- 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.
- We'll do this one by casting out 9's: 2
637583736455264957563; this leaves 2758375264575, or2758375264575 or 5835675, but now 5835675 leaves 58575. Adding up the digits gives 30; adding them up again gives 3. Thus the remainder is 3.
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.
- The answer is (41-2)*180 = 7020.
- 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.
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:
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
- 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.