Practice Quiz 2, M203E for Quiz 2 Wednesday Oct 19, 2011

Instructions: The quiz is open book (any books) and open notes (any notes or written material).

[1] Suppose a man has glossy paint which comes in three colors, and flat paint which comes in two colors. 

(a) How many different ways are there for him to paint his house? Explain your answer.

(b) How many different ways are there for him to paint his house and his daughter's house, if he wants glossy paint on his house and his daughter wants flat paint on her house? Explain your answer.

[2] Suppose a developer builds houses painted in one of 6 colors, using any of 4 floor plans, with either electric, gas or geothermal heat. 

(a) How many different kinds of houses can the developer build? Explain your answer. 

(b) In a subdevelopment of 50 houses, must there be at least two identical houses? Explain why or why not.

(c) Suppose the subdevelopment has 300 houses. What is the largest number of houses that you can be sure have to be identical knowing nothing more than your answer to (a) and that there are 300 houses altogether? Explain your answer.

[3] (a) Write 65 as a sum of Fibonacci numbers, no two of which are consecutive Fibonacci numbers.

(b) John and Jane play a game of Fibonacci nim, starting with a total of 65. John goes first. (Recall the rules: each player can take away as much of what is left as that player wants with two exceptions: the first player has to leave something for the second player, and after that each player can take at most twice what the previous player took. Whichever player gets stuck with 0 is the loser.)

	(i) How many should John take away in order to be sure to win? Explain your answer.

	(ii) If John takes away only 1, can Jane win? Explain.

[4] An arithmetic sequence of numbers is a sequence where the difference being consecutive numbers is always the same. For example, for 1, 2, 3, ..., 50, the difference is always 1 since each number is always 1 bigger than the previous number. For 4, 7, 10, ..., 70, 73, the difference is always 3, since each number is always 3 bigger than the previous number. Gauss's trick works to add up any arithmetic sequence of numbers. Recall how Gauss's trick works: write the sum and below it write it in reverse order.

1   + 2  + ... + 100
100 + 99 + ... + 1

Add up the numbers in columns (each column here gives 101), multiply by the number of columns (100*101=10100) and divide by 2, which gives in this case 5050.

(a) Explain how to use Gauss' trick to add up the whole numbers from 1 to 75.

(b) Explain how to use Gauss's trick to add up the whole numbers from 200 to 320.

(c) Explain how to use Gauss's trick to add up the whole numbers from 9 to 89, counting by 5's; i.e., add up 
9 + 14 + 19 + ... + 89.

[5] (a) What is the remainder if you divide 32117 by 9? Show your work or explain how you get your answer. 
 
(b) What is the remainder if you divide 32117 by 10? Show your work or explain how you get your answer. 

[6] (a) Find the postnet check digit for the math department zip code: 68588 0130. 

(b) Given that the postnet check digit is 4, fill in the following zip code's missing digit: 12574 3_11. 

[7] (a) Find the check digit (denoted here by x) for the following UPC code: 0 53000 15108 x. 

(b) Find the missing digit in the following UPC code: 0 15838 _0001 5.