Dr. Bennette Harris – Problem Set 2

Here is the continuation of Dr. Bennett Harris’ problems to warm us up before his lecture on September 27th. Congratulations to Dominic for being the first to answer the previous set of problems in the comments. Another happy discovery is the fact that WordPress, which hosts this site, supports the use of code!

Problems

A solution for each of the following should either give the correct answer, or a technique for determining the answer in reasonable time with the assistance of a calculator.

6. Crack this secret message: “uifsfaxjmmacfabmnptuaopaqsppgtaupebz”

7. Code the alphabet a=0, b=1, …, z=25, space=26. We will encrypt each letter of the alphabet by mapping it to the letter 3 farther along in the alphabet. Thus, a maps to d, b maps to e, and so forth. The message “a bat is fat” would be encrypted as “dcedwclvciodw” in this scheme. Find a mathematical function f(x) such that y=f(x) gives the correct encryption for every value x = 0,1,2,…,26. Test your function by encrypting “the toy boat floats”.

8. The purpose of these two questions is to provide points of comparison for the questions that follow:
How many particles are there in the known universe?
How many microseconds have elapsed since the beginning of the universe?

9. How long (approximately will do) would it take to calculate at 1,000,000 calculations per second? How many digits does the answer have? If p is 200 digits long, and if q is 200 digits long, about how many digits are there in pq?

10. The number – 1 is known to be prime. Estimate how long it would take a computer to demonstrate this by repeated division, at 1,000,000 divisions per second.

Read Full Post »

Dr. Bennette Harris – Problem Set 1

This Saturday’s (9/27) Math Circle lecture by Dr. Bennett Harris on “Computer Data Encryption – Decrypted” promises to be a great combination of number theory and applied math. To warm up the students, Dr. Harris forwarded some problems that I will post in two parts. Feel free to offer solutions in the comments or to just work them on your own. Full solutions will be made available at the next Math Circle.

Problems

A solution for each of the following should either give the correct answer, or a technique for determining the answer in reasonable time with the assistance of a calculator.

1. How many prime numbers are there? Prove your answer.

2. What is the largest number you must test to demonstrate that 83 is prime?

3. What is the smallest 5-digit prime? How could you find this number?

4. Is 1234567890 prime? What about 123456789?

5. Assume an alphabet with 26 letters (plus a blank space). A substitution cipher is a one-to-one mapping of this alphabet onto itself. How many such substitution ciphers are there?

Read Full Post »