The Law of Ruling Out Possibilities
Chapter 11
Lesson
11-1
Ruling Out Possibilities
BIG IDEA When the solution to a problem is one of a finite number of possibilities, an effective strategy may be to eliminate possibilities until only one is left.
The Law of Ruling Out Possibilities In the short story “A Scandal in Bohemia,” by Sir Arthur Conan Doyle, the famous fictional detective Sherlock Holmes remarks, “Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth.” For instance, if Sherlock Holmes knows that either the maid or the butler committed a crime, and he determined it was not the maid, then the butler must be guilty. We call this principle of reasoning the Law of Ruling Out Possibilities.
Law of Ruling Out Possibilities When statement p or statement q is true, and q is not true, then p is true.
Mental Math Fill in the Blanks a. If a triangle is not scalene, it must be ? . b. If an integer is neither negative nor positive, it ? must be . c. If a positive integer is prime and not odd, then the integer is ? .
This can be extended to “If you know that one of a set of possibilities must be true, and you can eliminate all but one, that one must be true.” The Law of Ruling Out Possibilities is used even by very young children. If a child knows that a toy is either in a parent’s right hand or in the parent’s left hand, and the right hand is opened and found empty, the child will know to look in the left hand for the toy. In mathematics, ruling out possibilities is often easy. For example, you know every angle in a triangle is either acute, right, or obtuse. If ∠A in ABC is not acute or right, then, using the Law of Ruling Out Possibilities, ∠A is obtuse. There is no other possibility. In real life, if words are not carefully defined, you may not be able to rule out possibilities so easily. For example, if a person is not young, that does not necessarily mean the person is old. Sudoku puzzles can be solved by repeated application of the Law of Ruling Out Possibilities.
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Lesson 11-1
Activity 1 Copy and solve this Sudoku puzzle. Remember that each of the numbers 1 through 9 must be in each row, each column, and each square of nine numbers only once.
9 4 7 2 5 6 2 8 1 5 7 5 2 2 3 7 8 3 4 1 1 2 4 6 7 5
Solving Logic Puzzles In the questions for this lesson, you are asked to solve some logic puzzles. These puzzles use the idea of ruling out possibilities again and again. Notice how little information is given and how much you can deduce. The same happens in geometry and other branches of mathematics.
4 3
3 8 5 1 9
1 1 9
4
6 2 2 8 7 6 5 3 9 4 2 1
Here are some hints for doing these puzzles:
1. Logic puzzles take a lot of time and analysis, so do not hurry. 2. Construct a grid and place Xs in squares whenever something cannot occur. Place an O in the square when the situation must occur.
Example Aaliyah, Marissa, Jordan, Hassan, and Ian each have a different hobby. Their hobbies are chess, fencing, sailing, hockey, and knitting. From the clues below, determine which hobby each student has. (1) (2) (3) (4) (5)
Aaliyah likes either sailing, hockey, or fencing. Jordan likes either hockey or knitting. Marissa does not like fencing, hockey, or sailing. Ian does not like sailing, knitting, fencing, or hockey. Hassan does not like fencing.
In fencing, points are scored when the tip of the blade touches a valid target area on the opponent.
Solution To solve this puzzle, the grid at the right can be used. The first clue tells you that Aaliyah does not like chess and does not like knitting. Two X1s in the third row of the grid show this. (We call it X1 so you can tell it comes from Clue 1.) The fourth clue tells you the hobbies Ian does not like. The four X4s in the fifth row show this. Of course, this means that Ian likes chess. We show this with an O. Now we know that no one else likes chess, so we place Xs in the rest of the cells in the column labeled “Chess.”
Hassan
X X5
Marissa
X X3 X3
Aaliyah
X1
Jordan Ian
X X2
X3 X1 X2
O X4 X4 X4 X4 s es ing key ing ing Ch enc Hoc nitt Sail F K
To continue, reread the clues and see what you can eliminate based on the clues and the grid on which you are recording your information. Look carefully at the grid. You now should know Aaliyah’s hobby and the hobbies Aaliyah doesn’t have. Put an O in the box showing that Aaliyah has fencing as a hobby and an X to show the things that are now ruled out as her hobbies. You are asked to complete this puzzle in Question 7.
Ruling Out Possibilities
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Chapter 11
Sometimes a single piece of given information may yield many conclusions, as in the following Activity.
Activity 2 When they were kids, Tina, Ernest, Charles, Lisa, and Jennifer were best friends. Each of them had a dream job. The jobs were actor, astronaut, writer, firefighter, and engineer. When they grew older they all got their dream jobs. These days they are called Ms. Apple, Mr. Leonards, Mr. Jameson, Ms. Nathan, and Ms. Willows, though not necessarily in that order. From the clues below, match each person’s first name, last name, and profession. (1) (2) (3) (4) (5)
Tina and Lisa are not actors, and neither is Ms. Willows. Jennifer is not the firefighter. The astronaut is either Tina or Ms. Apple. Neither Ernest nor Ms. Nathan is the firefighter or actor. Mr. Leonards, Ernest, and the writer have all recently traveled to Europe.
Step 1 Again, we use a grid. The grid should have a space for each last name-first name pair, and for each first name-job pair, and also each job-last name pair. Step 2 Look at Clue 1. Write five conclusions that you can make, looking for possibilities to rule out. Step 3 Mark the grid with Xs or Os for the conclusions made in Step 2. Step 4 Read Clue 2, make your conclusion(s), and mark them on the grid.
s s rd n n t er le na eso tha llow p au r ght eer i r o m a p n e a e N W o A L J fi in s. r. r. s. s. to tr rit re g M M M M M Ac As W Fi En
Tina Ernest Charles Lisa Jennifer Actor Astronaut Writer Firefighter Engineer
Step 5 Repeat Step 4 for the remaining clues and then start over again with Clue 1 until you have found each person’s last name and occupation.
Questions COVERING THE IDEAS 1. You examine a coin. The face that shows up is “heads.” You conclude that the face on the other side is “tails.” What principle of reasoning have you used? 2. Fill in the Blank If statement m or statement n is true, and m is not true, then ? must be true. 650
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Lesson 11-1
3. If A, B, and C are three different points, then either A, B, and C are collinear or there is a triangle ABC. a. If you know that A, B, and C are noncollinear, what can you conclude? b. What logical principle have you used in part a? In 4−6, refer to Activity 2.
4. From Clue 3 alone, who is known not to be the astronaut? 5. Write at least two conclusions that follow from Clue 1. 6. Write at least two conclusions that follow from Clue 5. 7. Finish the Example of this lesson. APPLYING THE MATHEMATICS In 8 and 9, make a conclusion from the given information.
8. Lines m and n in space are neither skew nor parallel. 9. Point A is neither on nor inside the circle with center C and radius 5. 10. The Trichotomy Law for real numbers says: “Of two real numbers a and b, either a < b, a = b, or a > b, and no two of these can be true at the same time.” You know that π ≠ 3.14159. What can you conclude by using the Trichotomy Law? 11. James Fitzgerald, sculptor and logician, was complimented on his ability to carve a lifelike tiger from a piece of wood. He replied, “It’s easy. You start with a block of wood and cut away all parts that do not look like a tiger.” What Law of Logic was he using? 12. Jasmine was given a parallelogram that is not a rhombus, a triangle, a kite that is not a rhombus, and a square, and told to paint each with yellow, blue, red, or green paint. Each is labeled as Polygon 1, Polygon 2, Polygon 3, or Polygon 4. Make a grid and tell Jasmine what color to paint and what number to mark on each shape using the clues below. (1) The parallelograms and Polygon 4 are not green. (2) Polygon 1 and Polygon 3 have diagonals that are perpendicular. (3) The regular quadrilateral is not yellow or red. (4) The yellow polygon has two pairs of opposite sides congruent. (5) Polygon 1 does not have four congruent sides. 13. In the puzzle at the right, each square is to contain a digit from 1 to 9. No digit can appear twice in the same row or in the same column. The sums of each row and column are given. Find one of the two solutions.
10 19 21 19 24 7
Ruling Out Possibilities
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Chapter 11
14. Three students (Chance, Paco, and Tallulah) were placed in a group to do a geometry project. Each student had a favorite geometric shape (hyperbola, sphere, or triangle) and each student had a favorite mathematician from history (Euclid, Nikolai Ivanovich Lobachevski (1792−1856), and Emmy Noether (1882−1935)). Further, each mathematician had a favorite geometric shape. Assume these five statements: (1) Chance is not the student that liked the hyperbola. (2) Noether favored the sphere. (3) Euclid liked the triangle. (4) Paco doesn’t like complicated figures such as the sphere or the hyperbola. (5) If a student favored a mathematician, and the mathematician favored a shape, then the student favored that shape. Write each triplet (student, shape, mathematician) of items that belong together. (You may wish to make one or more tables like the one below to record matches and mismatches.) Hyberbola
Triangle
Sphere
Chance Paco Tallulah
REVIEW 15. A cylinder has height h and bases with radius r. Two hemispheres are glued onto the two bases of the cylinder. A front view of this figure is shown at the right. a. Give a formula for the volume of the figure. (Lessons 10-6, 10-3)
b. Give a formula for the surface area of the figure. (Lessons 10-7, 9-9)
16. A container in the shape of a box is x feet wide, x feet long, and y feet high. The container is filled to the brim with water. How high would a container containing the same volume of water be, if it were y feet wide and y feet long? (Lesson 10-2) 17. True or False Any two nets for the same polyhedron will have the same area. (Lesson 9-8) 18. Name the special kinds of quadrilaterals in which the diagonals intersect each other and are perpendicular to each other. (Lesson 7-9)
19. Suppose r y-axis(x 2, 1) = (x 2 - 4, 1). Find x. (Lesson 4-2) 652
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Lesson 11-1
20. Simplify. (Lesson 1-1) a.
⎪5 __13 ⎥ + ⎪–12⎥
b.
⎪5 __13 − 12⎥
c.
( 5 __13 - 12 ) √ 2
EXPLORATION 21. Below are the 15 clues to one version of a famous and difficult logic puzzle called “Who Owns the Zebra?” People have reported that Lewis Carroll or Albert Einstein wrote it, but the earliest appearance seems to be in 1962, after these men had died. (1)
There are five houses in a row, each of a different color and inhabited by people of different nationalities, with different pets, drinks, and flowers.
(2)
The English person lives in the red house.
(3)
The Spaniard owns the dog.
(4)
Coffee is drunk in the green house.
(5)
The Ukrainian drinks tea.
(6)
The green house is immediately to the right (your right) of the ivory house.
(7)
The geranium grower owns snails.
(8)
Roses are in front of the yellow house.
(9)
Milk is drunk in the middle house.
(10) The Norwegian lives in the first house on the left. (11) The person who grows marigolds lives in the house next to the person with the fox. (12) Roses are grown at the house next to the house where the horse is kept. (13) The person who grows lilies drinks orange juice. (14) The Japanese person grows gardenias. (15) The Norwegian lives next to the blue house. Who drinks water? And who owns the zebra?
22. Try your hand at writing your own puzzle problem. Have a friend or family member try and solve your puzzle. 23. Logic puzzles like those in this lesson can be found in many puzzle books. Find an example of a logic puzzle different from the ones in this lesson, and solve it.
Ruling Out Possibilities
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Lesson
11-1
Ruling Out Possibilities
BIG IDEA When the solution to a problem is one of a finite number of possibilities, an effective strategy may be to eliminate possibilities until only one is left.
The Law of Ruling Out Possibilities In the short story “A Scandal in Bohemia,” by Sir Arthur Conan Doyle, the famous fictional detective Sherlock Holmes remarks, “Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth.” For instance, if Sherlock Holmes knows that either the maid or the butler committed a crime, and he determined it was not the maid, then the butler must be guilty. We call this principle of reasoning the Law of Ruling Out Possibilities.
Law of Ruling Out Possibilities When statement p or statement q is true, and q is not true, then p is true.
Mental Math Fill in the Blanks a. If a triangle is not scalene, it must be ? . b. If an integer is neither negative nor positive, it ? must be . c. If a positive integer is prime and not odd, then the integer is ? .
This can be extended to “If you know that one of a set of possibilities must be true, and you can eliminate all but one, that one must be true.” The Law of Ruling Out Possibilities is used even by very young children. If a child knows that a toy is either in a parent’s right hand or in the parent’s left hand, and the right hand is opened and found empty, the child will know to look in the left hand for the toy. In mathematics, ruling out possibilities is often easy. For example, you know every angle in a triangle is either acute, right, or obtuse. If ∠A in ABC is not acute or right, then, using the Law of Ruling Out Possibilities, ∠A is obtuse. There is no other possibility. In real life, if words are not carefully defined, you may not be able to rule out possibilities so easily. For example, if a person is not young, that does not necessarily mean the person is old. Sudoku puzzles can be solved by repeated application of the Law of Ruling Out Possibilities.
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Lesson 11-1
Activity 1 Copy and solve this Sudoku puzzle. Remember that each of the numbers 1 through 9 must be in each row, each column, and each square of nine numbers only once.
9 4 7 2 5 6 2 8 1 5 7 5 2 2 3 7 8 3 4 1 1 2 4 6 7 5
Solving Logic Puzzles In the questions for this lesson, you are asked to solve some logic puzzles. These puzzles use the idea of ruling out possibilities again and again. Notice how little information is given and how much you can deduce. The same happens in geometry and other branches of mathematics.
4 3
3 8 5 1 9
1 1 9
4
6 2 2 8 7 6 5 3 9 4 2 1
Here are some hints for doing these puzzles:
1. Logic puzzles take a lot of time and analysis, so do not hurry. 2. Construct a grid and place Xs in squares whenever something cannot occur. Place an O in the square when the situation must occur.
Example Aaliyah, Marissa, Jordan, Hassan, and Ian each have a different hobby. Their hobbies are chess, fencing, sailing, hockey, and knitting. From the clues below, determine which hobby each student has. (1) (2) (3) (4) (5)
Aaliyah likes either sailing, hockey, or fencing. Jordan likes either hockey or knitting. Marissa does not like fencing, hockey, or sailing. Ian does not like sailing, knitting, fencing, or hockey. Hassan does not like fencing.
In fencing, points are scored when the tip of the blade touches a valid target area on the opponent.
Solution To solve this puzzle, the grid at the right can be used. The first clue tells you that Aaliyah does not like chess and does not like knitting. Two X1s in the third row of the grid show this. (We call it X1 so you can tell it comes from Clue 1.) The fourth clue tells you the hobbies Ian does not like. The four X4s in the fifth row show this. Of course, this means that Ian likes chess. We show this with an O. Now we know that no one else likes chess, so we place Xs in the rest of the cells in the column labeled “Chess.”
Hassan
X X5
Marissa
X X3 X3
Aaliyah
X1
Jordan Ian
X X2
X3 X1 X2
O X4 X4 X4 X4 s es ing key ing ing Ch enc Hoc nitt Sail F K
To continue, reread the clues and see what you can eliminate based on the clues and the grid on which you are recording your information. Look carefully at the grid. You now should know Aaliyah’s hobby and the hobbies Aaliyah doesn’t have. Put an O in the box showing that Aaliyah has fencing as a hobby and an X to show the things that are now ruled out as her hobbies. You are asked to complete this puzzle in Question 7.
Ruling Out Possibilities
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Chapter 11
Sometimes a single piece of given information may yield many conclusions, as in the following Activity.
Activity 2 When they were kids, Tina, Ernest, Charles, Lisa, and Jennifer were best friends. Each of them had a dream job. The jobs were actor, astronaut, writer, firefighter, and engineer. When they grew older they all got their dream jobs. These days they are called Ms. Apple, Mr. Leonards, Mr. Jameson, Ms. Nathan, and Ms. Willows, though not necessarily in that order. From the clues below, match each person’s first name, last name, and profession. (1) (2) (3) (4) (5)
Tina and Lisa are not actors, and neither is Ms. Willows. Jennifer is not the firefighter. The astronaut is either Tina or Ms. Apple. Neither Ernest nor Ms. Nathan is the firefighter or actor. Mr. Leonards, Ernest, and the writer have all recently traveled to Europe.
Step 1 Again, we use a grid. The grid should have a space for each last name-first name pair, and for each first name-job pair, and also each job-last name pair. Step 2 Look at Clue 1. Write five conclusions that you can make, looking for possibilities to rule out. Step 3 Mark the grid with Xs or Os for the conclusions made in Step 2. Step 4 Read Clue 2, make your conclusion(s), and mark them on the grid.
s s rd n n t er le na eso tha llow p au r ght eer i r o m a p n e a e N W o A L J fi in s. r. r. s. s. to tr rit re g M M M M M Ac As W Fi En
Tina Ernest Charles Lisa Jennifer Actor Astronaut Writer Firefighter Engineer
Step 5 Repeat Step 4 for the remaining clues and then start over again with Clue 1 until you have found each person’s last name and occupation.
Questions COVERING THE IDEAS 1. You examine a coin. The face that shows up is “heads.” You conclude that the face on the other side is “tails.” What principle of reasoning have you used? 2. Fill in the Blank If statement m or statement n is true, and m is not true, then ? must be true. 650
Indirect Proofs and Coordinate Proofs
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Lesson 11-1
3. If A, B, and C are three different points, then either A, B, and C are collinear or there is a triangle ABC. a. If you know that A, B, and C are noncollinear, what can you conclude? b. What logical principle have you used in part a? In 4−6, refer to Activity 2.
4. From Clue 3 alone, who is known not to be the astronaut? 5. Write at least two conclusions that follow from Clue 1. 6. Write at least two conclusions that follow from Clue 5. 7. Finish the Example of this lesson. APPLYING THE MATHEMATICS In 8 and 9, make a conclusion from the given information.
8. Lines m and n in space are neither skew nor parallel. 9. Point A is neither on nor inside the circle with center C and radius 5. 10. The Trichotomy Law for real numbers says: “Of two real numbers a and b, either a < b, a = b, or a > b, and no two of these can be true at the same time.” You know that π ≠ 3.14159. What can you conclude by using the Trichotomy Law? 11. James Fitzgerald, sculptor and logician, was complimented on his ability to carve a lifelike tiger from a piece of wood. He replied, “It’s easy. You start with a block of wood and cut away all parts that do not look like a tiger.” What Law of Logic was he using? 12. Jasmine was given a parallelogram that is not a rhombus, a triangle, a kite that is not a rhombus, and a square, and told to paint each with yellow, blue, red, or green paint. Each is labeled as Polygon 1, Polygon 2, Polygon 3, or Polygon 4. Make a grid and tell Jasmine what color to paint and what number to mark on each shape using the clues below. (1) The parallelograms and Polygon 4 are not green. (2) Polygon 1 and Polygon 3 have diagonals that are perpendicular. (3) The regular quadrilateral is not yellow or red. (4) The yellow polygon has two pairs of opposite sides congruent. (5) Polygon 1 does not have four congruent sides. 13. In the puzzle at the right, each square is to contain a digit from 1 to 9. No digit can appear twice in the same row or in the same column. The sums of each row and column are given. Find one of the two solutions.
10 19 21 19 24 7
Ruling Out Possibilities
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Chapter 11
14. Three students (Chance, Paco, and Tallulah) were placed in a group to do a geometry project. Each student had a favorite geometric shape (hyperbola, sphere, or triangle) and each student had a favorite mathematician from history (Euclid, Nikolai Ivanovich Lobachevski (1792−1856), and Emmy Noether (1882−1935)). Further, each mathematician had a favorite geometric shape. Assume these five statements: (1) Chance is not the student that liked the hyperbola. (2) Noether favored the sphere. (3) Euclid liked the triangle. (4) Paco doesn’t like complicated figures such as the sphere or the hyperbola. (5) If a student favored a mathematician, and the mathematician favored a shape, then the student favored that shape. Write each triplet (student, shape, mathematician) of items that belong together. (You may wish to make one or more tables like the one below to record matches and mismatches.) Hyberbola
Triangle
Sphere
Chance Paco Tallulah
REVIEW 15. A cylinder has height h and bases with radius r. Two hemispheres are glued onto the two bases of the cylinder. A front view of this figure is shown at the right. a. Give a formula for the volume of the figure. (Lessons 10-6, 10-3)
b. Give a formula for the surface area of the figure. (Lessons 10-7, 9-9)
16. A container in the shape of a box is x feet wide, x feet long, and y feet high. The container is filled to the brim with water. How high would a container containing the same volume of water be, if it were y feet wide and y feet long? (Lesson 10-2) 17. True or False Any two nets for the same polyhedron will have the same area. (Lesson 9-8) 18. Name the special kinds of quadrilaterals in which the diagonals intersect each other and are perpendicular to each other. (Lesson 7-9)
19. Suppose r y-axis(x 2, 1) = (x 2 - 4, 1). Find x. (Lesson 4-2) 652
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Lesson 11-1
20. Simplify. (Lesson 1-1) a.
⎪5 __13 ⎥ + ⎪–12⎥
b.
⎪5 __13 − 12⎥
c.
( 5 __13 - 12 ) √ 2
EXPLORATION 21. Below are the 15 clues to one version of a famous and difficult logic puzzle called “Who Owns the Zebra?” People have reported that Lewis Carroll or Albert Einstein wrote it, but the earliest appearance seems to be in 1962, after these men had died. (1)
There are five houses in a row, each of a different color and inhabited by people of different nationalities, with different pets, drinks, and flowers.
(2)
The English person lives in the red house.
(3)
The Spaniard owns the dog.
(4)
Coffee is drunk in the green house.
(5)
The Ukrainian drinks tea.
(6)
The green house is immediately to the right (your right) of the ivory house.
(7)
The geranium grower owns snails.
(8)
Roses are in front of the yellow house.
(9)
Milk is drunk in the middle house.
(10) The Norwegian lives in the first house on the left. (11) The person who grows marigolds lives in the house next to the person with the fox. (12) Roses are grown at the house next to the house where the horse is kept. (13) The person who grows lilies drinks orange juice. (14) The Japanese person grows gardenias. (15) The Norwegian lives next to the blue house. Who drinks water? And who owns the zebra?
22. Try your hand at writing your own puzzle problem. Have a friend or family member try and solve your puzzle. 23. Logic puzzles like those in this lesson can be found in many puzzle books. Find an example of a logic puzzle different from the ones in this lesson, and solve it.
Ruling Out Possibilities
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