Chapter 5

1. Chapter 5 Look for a pattern

2. Patterns are all around us, but not everyone is able to see them. Those who can, will adapt to any new environment more quickly. Patterns can help us to solve problems and even predict the future.

3. What does a blinking Green light mean?

5. It is very difficult to define what a pattern is. In fact the word “pattern” can have many different meanings. From Wikipedia, the free encyclopedia, we can find that A pattern is a form, template, or model (or, more abstractly, a set of rules) which can be used to make or to generate things or parts of a thing, especially if the things that are generated have enough in common for the underlying pattern to be inferred or discerned, … But a pattern can also mean a customary way of operation or behavior, or a convention: something regarded as a normative example. In geometry, a pattern usually means the repetitive use of any form, object or color in a work.

6. Mathematics has been called the science of patterns. In fact, every theorem we learn in geometry class describes a certain pattern. A typical example is the Pythagorean theorem. As a problem-solving strategy, recognizing patterns enables you to reduce a complex problem to a pattern and then use the pattern to find a solution.

7. You can notice lots of patterns in driving, such as the ways that traffic lights change,the flow of traffic, and even places where policemen like to hide. For instance, if there is an accident in the front and one lane is blocked, which lane will you see cars moving faster?

8. IQ Test 1. Study the diagram below and decide what logically should be the missing section from the choices given.

9. 2. Look for a pattern and find the next figure. Choose from

10. Tiling is the most common form of geometric patterns. And most geometric patterns are repeating.

11. It took mathematicians several hundred years to find a non-repeating pattern that is made up of several congruent figures. The one on the right is called a Penrose pattern, discovered by Sir Roger Penrose, a British mathematician in 1970. It uses only four basic shapes and yet it can cover the plane without repeating exactly in arrangement.

12. It may not be obvious to you what kind of application the Penrose patterns can have. And it will not be a marketable idea for people redoing their bathrooms, it nevertheless is useful for making quilted bathroom tissue, which must be embossed with a design that never repeats itself. Otherwise, layers of the same pattern build up ridges and grooves and the roll becomes lumpy. Penrose’s design smoothes out the bumps. 1st UK toilet paper using Penrose pattern Penrose’s 1st pattern

13. Look for a pattern This technique is also heavily used in weather forecast, such as the track prediction of hurricanes. Katrina 2005 Year 200

14. School children use patterns to learn how to pronounce words and how to spell them.

15. Look for a pattern and guess the next 3 terms in the sequence 1, 2, 4, 8, 16, 32, 64, 128 4, 7, 10, 13, 16, 19, 22, 25 1, 4, 9, 16, 25, 36, 49, 64 3, 4, 6, 9, 13, 18, 24, 31, 39 5, 10, 9, 18, 16, 32, 29, 58, 54, 108 8 77, 49, 36, 18, ____ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 3, 4, 6, 10, 17, 25, 35, 48, 62, 78, 97, …

16. The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, … 1 pair 1 pair 2 pairs 3 pairs 5 pairs Starting with a male-female pair of baby bunnies, they become matured and productive in one month. By the end of the second month, they can produce another male-female pair of baby bunnies.

17. Fibonacci sequence in nature. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … Sunflower always have 21 counterclockwise spirals and 34 clockwise spirals.

18. One set of 5 spirals ascends at a shallow angle to the right, ... a second set of 8 spirals ascends more steeply to the left, ... and the third set of 13 spirals ascends very steeply to the right.

20. 2 3 5 8

21. Fibonacci was the professional name of an Italian Leonardo Pisano. Born in 1170, Pisa, Italy. Died in 1250, Pisa, Italy.

22. “Petals around the Rose” puzzle Five dice are rolled and the computer calls out the number of “petals” around the “rose”. For example, in the following roll, the answer is 2. The answer is always even, the smallest being 0, the largest being 20. Your mission is to look for a pattern and then find out how the computer calculates the number of “petals”. There is no use to know any advanced mathematics for this puzzle.

23. Petals around the Rose Dr. Richard Duke at the University of Michigan used to begin each of his gaming/simulation courses with this exercise. While some students would solve the problem right away, others would struggle all semester. It had taken Dr. Duke well over a year himself, and he would always explain that the smarter you were, the longer it took to figure it out. play this game on computer

24. What is the last digit of 763 ? 71 = 7 72 = 49 73 = 343 74 = 2401 75 = 16807 76 = 117649 77 = 823543 78 = 5764801

25. Goldbach’s conjecture (June 1742) every even number bigger than 2 is a sum of two primes 4 = 2 + 2 6 = 3 + 3 8 = 3 + 5 10 = 3 + 7 12 = 5 + 7 14 = 7 + 7 16 = 3 + 13 18 = 5 + 13 20 = 7 + 13 22 = 24 = 26 = 28 = 30 = 32 = 34 = 36 = 38 = It has been verified for all even numbers up to 1018 .

26. Every whole number is a sum of squares, but how many squares are needed? 0 = 02 1 = 12 2 = 12 + 12 3 = 12 + 12 + 12 4 = 22 5 = 12 + 22 6 = 12 + 12 + 22 7 = 12 + 12 +12 + 22

27. Change a repeating decimal to a fraction 1/9 = 0.1111… 2/9 = 0.2222… 3/9 = 0.3333… See a pattern? What is 0.555… equal to? 13/99 = 0.131313… 25/99 = 0.252525… 83/99 = 0.838383… What is 0.646464… equal to? Answer: 64/99 Answer: 5/9 More patterns 0.325325325… = 325/999 0.718718718… = ? Answer: 718/999 0.463946394639… = ? Answer: 4639/9999

28. Change a repeating decimal to a fraction More patterns 0.027272727… = 27/990 0.0038383838… = 38/9900 0.000545454… = ? Answer: 54/99000 Here is the hardest part, 0.2777777… = 25/90 0.5888888… = 53/90 0.345454545… = 342/990 0.164646464… = 163/990 What is 0.834343434… = ? Answer: 826/990

29. Look for a pattern and predict the answer. 51 × 51 = 2601 52 × 52 = 2704 53 × 53 = 2809 54 × 54 = 2916 55 × 55 = ? 56 × 56 = ? 57 × 57 = ? 58 × 58 = ? 59 × 59 = ? 3025 3136 3249 3364 3481

30. Look for a pattern and predict the answer. 35 × 35 = 1225 15 × 15 = 225 55 × 55 = 3025 25 × 25 = 625 45 × 45 = 2025 65 × 65 = ____ 75 × 75 = ____ 85 × 85 = ____ 95 × 95 = ____ 42 × 48 = 2016 36 × 34 = 1224 59 × 51 = 3009 78 × 72 = 5616 27 × 23 = ____ 54 × 56 = ____ 43 × 47 = ____ 61 × 69 = ____ 621 4225 3024 2021 5625 7225 4209 9025

31. Milk Lovers Alysia and Melissa and Dante and Melody loved milk. They convinced their older brother, Mark, who did all the shopping, to buy each of them their own gallon of milk because they each liked it so much. They all put their names on their gallons.

32. One day, they were really thirsty and each took 10 drinks according to a different pattern: Alysia started by drinking ½ of the milk in the container. Then she drank ⅓ of what was left. Then she drank ¼ of what was left, then 1/5 and so on. Melissa started by drinking 1/11 of her milk, then 1/10 of what was left, then 1/9 of what was left, and so on. ……. After each had taken 10 drinks, how much milk remained in each container?

33. 5. Roo and Tigger are having a 200 feet race. They have to run 100 feet forward and then another 100 feet back. Roo can make 3 two-foot jumps in the same amount of time as Tigger makes 2 three-foot jumps. Who will win the race?