Mt. Douglas Secondary

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1 Foundations of Math 11 Section 4.1 Patterns Patterns We have stated in chapter 2 that patterns are widely used in mathematics to reach logical conclusions. This type of reasoning is called inductive reasoning. This section will expand inductive reasoning to not only mathematics, but everyday life reasoning. Remember, your conclusion may be, but is not necessarily, true. Example 1 Find the patterns. a) 1 1 = = = = = b) = = = = = Solution: a) , b) 1111, 11111, = Example 2 Sixty-four teams enter the NCAA basketball championship. If a team loses, it is eliminated. How many games are played in the tournament? Solution: = 63 games Example 3 Draw the next shape in the pattern, and predict the number of pieces in the next two patterns. 3 pieces 9 pieces 18 pieces Solution: The next pattern would look like this: The next two patterns should have 30 and 45 pieces

2 168 Chapter 4 Analyze Puzzles and Games Foundations of Math 11 Example 4 Find the next two numbers in the following patterns: a) 3, 6, 8, 16, 18, 36, 38,, b) 2, 4, 5, 10, 12, 24, 27,, Solution: a) 3, 6, 8, 16, 18, 36, 38, 76, 78 (Number of terms is multiplied by 2 then 2 is added) b) 2, 4, 5, 10, 12, 24, 27, 54, 58 (Number is multiplied by 2, then 1 is added, then multiplied by 2 and 2 is added, then multipled by 2 and 3 is aded, ) Example 5 Which number(s) in the last group are zuts? Zuts Not zuts Which are zuts? Solution: Zuts are divisible by 3, thus 30 and 93 are zuts. Example 6 1 = = = = = n = Solution: = = 101, = 101, = 101,, = 101 There are 50 groups of 101, thus = n = 1+ n,2+ (n 1), 3 + (n 2),(1+ n) = 1+ n,1+ n,1+ n,,1+ n (each pair adds to 1+ n, and there are n 2 pairs) The sum of n terms is n(n +1) 2.

3 Foundations of Math 11 Section 4.1 Patterns Exercise Set 1. Study the pattern, and predict the missing values. a) = = = = = b) 9 2 = = = = = c) = = = = = d) 1 = = = = = e) 1 = = = = f) 2 = = = = = g) 1 = = = = = h) 2 = = = = = i) = = = = = j) = = = = =

4 170 Chapter 4 Analyze Puzzles and Games Foundations of Math Study the pattern, and predict the next two terms. a) 2, 3, 5, 8, 12,, b) 20, 25, 31, 38, 46,, c) 10, 7, 12, 9, 14,, d) 3, 6, 11, 18, 27, 38,, e) 2, 6, 15, 31, 56,, f) 2, 6, 12, 20, 30,, g) 15, 19, 25, 33, 43,, h) 1, 2, 5, 14, 41,, i) 3, 5, 11, 29, 83,, j) 59, 52, 55, 48, 51, 44, 47,,

5 Foundations of Math 11 Section 4.1 Patterns What pattern is observed in the following? (Hint: think about odd and even numbers.) a) = 12, = = 6, ( 17) + ( 41) = 58 b) = = 4,, = 58 ( 12) + ( 8) = 20 c) = = 9,, 14 + ( 17) = 3 ( 4) + ( 7) = 11 d) 3 6 = = 20,, 7 8 = 56 ( 9) ( 4) = 36 e) 3 5 = = 63,, 5 11 = 55 ( 1) ( 13) = 13 f) 18 3 = = 4,, 12 3 = = 10

6 172 Chapter 4 Analyze Puzzles and Games Foundations of Math Determine the number of matchsticks used in the 100 th pattern. a) b) c) d)

7 Foundations of Math 11 Section 4.1 Patterns Determine the number of squares in the 100 th pattern. a) 8 squares 16 squares 24 squares b) c) d)

8 174 Chapter 4 Analyze Puzzles and Games Foundations of Math Inductive Reasoning When we reach conclusions by observing diagrams and figures, we are using inductive reasoning. We will use inductive reasoning to establish, if possible, a general equation for different patterns. Example 1 Predict the n th term of the pattern 3, 6, 12, 24, Solution: 1 st term 2 nd term 3 rd term 4 th term The n th term is 3 2 n 1. Example 2 Predict the n th term of the pattern 2, 8, 14, 20, Solution: 1 st term 2 nd term 3 rd term 4 th term The pattern has a constant increase of 6. The n th term is 2 + (n 1)6= 2 + 6n 6= 6n 4. Example 3 Predict the n th term of the pattern 2, 6, 12, 20, 30, 42, Solution: 1 st term 2 nd term 3 rd term 4 th term 5 th term 6 th term The pattern does not have a constant increase, thus the n th term is not linear. The n th term is n(n + 1).

9 Foundations of Math 11 Section 4.2 Inductive Reasoning 175 Example 4 A Lazy S has a vertical line drawn through it. Guess how many pieces will be formed cut 4 pieces 2 cuts 7 pieces 3 cuts pieces 4 cuts pieces 100 cuts pieces n cuts pieces Solution: 3 cuts = 10 pieces 4 cuts = 13 pieces 100 cuts = = 301 pieces n cuts = 3n + 1 piece Counterexample If a statement is found for which the assumption is wrong by showing an example is called a counterexample. To show something is not true, it is sufficient to show one example that is not true. It is possible that the conclusion you reach in a statement might be proved incorrect if a counterexample were found. Thus, we must be cautious about reaching conclusions by inductive reasoning. Example 1 If possible, find a counterexample for each of the following assumptions: a) Every prime number is odd. b) A triangle drawn from two corners of a rectangle is half the area of the rectangle. Solution: a) 2 is a prime number. b) True Not true Example 2 Solution: Multiplying leads to larger number. Multiplying by zero leads to zero. Multiplying by 1 leads to the same number. Multiplying by a number between 0 and 1 leads to a smaller number.

10 176 Chapter 4 Analyze Puzzles and Games Foundations of Math Exercise Set 1. Study the pattern and predict the n th term. a) 1, 2, 3,, b) 1, 3, 5,, c) 2, 4, 6,, d) 3, 7, 11, 15,, e) 10, 17, 24, 31,, f) 0, 6, 12, 18,, g) 0, 2, 6, 12,, h) 1, 6, 15, 28,, 2. Determine the number of matchsticks used in the n th pattern. a) b) c) d)

11 Foundations of Math 11 Section 4.2 Inductive Reasoning Predict the missing terms. a) A 1 A 2 A 3 A 50 A n xx xxx xxx xxxx xxxx xxxx 2 stars 6 stars 12 stars stars stars 3. b) 1 cut 2 regions 2 cuts 4 regions 3 cuts 6 regions 50 cuts regions n cuts regions c) 2 points 2 regions 3 points 4 regions 4 points 8 regions Draw 5 points and count regions Draw 6 points and count regions regions regions Does inductive reasoning hold for all diagrams of 3 c)?

12 178 Chapter 4 Analyze Puzzles and Games Foundations of Math Number of diagonals of a polygon. a) 3 sides 0 diagonals 4 sides 2 diagonals 5 sides 5 diagonals 6 sides diagonals n sides diagonals 5. Study the pattern and predict the n th term. a) 8 squares 16 squares 24 squares b) c) d)

13 Foundations of Math 11 Section 4.2 Inductive Reasoning Inductive reasoning prediction. a) Substitute a positive integer into n 2 + n + 11 from 1 9. n n 2 + n + 11 b) What special type of number do you get? c) What inductive reasoning statement can you make based on your answer to 6 b)? d) Test your reasoning for n = 10 and n = 11. e) Is your inductive reasoning statement in 6 c) correct for all values of n?

14 180 Chapter 4 Analyze Puzzles and Games Foundations of Math When possible, find a counterexample for each of these assumptions. If not, write true. a) The acute angles in a right triangle are equal. b) A polygon has more sides than diagonals. c) The second power of any real number is positive. d) A real number to the zero power is one. e) For any real number, x > 0. f) For any real number x, x 2 > x. g) In a pentagon, the largest angle is opposite the largest side. h) An even number is any number which is not odd. i) For any real number, x = x if x < 0. j) x 1 cannot be factored. k) For any real number x, x is a negative number. l) For any real number n, n 2 + 4n + 4 = n + 2. m) For any real number n, n n = 1. n) For any natural number n, {1, 2, 3, }, n 2 n + 41 is a prime number.

15 Foundations of Math 11 Section 4.3 Deductive Reasoning Deductive Reasoning As stated in the last chapter, when doing two-column proofs, in deductive reasoning you arrive at conclusions from accepted facts. Each step in deductive reasoning represents a conclusion from the previous statement. If any statement has an error in it, the final conclusion is not correct. Here is an example of deductive reasoning. Premise All the planets revolve around the sun in an elliptical orbit. Mars is a planet. Conclusion: Mars moves around the sun in an elliptical orbit. Premise If n is a prime number greater than 3, then (n 1)(n + 1) is divisible by is a prime number greater than 3. Conclusion: 2208 is divisible by 24. Draw a conclusion from the following: Premise All English teachers like to read. Sam does not like to read. Conclusion: Answer: Sam is not an English teacher. Premise If a quadrilateral is a square, it is a regular polygon. A regular polygon has all sides and angles equal. Conclusion: Answer: A square has all sides and angles equal.

16 182 Chapter 4 Analyze Puzzles and Games Foundations of Math 11 Write whether each statement is true or false. If false, show a counterexample. Premise Every even number divisible by 6 is divisible by 3. Solution: True. Premise A number greater than 12 is divisible by 12 if it is divisible by both 2 and 3. Solution: False. (18 is divisible by 2 and 3 but 18 is not divisible by 12.) What deduction, if possible, can be made about each person? Premise A person must be 16 years old to have a driver s license. a) Tom has a driver s license. b) Sue drives a car. c) Sally is 20 years old. d) Bill is 12 years old. e) Len does not drive a car. Solution: a) Tom is 16 years or older. b) Nothing; we cannot say definitively that Sue has a driver s license. c) Nothing. d) Bill does not have a driver s license. e) Nothing. Which one(s) of the following statements are true? Premise a) If two angles are right angles, then they are equal. b) If two angles are equal, then they are right angles. c) If two angles are not right angles, then they are not equal. d) If two angles are not equal, then they are not right angles. Solution: a) True. b) False; angles could be both 70. c) False; angles cold be both 30. d) True.

17 Foundations of Math 11 Section 4.3 Deductive Reasoning Exercise Set 1. Tell whether each statement is true or false. If false, give a counterexample. a) If a triangle has two equal sides, then it has equal angles. b) All diagonals of a regular polygon are equal. c) If two triangles have equal perimeters, then they have equal sides. d) If the opposite sides of a quadrilateral are parallel, then they are equal. e) If x 2 > 0 then x > 0. f) The diagonals of an equilateral quadrilateral are perpendicular. g) The diameter is the axis of symmetry of a circle. h) A parallelogram can be formed from two equal triangles. i) A number is divisible by 4 if the last digit is divisible by 4. j) A number is divisible by 18 if it is an even number divisible by 9. k) A number is divisible by 12 if it is an even number divisible by 3. l) A number is divisible by 15 if it is an odd number divisible by 5.

18 184 Chapter 4 Analyze Puzzles and Games Foundations of Math Reach a conclusion using the following assumptions. a) All citizens of Calgary are Albertans. b) All Manitobans are fishermen. All Albertans are Canadians. Sue is a Manitoban. c) All rectangles are quadrilaterals. All squares are rectangles. d) All whales are mammals. All mammals can swim. e) If you study for the math exam you will pass. You study for the exam f) All Japanese are good at math. Aiko is Japanese. g) Paul is taller than Mike. Mike is 5'10". h) The diagonals of a rhombus are perpendicular to each other. A square is a rhombus. i) The product of a and b is negative. a is negative. j) a is greater than b. b is equal to c.

19 Foundations of Math 11 Section 4.3 Deductive Reasoning Use deductive reasoning to reach a conclusion based on the given assumption of a triangle. a) One angle is 80. b) One angle is 80 and the other 2 angles are equal. c) All 3 angles are equal. d) All 3 angles are consecutive integers. e) The middle angle is 10 more than the smallest angle, which is 1 the largest 2 angle. f) What is the sum of angles in a pentagon?

20 186 Chapter 4 Analyze Puzzles and Games Foundations of Math All members of the basketball team are over 6 feet tall. What, if anything, can you deduce with certainty about each person? 5. A person must be 12 years of age or over to have a fishing license. What deduction can be deduced with certainty about each person? a) Sue is on the basketball team. a) Sally has a fishing license. b) Tom is over 6 feet tall. b) Bill went fishing. c) Mary is 5'6" tall. c) Lora is 15 years old. d) Bert is not on the basketball team. d) George is under 12 years old. e) Tim does not fish.

21 Foundations of Math 11 Section 4.3 Deductive Reasoning Find the error in the following proofs. a) x = 2 x(x 1)= 2(x 1) x 2 x = 2x 2 x 2 2x = x 2 x(x 2)= x 2 x = 1 b) Let a = b = 1 Then a = b a 2 = ab a 2 b 2 = ab b 2 (a b)(a + b) = b(a b) a + b = b = 1 2 = 1 c) Let A = amount of studying done per week B = amount of studying needed to get a good grade C = increase in hours needed to get good grades with C > 0. Then A + C = B A=B C A(A B) = (B C)(A B) A 2 AB=AB AC+ BC B 2 A 2 AB+AC=AB B 2 + BC A(A B + C) = B(A B + C) A=B

22 188 Chapter 4 Analyze Puzzles and Games Foundations of Math Decide whether the process used is inductive, or deductive reasoning. a) Show the sum of two even numbers is even by showing several examples. b) No mathematician is boring. Ann is a mathematician. Therefore, Ann is not boring. c) One counterexample proves that a conjecture is false. d) You show why your statement makes sense. e) You give evidence that your statement is true. f) Six other examples to show that your conjecture is true. g) What three coins have a value of 60? h) Cut out several triangles, then tear off the corners of each triangle and fit them together to conclude that the angles of any triangle add up to 180.

23 Foundations of Math 11 Section 4.3 Deductive Reasoning Use deductive reasoning to reach the conclusion of the following: a) Choose a number, add five, double the result, subtract four, divide by two, subtract the original number. The result is three. b) Choose a number, triple the number, add 6, subtract the original number, divide by 2, subtract 3. The result is the original number. c) Choose a positive number, square the number, add one more than twice the original number, take the square root of the number, subtract one. The result is the original number. d) Pick a number between 1 and 9. Triple it and add 7 to the number. Multiply the result by 5 and subtract 32. Then divide your answer by 3. Finally, double your result and remove the 2 from your answer. The number left is the original number.

24 190 Chapter 4 Analyze Puzzles and Games Foundations of Math All people wearing toques have blonde hair. Some of the people have brown hair. All people who have blonde hair like dancing. People who have brown hair like sports. Ray has blonde hair. Of the following, which statement must be true? Ray likes dancing. Ray likes brown hair. Ray likes sports. Ray is wearing a toque. 10. Art, Bill, Cecil and Don live in the same apartment. They are a manager, teacher, artist and musician. Art and Cecil watch TV with the teacher. Bill and Don go to the hockey game with the manager. Cecil jogs with the manager and teacher. Who is the manager? 11. Ann, Beth and Cindy are a doctor, lawyer and mathematician, but not necessarily in that order. The doctor examines Cindy. Beth hired the lawyer to do legal work. Ann earns less than the mathematician, but more than Beth. Match the occupation with the person. 12. Al, Bob, Cal and Dave are on four sports teams. Each play on just one team. They play football, basketball, baseball and hockey. Bob is a goalie. The tallest player plays basketball, and the shortest baseball. Call is taller than Dave, but shorter than Al and Bob. What sports does each play?

25 Foundations of Math 11 Section 4.3 Deductive Reasoning Use deductive reasoning to reach the conclusion: a) No student is lazy. Jerry is a musician. All musicians are lazy. Conclusion L = lazy M = musician J = Jerry S = student (Diagrams may help) L M J S b) All doctors are rich Musicians are temperamental. Sanjeet is a doctor. No temperamental person is rich. Conclusion R = rich D = doctor S = Sanjeet T = temperamental M = musicians R D S T M c) Everyone who is sane can use logic. No insane person can play chess. None of your sons can use logic. Conclusion d) No one takes the newspaper who is well educated. No elephant can read. Those who cannot read are not well educated. Conclusion e) No experienced woman is incompetent Sue is making mistakes. No competent person makes mistakes. Conclusion

26 192 Chapter 4 Analyze Puzzles and Games Foundations of Math Puzzles and Games of Strategy The peculiar beauty of mathematics lies in deduction, in the dependency of one fact upon another. The less expected a dependency is, the simpler the facts on which the deduction is based the more beautiful the result. Alexander Bogomolny This section will show the beauty of both inductive and deductive reasoning in solving puzzles and games of strategy. Some of the questions are not very difficult, and some are very challenging. Do not look at the solution right away; allow your thought process to develop! Example 1 Put the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9 in the circles so that each edge adds up to the same number. Solution: There are many solutions to this problem. Here is one of them: 8 7 Put the three largest numbers in the 3 corners. The six remaining numbers = 21, = 15, = 16, = Thus, we must pick 3 pairs of numbers that differ in sums by one or 5 6 two; = 21. Now, choose pairs from numbers 1, 2, 3, 4, 5, 6: = 6, = 7, = Example 2 If this pattern continues, what is the last number in the 10 th row? Solution: First row has 1 number. Second row has 2 numbers. The last number in each row is the sum of how many numbers were in each row (e.g., = 10). Thus, the last number in the 10 th row is = 55.

27 Foundations of Math 11 Section 4.4 Puzzles and Games of Strategy Exercise Set Use inductive and deductive reasoning to solve the following. Do not look up the answer until you have really tried to solve the problem! 1. Use the numbers 1, 2, 3, 4, 5 and 6 for the multiplication problem 2. Use four 9 s in a math equation that equals Use the numbers 1, 2, 3, 4, 5, 6 and 7 such that each straight line adds up to the same total. 4. Can you move just two toothpicks and create seven squares?

28 194 Chapter 4 Analyze Puzzles and Games Foundations of Math Can you move just 3 pennies and flip this triangle upside down? 6. How many way can you arrange 5 different books on a shelf? 7. Use 1, 2, 3, 4, 5, 6, 7, 8 and 9 just once each to produce the largest product possible from multiplying two numbers together. 8. Every minute a cell splits in two. If at 4:00 p.m. we have one cell in a jar and at 5:00 p.m. the jar is full of cells, when was the jar half full of cells?

29 Foundations of Math 11 Section 4.4 Puzzles and Games of Strategy Put the numbers 8, 9, 10, 11, 12, 13, 14, 14, 16, 18 in the circles such that each row adds up to the same number. 10. Put the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9 in the circles so that each edge adds up to the same number. 11. How many squares are in the following pattern? (up to grid)

30 196 Chapter 4 Analyze Puzzles and Games Foundations of Math Put the numbers 1 to 8 in each square so that each side adds up to the middle term Without lifting your pencil, connect all of the dots below with four straight lines. 14. If you have 20 people in a class and everyone shakes hands with each person in the class once, how many handshakes are made?

31 Foundations of Math 11 Section 4.4 Puzzles and Games of Strategy A cat is at the bottom of a 25-foot well. Each day, she climbs up 5 ft and slides down 4 ft each night. How many days will it take her to reach the top? 16. A mother has two children. One of them is a boy. What are the odds the other is a boy? 17. You have 2 coins. One is heads on both sides and the other is heads on one side and tails on the other. A coin is selected at random and the face showing is heads. What are the odds the other side is heads? 18. One of three cards is black on both sides, another is white on both sides, and the other is black on one side and white on the other. One card is selected at random, and the side up is white. What are the odds the other side is black?

32 198 Chapter 4 Analyze Puzzles and Games Foundations of Math You have a boat to take a fox, chicken and some corn across the river. You can only take one at a time. The fox will eat the chicken if left alone. The chicken will eat the corn if left alone. How do you get them across in the least amount of trips? 20. Three men walk into a hotel and ask for a room. The desk clerk says a room is $30, so each pay $10. Later, the clerk says there is a special rate and the price for the room is $25. The three men decide to keep $1 each and give $2 to the desk clerk. Thus, each paid $9 for the room and the desk clerk got $2, totalling $29. But the original charge was $30 so where did the extra $1 go? 21. Here is the classic 3-door problem from Monty Hall s Let s Make a Deal game show. Behind one of the doors is a car; behind the other two are goats. The contestant does not know where the car is but Monty Hall does. The contestant picks a door and Monty opens one of the remaining doors to reveal a goat. After opening the door with a goat behind it, the contestant is then given the choice of keeping the door that was originally picked or of switching doors. What is the contestant s best choice for winning? 22. The black and white buttons can only move diagonally. What is the least number of moves to get the black buttons to where the white buttons are and the white buttons to where the black buttons are? You are not allowed to jump pieces or have them land on each other.

33 Foundations of Math 11 Section 4.4 Puzzles and Games of Strategy A person has an 8 litre jug of water and two empty jugs that hold 5 litres and 3 litres of water. How does he divide the 8 litre jug of water into 2 jugs of 4 litres each? 24. You are in a jail cell with two doors. One leads to freedom the other to the dungeon. There are two guards, one always tells the truth while the other always lies. You can ask just one question to either guard. What is your question, and which guard will you ask?

34 200 Chapter 4 Analyze Puzzles and Games Foundations of Math Chapter Review Section Study the pattern, and predict the missing value. a) = = 111 = 1111 = b) 9 2 = = 9801 = = 2. Study the pattern, and predict the next two terms. a) 2, 6, 15, 31, 56, 92,, b) 59, 51, 55, 46, 50, 40, 44,, 3. Determine the number of line segments in the 100 th pattern. a) b)

35 Foundations of Math 11 Section 4.5 Chapter Review 201 Section Study the pattern and predict the n th term. a) 2, 6, 12, 20, b) 2, 5, 10, 17, c) 0, 1, 4, 9, d) 0, 1 2, 2 3, 3 4, 5. Determine the number of line segments in the n th pattern. a) b)

36 202 Chapter 4 Analyze Puzzles and Games Foundations of Math Make an inductive conclusion about the n th term in the following patterns. a) = 2 3 b) = = = = = = = c) = = = = d) = = = = e) 1 2 = = = = f) = = = =

37 Foundations of Math 11 Section 4.5 Chapter Review 203 Section Reason deductively who is older, Mike or John, if possible. a) Mike is 21 years old; John is a teenager. b) Mike is younger than Sam; John is older than Sam. c) Both Mike and John are older than Paul. d) Mike is older than Sue; John is older than Ann. e) Mike is older than Sue; Sue is older than John. f) The product of Mike and John s age is 360, and John is 19.

38 204 Chapter 4 Analyze Puzzles and Games Foundations of Math Make a deduction from the following information. a) x is an integer between 17.3 and b) x is a negative number and x 2 = 4. c) A, B and C are collinear points and AC + BC = AB. d) A, B and C are collinear points and AB = AC BC. e) Point A lies on line BC, and AB = 1 AC. f) B lies between A and C. C lies between 2 B and D. AC = BD. g) B lies between A and C. C lies between B and D. AB = CD. h) B lies between A and C. DE + BC = AC.

39 N Foundations of Math 11 Section 4.5 Chapter Review 205 Section Put the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9 so the total of each row, column and diagonal add up to the same number. 10. If A, B, C and D are different non-zero digits, find the number such that A B C D 4 D C B A 11. Simplify (x a)(x b)(x c) (x z). 12. Four coins with tails up. 5 CENTS A C N A D A A D A 25 CENTS 1 C A N A D A CANADA CENTS 1999 A C 1 D O L L A R Turning any three coins over counts as one move. How many moves does it take to get all coins to show heads?

40 206 Chapter 4 Analyze Puzzles and Games Foundations of Math How many triangles can you find? 14. There are three light switches in one room controlling three light bulbs, which you can t see, in another room. You don t know which is connected to which bulb. You can make one guess to figure it out. How do you do it? (There is nobody else at home to help you.)