Assignment #3, due Friday, November 5th

Please write your answers on one side of the page only and staple the pages together. Show your work.

1. Pascal’s Triangle

1
1	1
1	2	1
1	3	3	1
1	4	6	4	1
1	5	10	10	5	1
1	6	15	20	15	6	1
1	7	21	35	35	21	7	1
1	8	28	56	70	56	28	8	1
1	9	36	84	126	126	84	36	9	1
1	10	45	120	210	252	210	120	45	10	1
1	11	55	165	330	462	462	330	165	55	11	1
1	12	66	220	495	792	924	792	495	220	66	12	1
1	13	78	286	715	1287	1716	1716	1287	715	286	78	13	1
1	14	91	364	1001	2002	3003	3432	3003	2002	1001	364	91	14	1
1	15	105	455	1365	3003	5005	6435	6435	5005	3003	1365	455	105	15	1
1	16	120	560	1820	4368	8008	11440	12870	11440	8008	4368	1820	560	120	16	1
1	17	136	680	2380	6188	12376	19448	24310	24310	19448	12376	6188	2380	680	136	17	1
1	18	153	816	3060	8568	18564	31824	43758	48620	43758	31824	18564	8568	3060	816	153	18	1
1	19	171	969	3876	11628	27132	50388	75582	92378	92378	75582	50388	27132	11628	3876	969	171	19	1

This famous triangle has been discovered by many cultures around the globe—later in the semester, we’ll see how it was discovered by Ancient Indian poets. (If you went to my talk “Math for Poets and Drummers” you’ll know that it was also discovered by my dad!) It has a lot of interesting properties.

a. Explain how you get the next row in the triangle (you don’t need to do it!)

b. Color every odd number black and leave the even numbers white.

c. Find a picture of Sierpinski’s Triangle on the web. It turns out that the pattern of odd and even numbers in Pascal’s Triangle looks very like Sierpinski’s Triangle!

2. Exploration of multiplication patterns: Color each square according to equivalence mod 8. That is, color the numbers equivalent to 0 one color, color those equivalent to 1 another color, etc. Complete the two rows on the bottom, and comment on any interesting patterns or relationships.

	1	2	3	4	5	6	7	8	9	10
	2	4	6	8	10	12	14	16	18	20
	3	6	9	12	15	18	21	24	27	30
	4	8	12	16	20	24	28	32	36	40
	5	10	15	20	25	30	35	40	45	50
	6	12	18	24	30	36	42	48	54	60
	7	14	21	28	35	42	49	56	63	70
	8	16	24	32	40	48	56	64	72	80
	9	18	27	36	45	54	63	72	81	90
	10	20	30	40	50	60	70	80	90	100
number of colors	8
gcd(column header, 8)		2					1

3. We actually use mod all the time without knowing it.  When we tell time (assuming we’re using the American system, where 1:00 is used for both 1 a.m. and 1 p.m.),

13:00 = 1:00

20:00 = 8:00

23:00 + 20 hours = 7:00

6:45 + 246 hours = 12:45

All we’re doing is computing the value mod 12.  That is,

13 ≡ 1 (mod 12)

20 ≡ 8 (mod 12)

23 + 20 ≡ 11 + 8 (mod 12) ≡ 19 (mod 12) ≡ 7 (mod 12)

6:45 + 246 ≡ 6:45 + 6 (mod 12) ≡ 12:45

We can also use modular arithmetic to solve problems about weeks, minutes, or seconds. Solve the following using mod arithmetic and write which number you’re using to find the mod equivalence.

a. 8:00 + 45 hours =

b. If today is Thursday, what day of the week will it be 100 days later?

c. If it is 3:47 right now, what time will it be 279 minutes later?

4. Each of the following multiplication problems contains a common mistake. Identify the mistake that was made. Can casting out 9s always identify this type of mistake? sometimes? never? Explain.

a. 25 ×63 75 150 225 Answer: 225

b. 25 ×63 65 1500 1565 Answer: 1565

c. 25 ×63 75 1500 1575 Answer: 1557

5. If X, Y, and b are any counting numbers, then

X × Y (mod b) ≡ X (mod b) × Y (mod b)

For example,

343 × 259 (mod 3) ≡ 343(mod 3) × 259 (mod 3) ≡ 1 × 1 ≡ 1 (mod 3)

Use that fact to solve the following problems.

a. 1008 × 326 (mod 5) ≡ _______ (mod 5)

b. 295 × 1588 (mod 2) ≡ _______ (mod 2)

c. (any odd number) × (any odd number) ≡ _______ (mod 2)

d. Explain how your answer in c. shows that the product of two odd numbers is an odd number.

6. A UPC code is a 12-digit number you find embedded within the bar code (in fact, the bar code is just a representation of the UPC number). For example, the UPC code for a box of Kleenex is

0 3 6 0 0 0 2 8 5 1 0 9

Every UPC code has this special property: if

a₁ a₂ a₃ a₄ a₅ a₆ a₇ a₈ a₉ a₁₀ a₁₁ a₁₂

are its digits, then the digits must satisfy the “UPC code equation”:

3a₁ + a₂ + 3a₃ + a₄ + 3a₅ + a₆ + 3a₇ + a₈ + 3a₉ + a₁₀ + 3a₁₁ + a₁₂ ≡ 0 (mod 10)

For example, the Kleenex code works:

(3 × 0) + 3 + (3 × 6) + 0 + (3 × 0) + 0 + (3 × 2) + 8 + (3 × 5) + 1 + (3 × 0) + 9 = 50 ≡ 0 (mod 10)

However, if you made a mistake and entered the number as

0 3 6 0 0 0 2 5 8 1 0 9

the scanner would detect an error, because

(3 × 0) + 3 + (3 × 6) + 0 + (3 × 0) + 0 + (3 × 2) + 5 + (3 × 8) + 1 + (3 × 0) + 9 = 64 ≠ 0 (mod 10)

a. Show that 048500 001769 satisfies the UPC code equation.

b. Explain why the number 080161 850191 is not a valid UPC code.

b. What last digit X makes the UPC code 080161 85019X valid?

c. What type of errors is the UPC code equation designed to catch?

7. The following comes from www.datingmatchmakers.com (I don’t endorse this site, but it came up on a web search!)

There are four elements in Astrology and three signs in each:

· FIRE covers Aries, Leo, and Sagittarius. They are primarily energetic, enthusiastic and impulsive (fiery).

· EARTH covers Taurus, Virgo, and Capricorn. They are primarily materialistic and practical (earthly).

· AIR covers Gemini, Libra, Aquarius. They are primarily communicative, oriented on the mental plane and social (airy).

· WATER covers Cancer, Scorpio, and Pisces. They are primarily responsive, emotional and sensitive (watery).

These signs are also divided based on basic modes of energy and modes of operating:

· Cardinal Signs are outgoing, initiatory and active. They represent the force of beginning. Aries, Cancer, Libra and Capricorn are the four cardinal signs.

· Fixed Signs are stable, resolute and determined. They represent the force of holding steady. Taurus, Leo, Scorpio and Aquarius are the four fixed signs.

· Mutable Signs are adaptable and versatile. They represent the force of fluidity. Gemini, Virgo, Sagittarius and Pisces are the four mutable signs.

In order, the signs are (1) Aries, (2) Taurus, (3) Gemini, (4) Cancer, (5) Leo, (6) Virgo, (7) Libra, (8) Scorpio, (9) Sagittarius, (10) Capricorn, (11) Aquarius and (12) Pisces. Comment on the role of modular arithmetic in determining your “element” and “mode of energy.”