Lesson Advanced logical problems

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Advanced logical problems

Problem 1

James, Alex, and Linda were playing inside of their classroom.
One of them broke the window. When their teacher asked who broke the window, she got these replies:
    Linda:     James broke the window!
    James:   I didn't break it!
    Alex:      Neither did I!
If two of the children lied, then who broke the window?

Solution

(1)  Let's assume Linda broke the window.
 
     Then James and Alex are truthful, which contradicts to the condition; hence, this assumption is wrong.



(2)  Let's assume that James broke the window.

     Then Linda and Alex are truthful, which contradicts to the condition, again;  hence, this assumption is wrong.



(3)  So, if the problem is built in logically correct way, then the only possible version is that Alex broke the window.



(4)  Nevertheless, we should check, if the problem is REALLY built in logically correct way.

     So, let's assume that Alex broke the window.


     Then Linda and Alex are liars and James is truthful, which is consistent with the condition.



The problem is just solved, and the ANSWER is:  Alex broke the window.

Problem 2

Brianna, Ryan, Tyler, and Ashley were recently elected as the new class officer (president, VP, Secretary, Treasurer).
From the following clues, determine which position each holds.
    A.   Ashley is younger than the president but older than the treasurer.
    B.   Brianna and the secretary are both same age, and they are the youngest member of the group.
    C.   Tyler and the secretary are next-door neighbors.

Solution

1.   First portion of logical arguments

    From A), Ashley is not the president and is not the treasurer.

    From A), Ashley is not the youngest of the group;

             hence, from B), Ashley is not the secretary.

    Thus the only one last opportunity for Ashley is to be the VP.

2. Second portion of logical arguments

    From #1, Ashley is not the secretary (he/she is just the VP !);

    From C), Tyler is not the secretary;

    From B), Brianna is not the secretary.

    It leaves only one (last) opportunity for the secretary:  Ryan is the secretary.

3. Third portion of logical arguments

    From #2, Ryan is not the president (he is just the secretary !);

    From #1, Ashley is not the president (he/she is just the VP !);

    From A), Brianna is not the president (since she is too young !);

    
    It leaves only one (last) opportunity for the president:  Tyler is the president.

4. The last portion of logical arguments

    Brianna is the treasurer (last vacancy !).

--------------------

One small secret about solving similar logical problems - as a conclusion.

    It is useful to make a matrix of situations.

    In this problem this matrix looks like this:



        President    VP    Secretary  Treasurer
        
    B

    R

    T

    A



Vertical input column (the most left) contains the names of participants;

The most upper horizontal input line contains the names of positions.


Then you fill the cells of the matrix by the signs " + " or " - " according to your logical conclusions.


It helps you to document the process of your thinking when you solve the problem step by step.


It even helps you sometime to find your next step and/or to find the next set of logical arguments.


From the other side, such a matrix only HEPLS you; but it is not an engine.

The engine is your mind / (your brain).

Problem 3

Alfred, Roger, Clay, Stan and Bob are the starters on the local basketball team.
Two of them are left-handed and three are right-handed.
Two are over 6 feet tall and 3 are under 6 feet tall.
Alfred and Clay are of the same handedness whereas Stan and Bob use different hands.
Alfred and Clay are of the same height whereas Clay and Stan are of different hands.
Stan is shorter than Clay.
The man who plays center is over 6 feet tall and is left-handed. What is his name?

Solution

From the condition, we have these three statements:

    (1)  Alfred and Clay are of the same handedness.

    (2)  Stan and Bob use different hands.

    (3)  Clay and Stan are of different hands.


Comparing statements (2) and (3), we see that Bob and Clay use the same hand (both are opposite to Stan).

Adding statement (1) to it, we see that the three, Bob, Clay and Alfred use the same hand; 

    hence, these three, A, B and C all are right-handed.


From it, we conclude that the left-handed are remaining two, Roger and Stan.

So, only Roger and Stan are potential candidates for the role of playing center.


Now we compare the height of the participants. 

Stan is shorter than Clay. Since Alfred and Clay are of the same height, Stan can not be anong these two, who are over 6 feet tall.

So, the only player who is left-handed and over 6 feet tall is Roger, ONLY.

Problem 4

Each of the four children Dino, Sonny, Nikko, and Xian,
holds a different object (fan, car, dart, and slime). From the following clues, determine who owns which object.
    (a)   Sonny is older than his friend who owns the car and younger than his friend who owns the dart.
    (b)   Nikko and his friend who owns the slime both have the same age. They are the youngest members of them.
    (c)   Dino is older than his friend who owns the fan.

Solution

First, let's order the children on the time-line according their age and their objects.


Regarding the age, from (a) and (b) we have this time-line of increasing age

   ------------------------------------------------------------------------------------>
    Nikko and his friend(slime)              Sonny            Sonny's friend(dart)
    are youngest. They have the same age     


From this info, and adding info from (c), we conclude that  Sonny's friend(dart) is Dino.


So, we can update our time-line in this way

   ------------------------------------------------------------------------------------>
    Nikko and his friend(slime)              Sonny            Dino(dart)
    are youngest. They have the same age     


Looking at this time-line, we can conclude that the Nikko's friend(slime) is Xian, since there is no other opportunity.


So, we can update our time-line in this way

   ------------------------------------------------------------------------------------>
    Nikko and Xian(slime)                    Sonny            Dino(dart)
    are youngest. They have the same age     


Looking at the last time-line, we can conclude from (a) that Nikko owns the car.


Thus the ANSWER  is:  Nikko has a car;  Xian has a slime;  Sonny has a fan;  Dino has a dart.

Problem 5

Because of the effect of the TRAIN LAW, Anna, Jennifer, and Khaeh each decided to take two jobs,
but no two have the same occupation. The occupations are nurse, accountant, singer, architect, cartoonist,
and interior designer. Given the information below, determine the occupations of each individual.
    (a)   The nurse had lunch with the singer.
    (b)   The singer went fishing with Anna, who is not a cartoonist.
    (c)   The interior designer is related to the accountant.
    (d)   The nurse hired the interior designer to do a job.
    (e)   Jennifer lives next door to the cartoonist.
    (f)   Khaeh beats Jennifer and the interior designer at tennis.
    (g)   Khaeh is not the nurse.

Solution

(0) Start preparing this 3x6 table

nurse accountant singer architect cartoonist interior designer
Anna
Jennifer
Khaeh

For now, this table is empty.
We will move forward step by step, making logical conclusions and filling the table gradually
by placing "+" or "-" signs in the cells.
Notice that each column must have precisely one "+" and each row must have precisely two "+".

(1) From (b), Anna is not the cartoonist.
From (e), Jennifer is not the cartoonist.
Hence, Khaeh is the cartoonist.