f. Simultaneously in both the rectangle and triangle but not in the square 1 2

g. Simultaneously in both the rectangle and square 2 11

h. Simultaneously in the rectangle, square AND triangle 3 5

Homework Problem 2 A class has 38 students. Everyone plays at least one of the two instruments, harmonica or keyboard. Suppose 22 people can play harmonica and 25 people can play keyboard, answer the following questions. How many students can play harmonica alone and can’t play keyboard? – 38 = 9 (Plays both instruments) 4

Homework Problem 2 How many students can play harmonica alone and can’t play keyboard? 22 – 9 = 13 5

Homework Problem 2 How many students can play keyboard alone and can’t play harmonica? 25 – 9 = 16 6

Homework Problem 2 How many students can play both instruments? 9 7

In the following graph, circle A represents animals that can swim, circle B represents animals that can climb a tree, and circle C represents animals with four legs. Please place fish, sheep, cat, woodpecker, rodent, snake, and frog in the appropriate places. 8 A B C fish WP snakefrog sheepcat rodent

Homework Problem 4 Each red pencil costs 3 cents and each blue pencil costs 4 cents. If Jennifer uses all her money to buy red pencils there will be one cent left over. If she uses all her money to buy blue pencils there will be one cent left over. At least how much money does she have? How many red pencils or blue pencils can she buy? By observing the following money figure, can you find the solution? 9

Homework Problem 4 For her to buy red pencils and still have 1¢ left, the money may be one of the following: 4, 7, 10, 13, 16, 19, 22, 25, 28, …… For her to buy blue pencils and still have 1¢ left, the money may be one of the following: 5, 9, 13, 17, 21, 25, 29, 33, 37, …… 13¢ is the smallest common number. 10

Olympiad Math III Lesson 9 Arrays Boundaries 11

Commentary These types of problems are not hard. You might think they are too easy but you might make a mistake. Careful critical thinking is required to get the correct answer. 12

Logging competition In a logging competition, it took a man 12 minutes to evenly cut a log into 4 pieces. At this rate, how long would it take him to cut the same log into 6 pieces? The answer is 20 minutes why? 13

Vacation days I took a vacation from 8/7 to 8/19. How many days was this vacation? Why is the answer not 12? If I took a vacation from 8/10 to 8/11, how many days was this vacation? It is obviously 2 days. You use 11 – The vacation from 8/7 to 8/19 is 13 days 19 –

Generalization Here is the general rule: If you know the beginning number and the end number of a sequence that change by 1 between the numbers The count is the difference between the beginning number and end number …… and plus 1 15

People in a square array A group of people lined up and stand in a square array. We know that the number of people standing on the outside perimeter is 80. How many people are in the array? 16

Before you answer Before you answer “400” quickly, consider these two smaller problems If the number of people standing on the outside perimeter is 8, how many people are in the array? If the number of people standing on the outside perimeter is 12, how many people are in the array? 17

What is the general rule? When you divide the number of people on the outside perimeter by 4 what do you get? One less than the people standing on one side of the perimeter 18

Back to the original question If the number of people standing on the out side edge is 80, the number of people in a line (or column) is 21 The total number of people in the array is 21x21 =

Hollow Array Many soldiers stand in a hollow square array that is 3 people thick. On the outside perimeter each line has 42 soldiers. How many soldiers are on the inside perimeter? Totally how many soldiers are there? 20

Part 1 As you step into each layer, the number of soldiers in a line drop by 2. The inside perimeter has = 38 soldiers on each line. The number of people on the inside perimeter is (38-1)x4 =

Part 2 Total number of soldiers: (42-1)x4 + (40-1)x4 + (38-1)x4 = ( )x4 =

Part 2 Another way to look at the problem in a smaller scale 23

Part 2 Each of the four rectangles length is the side minus 3 Total number of soldiers is (42-3)x3x4 =

Flowers The flowers in a garden is arranged, as shown in the diagram, into 4 little triangles that also made a big triangle. Each side of the little triangles has 10 flowers. How many flowers are on the big triangle? How many flowers total? 25

Big triangle Note that the flower in the middle is shared Big triangle has 19 flowers on each side The total flowers on the Perimeter of the big triangle is (19-1)x3 = 54 26

Total flowers The total number of flowers is simply the flowers on the perimeter of the big triangle plus the flowers in the center small triangle without the corners. Total = x 8 = 78 27

Alternating Trees An orchard grew apple trees and pear trees in a square array. The trees are grown in an alternating fashion: apple, pear, apple, pear, etc. If the array is 9x9, how many trees of each kind is in the outside edge of the orchard? What’s the number of each trees total? 28

Alternating Trees If the array is even in size, there will be equal numbers of apple trees and pear trees. If the array is odd the trees are not equal Along any square perimeter there are equal number of two trees except the middle 29

Alternating Trees If the array size is odd the number of the two kind of trees differ by 1 (The middle tree) Since 9x9 array has 81 trees, assuming the apple tree is in the middle, there are 41 apple trees and 40 pear trees. 30

Number of skaters In a skating show skaters are grouped into 6 triangles with 7 skaters on each side. Together they made a hexagon as shown. How many skaters are on the perimeter of the hexagon and how many skaters are there totally? 31

Number of skaters On the hexagonal perimeter: (7-1)x6 = 36 On the inside arms (not counting the middle) there are 5x6 = 30 skaters The grand total of skaters: = 67 32

Divisible by 9 How many numbers between 200 and 300 that are divisible by 9 The smallest number in this range is 207 The largest number in this range is = 23x9, 297=33x9 The count is = 11 33

Puzzle time The cups on the next page formed a square array with some amount of water in each cup in oz. Take only one cup and pour the water (some or all) into others such that each row, column, and the two diagonals cups have the same amount of water. Explain your strategy. 34

Puzzle time

Puzzle time The total amount is 40 oz. Each column, row and diagonal should measure 10 oz. Currently, there is only one row over this and only one column over this. Cup on 4 th row 3 rd column should be poured to others. 36

Puzzle time High value target

Puzzle time