theorem

Use the diagram on page 122 of Journey Through Genius assuming that ACB is a right angle, and prove the Pythagorean Theorem, by computing the area of triangle ABC in two different ways. (Note: Do NOT follow the book's idea on page 128. I am asking for something easier and different than what he does). (Reference: Letters in Mathematics Teacher, Vol. 84, No. 2, pp. 148, February 1991).

9. (10 points) Sums - Borrowing Methods

Johann Bernoulli's proof of the divergence of the Harmonic series (H) starts with the reciprocals of the triangle numbers which we know add up to 1, and he proceeds to construct an infinite number of infinite series, and adds them all up on both sides to get that H = 1 + H, after which he concludes that H is infinite.

Use Bernoulli's method starting with the geometric series 1/2 + 1/4 + 1/8 + ... which also adds up to 1. This time we don't get H=1+H. What result DOES do we get?

10. (10 points) Probability and Counting

What is the chance given 10 people in a room, that three or more were born on the same day? Explain all your resoning.