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Exercise 1:
Your mind may be spinning after your
introduction to mathematical induction. To get a better understanding of what is really
happening with these proofs, check out this real-life example... It's a game called Splitting Piles. Get
together with one of your classmates and start with several different piles of toothpicks
to see if this game really works. Once you believe it does, read and reason through the
proof. |
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Exercise 2:
Pascal's triangle was invented by the Chinese,
but mathematician and philosopher Blaise Pascal (often called the Father of Probability)
found so many patterns in the triangle of numbers that it has since been named after him.
Let's see how you do at finding some patterns! Go to this Pascal's
Triangle site and answer all the questions, then give # 60 in 8.4a try! (Note: This might be a good exercise to do
with a group of your classmates.) |
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Exercise 3:
Some calculators (mostly the basic scientific
type) can only handle up to 69!, while others (like graphing calculators) can compute up
to 449! There's actually a site that is appropriately called 999 Factorials. To gain a
true appreciation for how large these numbers get (and how fast they grow), check this
site out and answer this question:How many digits is in 999!? |
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Exercise 4:
Figuring out puzzles and sequences can be
great fun -- when you're not being tested on it! Go to Sloane's On-Line Encyclopedia of
Integer Sequences and try some of their Puzzle Sequences.
Warning: They can be addictive. |
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Exercise 5:
You are now living in the age of the computer
-- You're even staring at one right now! What goes along with computers? Passwords.
Whether you are at home or at school, you had to enter a password to access the Internet
to do this assignment. Go read about Unix
computer system security and their recommendations on passwords. (For the following, assume there is a
maximum password length is 10 and the system is case-sensitive: "a" is read
differently than "A".)
How many possible passwords are
there if you are allowed:
| a) |
Only upper-case letters? |
| b) |
Only upper and lower case letters? |
| c) |
Only upper and lower case letters
and the digits 0-9? |
| d) |
The conditions in part c with
blanks allowed? |
| e) |
The conditions in part d with
characters: @, #, $, %, &? |
(answer)
Which conditions would you rather
have your system use?
Which conditions would a hacker rather you use? |
