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Chapter 3 Elimination of possibilities

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“Once you have eliminated the impossible, then whatever left, no matter how improbable, must be the solution”

— Sherlock Holmes

Eliminating possibilities can be a powerful technique in answering multiple-choice questions, such as

IQ test question

The map below shows the places listed below. Using only your eyes, imagine a line joining Dallas to Minneapolis. Then imagine a line going from Washington D.C. to San Francisco. What city will be the nearest to the intersection of these lines?

a) Atlanta

b) Chicago

c) Portland

d) Denver

e) El Paso

f) Kansas City

g) Los Angeles

Example 1

When will the equation be true (given that c and f are non-zero numbers)?

- Always
- Only when c and f are both 1.
- Only when c = f.
- Only when | c | = | f |
- Never.

Example 2 Who is lying?

Jim tells lies on Fridays, Saturdays, and Sundays. He tells the truth on all other days. Freda tells lies on Tuesdays, Wednesdays, and Thursdays. She tells the truth on all other days.

If on one day they both said, “Yesterday I lied,” then what day did they say that?

Example 2 Who is lying?

Jim tells lies on Fridays, Saturdays, and Sundays. He tells the truth on all other days. Freda tells lies on Tuesdays, Wednesdays, and Thursdays. She tells the truth on all other days.

If on one day they both said, “Yesterday I lied,” then what day did they say that?

Example 3

Find the square root of 5329 with “elimination of possibilities” but without a calculator.

ends in 1

not the answer

ends in 4

not the answer

72

ends in 9

probable

73

ends in 6

not the answer

74

not the answer

ends in 5

75

ends in 6

not the answer

76

probable

ends in 9

77

78

ends in 4

not the answer

79

ends in 1

not the answer

Three intelligent ladies were standing in row, one behind the other, all facing forward. They were all blindfolded. “Here are 5 hats’” said a man, “2 are red and 3 are white. I shall place one hat on each of your heads and put the other two in a box. You may then remove your blindfolds. But, you must stand still and cannot turn your head. The one who can correctly tell me the color of her hat will receive a prize.”

After the man hid the remaining 2 hats, he told the ladies to remove the blindfolds.The lady at the end of the row said: “There is not enough information for me to deduce the color of my hat.”The middle lady next said: “Same for me.”

The lady in the front then said: “Now I know the color of my hat.” And her answer was right!Assuming that they were really smart, how could the lady in the front deduce the color of her hat even if she could not see the other two hats?

Let’s list all possible combinations.

This is forward

2 red hats and 3 white hats

Now, if A is wearing a red hat, then B can determine that B herself is wearing a white hat.

A more challenging version of the same riddle

Four intelligent men were sitting in a circle, blind folded. The host of the game brought out 3 red hats and 4 green hats. She then carefully put one hat on each man’s head and hided the extra hats.

Next she removed all blind folds and asked each of the 4 men to deduce the color of his own hat.

On the next slide, you will see the conversations. And remember, each man knew that there were at most 3 red hats and at most 4 green hats, yet no man can see the color of his own hat.

Let’s list all possible combinations

(3R 4G)

A: “There is not enough information for me to determine.” (Click to eliminate)

B: “There is still not enough information for me to determine.” (Click to eliminate)

C: “There is still not enough information for me to determine.” (Click to eliminate)

D: “Now I know the color of my hat”. D is right, but he is completely color blind, the host is aware of this but other 3 people are not color blind and are unaware of this.

What is the color of Mr. D’s hat and how did he determine that?

Let’s list all possible combinations

Host: “Okay, I’ll give you a hint – there is at least one red hat being worn.”

A,B,C protested together: “I knew that a long time ago!”

Host: “Alright, then if anyone of you can tell me the color of your hat correctly, you can still win a prize.”

After a long silence, they all say that they have the answer – Red.

Cryptarithmetic Problems

In the addition problem below, each letter represents a digit and different letters represent different digits. Can you decode the math problem?

T O P

+ T O T

O PT

Cryptarithmetic Problems

In the addition problem below, each letter represents a digit and different letters represent different digits. Can you decode the math problem?

S H E

+ E E L

E L S E

Cryptarithmetic Problems

In the addition problem below, each letter represents a digit and different letters represent different digits. Can you decode the math problem?

BOYS

+ BOYS

SI LLY

The New Monopoly Game

Example 4 The secret to MonopolyTM

The victor in a MonopolyTM game often depends on who has the most houses and hotels. You might say that the person with the most houses and hotels controls the game.

In the following addition problem, each letter stands for a different digit, 0 through 9. No two different letters stand for the same digit.

H O U S E S

+H O T E L S

C O N T R O L

Hint: O = 7

H O U S E S

+H O T E L S

C O N T R O L

Hints:

- What can you say about C?
- What can you say about H?
- What can you say about N?
- What can you say about U?
- What can you say about S?

8 7 U S E S

+8 7 T E L S

1 7 N T R 7 L

Final Answer

H O U S E S

+H OT E L S

C O N T R O L

8 7 9 6 4 6

+ 8 7 3 4 2 6

1 7 5 3 0 7 2

Knights and Knaves

There is an island containing two (and only two) types of people: knights who always tell the truth and knaves who always lie.

One day you visit the island and are approached by a group of six (6) natives, A, B, C, D, E, and F who speak to you as follows:

A says: None of us is a knight.

B says: At least 3 of us are knights.

C says: At most 3 of us are knights.D says: Exactly 5 of us are knights.

E says: Exactly 2 of us are knights

F says: Exactly 1 of us is a knight.

Determine who are knights and who are knaves.