Professor Simonson History of Math Ingenuity
Spring 1998 MA 149
Write your solutions to these problems as a group. They will be
discussed in detail during next class. If you cannot solve a
problem then report on your group discussion and explain where
you got stuck. You will always have the opportunity of rewriting
an assignment after class discussion.
1. In Aristotle's proof that the square root of 2 is irrational (cannot be written as a fraction), he uses a "lemma" (helper theorem) that if a number squared is even then the original number is even.
How many times does he use this lemma and where in the proof is it used?
Prove this lemma by a clear convincing explanation in English.
2. Use the Pythagorean theorem to solve the following puzzle:
Given a square, draw a square with exactly twice the area.
3. There is a game that is used to teach kids odd and even numbers, multiplication and addition. It goes like this:
a. Start with a number equal to the kid's age.
b. If the number is even divide it by 2 to get a new number.
c. If the number is odd multiply it by 3 and add 1 to get a new number.
d. Go to step b.
Experiment and guess a theorem about this game.
4. Recall that a triangular number is one of the form 1+2+3+...+n. For example, the first four triangular numbers are 1, 3, 6, 10.
a. Can every square be expressed as a sum of two triangle numbers? If so, explain why and prove it. If not, give a counterexample.
b. Can every integer (whole number) be written as a sum
of two triangular numbers? If so, explain why and prove it.
If not, give a counterexample.
5. We discussed an elegant proof due to Euclid that there are an infinite number of prime numbers. In this proof by contradiction, he constructs a new number.
a. Is his new number odd or even? Use this to prove that there are an infinite number of odd prime numbers.
All integers can be divided into 3 groups:
i. Multiples of 3.
ii. Multiples of 3 remainder 1.
iii. Multiples of 3 remainder 2.
b. Which of the three groups do you guess contain an infinite number of prime numbers?
c. In which one of the three groups does the number constructed by Euclid, belong? Explain.
d. Try to use Euclid's method to prove that there are an infinite number of primes in this group. Explain what part of the logic doesn't work.
e. Modify Euclid's proof so that in constructing the number,
instead of adding 1, we subtract 1. Then answer problems c and
d again. This time Euclid's proof is salvagable. Try to work
out the details.