Chapter 9 Solve an easier related problem

1. Chapter 9Solve an easier related problem Some problems are just too complex and too challenging. It is necessary to solve an easier version first to gain experience or to test a method. Some times, you may find out that it is better (or necessary) to use a totally different approach.

2. “We choose to go to the moon, not because it is easy, but because it is hard.” JFK, 1962

3. Painting a swimming pool is a pretty challenging task, so it is better to solve an easier but similar task first, such as painting the spa. This will give you the necessary experience.

4. Example 0 Jack and Jill are both in the same line for the Soarin’ over California ride in Disney World. Jack is in the 25th place and Jill is in the 125th. How many people are between them?

5. Example 1 How many squares (of all possible sizes) are there in the following 8 × 8 checker board? We need to do some easier and related problems.

6. 1. How many squares are in this baby checker board? Answer: 4 + 1 = 5 2. How many squares are in this “toddler” checker board? How many squares of size 1 × 1: 9 How many squares of size 2 × 2: 4 How many squares of size 3 × 3: 1 Therefore there are totally 1 + 4 + 9 squares.

7. 3. How many squares are in this “junior” checker board? How many squares of size 1 × 1: 16 How many squares of size 2 × 2: 9

8. 3. How many squares are in this “junior” checker board? How many squares of size 1 × 1: 16 How many squares of size 2 × 2: 9 How many squares of size 3 × 3: 4 How many squares of size 4 × 4: 1 Therefore there are totally 1 + 4 + 9 + 16 squares.

9. Now return to the original question: How many squares (of all possible sizes) are there in the following 8 × 8 checker board? Answer: 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 = 204

10. Example 2 How many paths are there from corner A to corner B if you have to stay on the black lines, and you can only go up or right? B A

11. Example 3: There are 4 different containers, and 10 identical balls. If the containers are all big enough to hold 10 balls, how many ways can you put these 10 balls into some or all of these containers?

12. Solve some smaller problems first and then find a pattern.

13. Solve an easier and related problem Example 4. Following recess, the 1000 students of a school lined up and enter the school as follows: The 1st student opened up all of the 1000 lockers in the school. The 2nd student closed all lockers with even numbers. The 3rd student “changed” all lockers that were numbered with multiples of 3 (by closing those that were open and opening those that are closed). The 4th student “changed” all lockers that were numbered with multiples of 4, and so on. After all 1000 students had entered the building in this fashion, which lockers were left open?

14. An easier related problem is to consider the same situation with only 26 lockers. The following are 26 locker doors. Click to open and click to close. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 (Use Control A to show the cursor )

15. Example 5 What is the sum of ?

16. Example 6. The Tower of Hanoi Puzzle • In the beginning there are 64 discs on the left pole. The goal is to move all the discs from the left pole to the right pole while following the rules: • Only one disc can be moved at each step, • Smaller discs must be on top of bigger discs, • Each disc must be on any one pole at the end of each step. How many steps are required to move all the discs to the right pole? (we only show 8 discs for simplicity.)

17. 3 7 15 31 63

18. The Tower of Hanoi Puzzle • The goal is to move all the discs from the left pole to the right pole while following the rules: • Only one disc can be moved at each step, • Smaller discs must be on top of bigger discs, • Each disc must be on any one pole at the end of each step. If there are 64 discs on the left pole, how many steps are required to move the discs over? (we only show 3 discs for simplicity.) (Please click to see the moves)

30. Example 7. Counterfeit Coin In front of you are 12 identical looking gold coins, but one is counterfeit. The only difference is its weight - it may be heavier or lighter than a real one. If you are given a pan balance, and are allowed to use it only 3 times, how can you determine which one is fake, and whether it is heavier or lighter?

31. Method 2 Fix some variables or parameters Using a batting tee to practice your swing is example of fixing some parameters.

33. Method 2 Fix some variables or parameters Putting a pair of training wheels on a bike is an example of “solving an easier related problem”.

34. Method 3 Use a totally different approach.

35. The Invention of Dynamite The explosive chemical in dynamite is Nitroglycerin, which was first invented by Italian chemist Ascanio Sobrero in 1846. In its natural liquid state, nitroglycerin is very volatile. It is very dangerous to manufacture and to be transported. In 1860, Nobel started experimenting with chemical additives to tame the explosive power of nitroglycerine. None of the additives works. Alfred Nobel Then, one day in 1866, at age 33, Nobel had produced a full test tube of the substance--enough to easily blow up his laboratory. He was just ready to pour a drop into another test tube. He was very nervous, when suddenly the test tube full of nitroglycerin slipped out of his hands and fell to the floor!

36. Luckily, the tube fell into a packing box filled with sawdust. If it would have hit the floor, there probably would have been a great explosion, killing Nobel and others around him. The nitroglycerin ran out of the test tube and was absorbed in the sawdust. Not letting the expensive material go to waste, Nobel started to test the mixture and found that it still had great explosive power but could be handled easier. He then realized that the approach to the problem is not adding chemical substances, but mixing nitroglycerin with absorbent and chemically inert substances! His final choice was a porous siliceous earth called kieselguhr .

37. Example 8. What is the sum of 1 + 2 + 3 + 4 +    + 100 ?

38. Example 8 Johann Carl Friedrich Gauss (1777 – 1855) was a German mathematician and scientist who contributed significantly to many fields. He is known as “the Prince of Mathematicians”. There are several stories of his early genius. According to one, his gifts became very apparent at the age of three when he corrected, mentally and without fault in his calculations, an error his father had made on paper while calculating finances. Another famous story, has it that at the age of 8, his primary school teacher, J.G. Büttner, tried to occupy his pupils by making them add a list of whole numbers: 1 + 2 + 3 + 4 + 5 + ∙∙∙ ∙∙∙ + 95 + 96 + 97 + 98 + 99 + 100 The young Gauss reputedly astonished his teacher and assistant Martin Bartels by producing the correct answer within seconds. How did he solve the problem so quickly?

39. Here is Gauss’s secret: 5 + · · · · · + 95 + 98 97 100 4 + + + 2 + 3 + 96 + + 1 + 99 = 101 = 101 = 101 = 101 . . . 50 + 51 = 101 Therefore, there will be 50 copies of 101, hence the sum is 50 × 101 = 5050.

40. Exercise Find the sum of 7 + 10 + 13 + 16 +    + 103 Formula:

41. Example 9 Find the area of the following triangle 5 inches 6 inches

42. Example 10 A 5 km long straight tunnel is pointing roughly east-west direction. A bicycle enters the west end exactly when another bicycle enters the east end. A fly is flying back and forth between the two bicycles at 16km per hour, leaving the eastbound bicycle as it enters the tunnel. If the bicycles are both traveling at 10 km per hour, how far has the fly traveled in the tunnel when the bicycles meet?

43. Example 11 Good Luck Goats In the mythical land of Kantanu, it was considered good luck to own goats. Basanta owned some goats at the time of her death and willed them to her children. To her first born, she willed one-half of her goats. To her second born, she willed one-third of her goats. And last she gave one-ninth of her goats to her third born. As it turned out, when Basanta died, she had 17 goats. Barring a Solomonic approach, how should the goats be divided?

44. How many triangles (of all possible sizes) are in the following diagram?

45. Look for a pattern 1 triangle 1 + 3 + 1 triangles (red means upside-down) 1 + 3 + 6 + 3 triangles (red means upside-down)

46. 1 + 3 + 6 + 10 + 1 + 6 triangles

47. There are 1 + 3 + 6 + 10 + 15 + 21 + 1 + 6 + 15 = 78 Triangles.

48. Diagonals • A certain convex polygon has 25 sides. How many diagonals can be drawn?

49. 2. Sum of Odds Find the sum of the first 5000 odd numbers.

50. 3. TV Truck Theotis has to load a truck with television sets. The cargo area of the truck is a rectangular block that measures 8 ft by 21 ft by 11 ft. Each television set measures 1 1/2 ft by 1 2/3 ft by 1 1/3 ft. What is the maximum number of TV sets that can be loaded into the truck?