Practice Quiz 1 covering Chapters 1 and 2

Quiz 1 Friday January 26 will consist of 6 questions. Each one will be like one of the 9 questions below. Each question is 5 points (for a total of 30 points on Quiz 1).

Instructions: Answer each question, and explain your answer. An answer alone is not enough for full credit. Your explanation must be clear and show how to get the answer. The actual exam will have a selection of 6 of the following problems, with some changes in the details of the problem.

1. The UPC code on a Kleenex cents off coupon is 5 36000 51031 "?" (the 5 means it's a coupon, the 36000 gives the manufacturer, the 51031 is item specific and the "?" is supposed to be the check digit). What should the check digit be? (Remember: the UPC code is such that when you add every other digit starting with the first, triple the result and then add the remaining digits, you get an even multiple of 10.)
• First, let's add every other digit starting with the first, triple the result and then add the remaining digits: 3(5+6+0+5+0+1)+3+0+0+1+3 = 61. Since we want to get an even multiple of 10, we need a check digit of 9; thus ? = 9.
• (a) Determine the remainder when 145 is divided by 9.
• Recall that the remainder can be found by casting out 9's, so the remainder is 1. Alternatively, 9 goes into 145 evenly 16 times, so the remainder is what's left over; i.e., 145 - 9*16 = 1. Yet another way is to round down the fraction: 145/9 = 16.111... This rounds down to 16, so the remainder is 145 - 9*16 = 1.
• (b) Determine the remainder when -145 is divided by 8.
• Round down the fraction: -145/8 = -18.125. This rounds down to -19 (remember, -19 is less than -18.125), so the remainder is -145 - 8*(-19) = 152 - 145 = 7.
2. Determine the remainder when 2637583736455264957563 is divided by 9.
• We'll do this one by casting out 9's: 2637583736455264957563; this leaves 2758375264575, or 2758375264575 or 5835675, but now 5835675 leaves 58575. Adding up the digits gives 30; adding them up again gives 3. Thus the remainder is 3.
3. The Postnet bar code shown here has a single error in which either one vertical bar which should be long is short or vice versa:

||...||..|.|.......||..|.|...||||...|..|...|.|.|..||

Express the correct zip code in ordinary characters. (Remember: the outside bars are framing bars which you ignore; also, the digit Postnet bar codes are as follows:
1: ...|| 2: ..|.| 3: ..||. 4: .|..| 5: .|.|. 6: .||.. 7: |...| 8: |..|. 9: |.|.. 0: ||... Also, the digits in a valid Postnet code must sum to an even multiple of 10.)
• First let's convert the postnet code into digits:
```|...|  |..|.  |....  ...||  ..|.|  ...||  ||...  |..|.  ..|.|  .|..|
7      8      ?      1      2      1      0      8      2      4
```
We know there is an error for the third digit, since no digit is represented by code with only one long bar. We can find out what the third digit should have been by picking it so that the sum of the digits is an even multiple of 10: 7 + 8 + ? + 1 + 2 + 1 + 0 + 8 + 2 + 4 = 33 + ?, so ? must be 7.
4. What is the sum of the measures of the vertex angles of a regular polygon with 41 sides?
• The answer is (41-2)*180 = 7020.
5. What is n for a regular n-gon if the sum of the measures of all but one of the n vertex angles is 3078 degrees?
• We know that the sum of all of the vertex angles is (n-2)180, so one angle is (n-2)180/n, and thus all but one of them is (n-1)(n-2)180/n, and this is supposed to be 3078. Thus 3078 = (n-1)(n-2)180/n, or (n-1)(n-2)/n = 3078/180 = 17.1. Thus n-2 (whatever it is) times (n-1)/n is 17.1, but (n-1)/n is a bit less than 1, so 17.1 = (n-1)(n-2)/n = ((n-1)/n)(n-2) is a bit less than 1*(n-2) = n-2. Thus n-2 is at least 18, so n is at least 18+2 = 20. We can now use guess and check; we'll try n=20, n=21, etc, until we find the n that works. But it turns out that n=20 works: for a regular 20-gon, one vertex angle measures (20-2)180/20 = 162, so the sum of the measures of all but one of the angles is 19*162 = 3078.
6. Show how to tile the plane with the following quadrilateral:

Do this with a drawing in which the given polygonal tile is surrounded by 8 additional tiles, one for each vertex and side. (Use a separate piece of paper, tracing the given tile by placing the paper over it, and sliding the paper to a new position and tracing again, etc.)
• Just keep turning the paper around, matching each side of the polygon with itself after rotating it 180 degrees:

7. Determine the symmetries of the following strip pattern (which you should imgain as extending indefinitely to the right and left), and use crystallographic notation to classify the pattern:
```       8              8              8              8              8              8
8              8              8              8              8              8
```
• This has translational symmetry, of course: just slide it sideways until the 8's match up. It also has vertical reflection symmetry (across any vertical line through the center of any of the 8's). And it has a glide reflection: reflect across the central horizontal line, adn then slide it over until the 8's match up. Thus the crystallographic notation is pma?. The "?" is a 1 if there is no rotational symmetry, and a 2 if there is a rotational symmetry. But from the classification of strip patterns on p. 98, we see there is only one which is consistent with pma?, and this one is pma2. Thus there must be a rotational symmetry, and indeed there is: you can rotate the strip pattern 180 degrees around the point approximately given by the dot in the figure below:
```       8              8              8   .         8              8              8
8              8              8              8              8              8
```
8. Problem 7 on page 125 of the text: Find the value of the 26th Fibonacci number if the 25th is 75025 and the 27th is 196418.
• If the 26th Fibonacci number is f26, then the rule tells us that f25 + f26 = f27; i.e., 75025 + f26 = 196418, and hence that f26 = 196418 - 75025 = 121393.