Course:MATH110/Archive/2010-2011/003/Homework/4/Solutions

Here are some good solutions for homework 4, chosen from the groups.

Problem 1

Five persons named their pets after each other. From the following clues, can you decide which pet belongs to Suzan's mother?

• Tosh owns a cat,
• Bianca owns a frog that she loves,
• Jaela owns a parrot which keeps calling her "darling, darling",
• Jun owns a snake, don't mess with him,
• Suzan is the name of the frog,
• The cat is named Jun,
• The name by which they call the turtle is the name of the woman whose pet is Tosh,
• Finally, Suzan's mother's pet is Bianca.

Solution by Group 13

The best way to tackle a problem such as this would be to go through each piece of information given and organise the data into a form much easier to go back to and make adjustments to. Below is a simple person to pet chart I drew up from the information given.

Tosh ---> Cat (Jun)

♀Bianca ---> Frog (Suzan)

♀Jaela ---> Parrot (No name)

♂Jun ---> Snake (No name)

♀Suzan ---> Turtle (No name)

With ♀ denoting female and ♂ denoting male, we can see in the table that the information has given us enough to start off with. We know Bianca is female from the statement "Bianca owns a frog that she loves" and we know that Jaela is also female from the statement "Jaela owns a parrot which keeps calling her 'darling, darling'". We also know that Jun is male from the statement "Jun owns a snake, don't mess with him", with him referring to the owner and not the pet, otherwise it would be saying don't mess with it.

We can also assume that Suzan's pet is the turtle since both are the only owner and pet that do not have a match and since the problem states that there are only five people then we can assume that there are also only five pets.

Next, we can assign the pet name Tosh to the only other female aside from Suzan, Jaela. We can do this because we know that Suzan's pet is the turtle and if the name by which they call the turtle is the name of the woman whose pet is Tosh then naming the turtle Tosh would mean that the turtle's name needs to be Suzan which is all very confusing and counter-intuitive.

♀Jaela ---> Parrot (Tosh)

This piece of information also allows us to name Suzan's pet because again the name of the turtle would be the name of the person whose pet is called Tosh. Thus we receive the next bit of information:

♀Suzan ---> Turtle (Jaela)

That only leaves the last part of the problem, which would be Jun's pet:

♂Jun ---> Snake (Bianca)

Though the logic may fit, this solution is wrong as Jun is supposedly male and Susan's mother cannot possibly be male unless the label "mother" is semantically misleading (i.e. nickname, etc.).

There are then two possible solutions after this error. The first is that the statement "Jun owns a snake, don't mess with him" has the pronoun 'him' refer to the snake, which can therefore make the solution of Jun owning the snake Bianca correct (although the male snake would be awkwardly named Bianca). This solution ends up with Jun being Suzan's mother and the pet Bianca being her pet.

The second solution could be that the turtle is not owned by Suzan. This would allow any other person to own the turtle, allowing the turtle to then be named Suzan and therefore allowing Suzan's pet to be Tosh and lastly having Jaela's pet to be Bianca (and thus having Jaela as Suzan's mother). This scenario assumes that Jun is male and assumes that since the definition of the problem never assigns a finite number of pets within the group or never limits the amount of pets that the group can have (or limits the doubling of names within those pets, etc.) that there is the possibility that each person can own more than one pet. The problem never states that the pets listed are the only pets that these individuals own and thus can be seen as a possible solution, and a more likely one then the previously aforementioned. In this case, it is not realistically possible to know who Suzan's mother is with the given information.

Problem 2

Bohao, Stewart, Dylan, Tim and Chan are the five players of a basketball team. Two are left handed and three right handed, Two are over 2m tall and three are under 2m, Bohao and Dylan are of the same handedness, whereas Tim and Chan use different hands. Stewart and Chan are of the same height range, while Dylan and Tim are in different height ranges. If you know that the one playing centre is over 2m tall and is left handed, can you guess his name?

Solution by Group 13

First, simplify the information:

Bohao, Stewart, Dylan, Tim and Chan

2 are left-handed, 3 are right-handed

Bohnao and Dylan = Same Hand

Tim and Chan = Different Hand

2 are over 2m, 3 are under 2m

Steward and Chan = Same Height

Tim and Dylan = Different Height

Need to find: player that is over 2 meters tall and left-handed

Now draw a diagram to help solve the first part of the problem (Left/Right Handed).

Now draw another diagram to help solve the final part of the problem (Above/Under 2 meters in height).

Now, using these two diagrams find a player that is over 2 meters tall and left-handed

By process of elimination we know that it can't be:

Bohao (right-handed)

Dylan (right-handed)

Stewart (Under 2 meters)

Chan (Under 2 meters)

Therefore it COULD only be TIM

Therefore the player that is over 2 meters tall and left-handed is TIM

Problem 3

Adam, Bobo, Charles, Ed, Hassan, Jason, Mathieu, Pascal and Sung have formed a baseball team. The following facts are true:

• Adam does not like the catcher,
• Ed's sister is engaged to the second baseman,
• The centre fielder is taller than the right fielder,
• Hassan and the third baseman live in the same building,
• Pascal and Charles each won $20 from the pitcher at a poker game,
• Ed and the outfielders play cards during their free time,
• The pitcher's wife is the third baseman's sister,
• All the battery and infield except Charles, Hassan and Adam are shorter than Sung,
• Pascal, Adam and the shortstop lost $100 each at the race track,
• The second baseman beat Pascal, Hassan, Bobo and the catcher at billiards,
• Sung is in the process of getting a divorce,
• The catcher and the third baseman each have two legitimate children,
• Ed, Pascal Jason, the right fielder and the centre fielder are bachelors, the others are all married
• The shortstop, the third baseman and Bobo all attended the fight,
• Mathieu is the shortest player of the team,

Determine the positions of each player on the baseball team.

Note: On a baseball team there are three outfielders (right, centre and left), four infielders (first baseman, second baseman, third baseman and shortstop) and the battery (pitcher and catcher).

Solution by Group 2

This question took me a reallllllly long time. There are a few things to keep in mind:

1. Don't assume anything (all the facts are there)

2. If someone has a wife, that means their marrid (sung is married)

3. Don't assume that Ed and the outfielders were present at the poker game

4. It might help if you tart off with a table like this:

I followed these steps to get to the solution. Here we go:

Hassan and the third basemen live in the same building

           Therefore Hassan is not the third basemen

Pascal and Charles won $20 some the pitcher at the poker game

           Pascal and Charles are not the pitcher

Ed and outfielder play cards on their free time

           So Ed is not an outfielder

The pitchers wife is the third baseman’s sister

           So the pitcher is married

All battery and infield except Charles Hassan and Adam are shorter then Sung

           Charles Hassan and Adam are part of the battery and infield. Not making them outfielders

Pascal Adam and Shortstop lose $100 at the race track

           Therefore Pascal and Adam are not shortstop

The second basemen beat Pascal, Hassan, Bobo and the catcher at billiards

            So Pascal, Hassan and Bobo are not the second basemen and the catcher

Sung is in the process of getting a divorce

             Meaning that Sung is married (he still has a wife and can’t be considered a bachelor)

The catcher and the third basemen each have two legitimate children

         Making the catcher and the third basemen married

          From all this information, it is concluded that: Ed, Pascal, Jason, right fielder and the outfield center are bachelor

         Ed, Pascal, Jason, the right field and the center field are bachelors

         That means Ed, Pascal and Jason are not the right fielder and center fielder

         From the information we can say that: the outfield right, the infield second and the outfield center are bachelors. 
         The infield third, the pitcher and the catcher are married. 

          Ed is not the infield third the pitcher or the catcher. 
          Jason is no the infield third the pitcher or the catcher. 
           Sung is not out field right out field center and infield second

The third basemen, the shortstop and Bobo attended the fight

         Concluding that Bobo is not the shortstop or the third baseman. 

       Mathieu is the shortest player of the team

         Meaning that Mathieu cannot be center field because it is stated that the center fielder is taller than the right fielder 
         (We crossed Mathieu out of this position earlier on, just double checking)

      Following all these steps we can conclude that Bobo is the outfield center. So Bobo is not any of the other positions

      From this elimination, we can find that Mathieu is outfield right.  This eliminates him from the rest of the positions

          We can conclude that Bobo is a bachelor because he is outfield center

       We can conclude that Mathieu is a bachelor because he is outfield right

       Only a bachelor can be infield second so that leaves us with Jason. 
       Jason is the infield second and is eliminated from the other positions

      We can conclude that Adam, Charles, Hassan, Sung are married

       Sung is the outfield left because he is not the battery or the infield since they are shorter than him. 

       Eliminating the rest of the people being outfield left.

       When this elimination is made: Pascal is the infield first.  This eliminates everyone else from that position. 

      When this elimination is made: Ed is the infield shortstop this means that no one else is shortstop. 
      This eliminates everyone from short stop. 

       When this elimination is made: Hassan is the pitcher. This means no one else can be the pitcher. 

      When this happens we can conclude that Adam is the infield third. So everyone is eliminated from the position

So…. Charles is the catcher.

ADAM= infield third

CHARLES= battery catcher

HASSAN= battery pitcher

PASCAL= infield first

ED= infield shortstop

JASON= infield second

SUNG= outfield left

BOBO= outfield center

MATHEIU= outfield right

THIS IS A LOT OF INFORMATION.... but at the end you feel soooooo good :)

Problem 4

Six players - Petra, Carla, Janet, Sandra, Li and Fernanda - are competing in a chess tournament over a period of five days. Each player plays each of the others once. Three matches are played simultaneously during each of the five days. The first day, Carla beats Petra after 36 moves. The second day, Carla was again victorious when Janet failed to complete 40 moves within the required time limit. The third day had the most exciting match of all when Janet declared that she would checkmate Li in 8 moves and succeeded in doing so. On the fourth day, Petra defeated Sandra. Who played against Fernanda on the fifth day?

Solution by Group 12

From the description, we know that Carla had a match against Petra. We also know that there are three matches played at the same time during each of the five days. So, from these statements, we know that these matches could be a possibility:

• Janet vs Sandra
• Janet vs Fernanda
• Sandra vs Li
• Sandra vs Fernanda
• Fernanda vs Li

On the second day, we know that Carla had a match against Janet. Leaving us with these possible match-ups:

• Petra vs Janet
• Petra vs Li
• Petra vs Fernanda
• Sandra vs Li
• Sandra vs Fernanda
• Fernanda vs Li

On the third day, we know that Janet had a match against Li. Leaving us with these possible match-ups:

• Petra vs Li
• Petra vs Fernanda
• Carla vs Sandra
• Carla vs Li
• Carla vs Fernanda
• Sandra vs Li
• Sandra vs Fernanda
• Fernanda vs Li

On the fourth day, we know that Petra had a match against Sandra. Leaving us with these possible match-ups:

• Carla vs Li
• Carla vs Fernanda
• Janet vs Petra
• Janet vs Fernanda
• Li vs Fernanda

Using the possible match-ups above, we can conclude the following possible match-ups:

Day One

• Janet vs Sandra (This possible match-up only exists on Day 1)
• Fernanda vs Li (This possible match-up exists across all four days whereas other possible match-ups don't. Since this is universal, it has the be this match-up)

Day Two

• Petra vs Li (This possible match-up exists up until Day 4, so this match-up has to occur either on this day or the next; whether on this day or the next does not matter)
• Sandra vs Fernanda (This possible match-up exists up until Day 4, so this match-up has to occur either on this day or the next; whether on this day or the next does not matter)

Day Three

• Petra vs Fernanda (Has to occur because this match-up does not exist on Day 4.)
• Carla vs Sandra (Has to occur because this match-up does not exist on Day 4.)

Day Four

• Fernanda vs Janet (Remaining match-ups from information given)
• Carla vs Li (Remaining match-ups from information given)

From the listed match-ups above, we can safely conclude that Carla had the match against Fernanda because of the face that each member plays each other once.

Problem 5

Homer finally had a week off from his job at the nuclear power plant and intended to spend all nine days of his vacation (Saturday through the following Sunday) sleeping late. But his plans were foiled by some of the people who work in his neighbourhood.

On Saturday, his first morning off, Homer was wakened by the doorbell; it was a salesman of magazine subscriptions.

On Sunday, the barking of the neighbour's dog abruptly ended Homer's sleep.

On Monday, he was again wakened by the persistent salesman but was able to fall asleep again, only to be disturbed by the construction workers next door.

In fact, the salesman, the neighbour's dog and the construction workers combined to wake Homer at least once each day of his vacation, with only one exception.

The salesman woke him again on Wednesday; the construction workers on the second Saturday; the dog on Wednesday and on the final Sunday.

No one of the three noisemakers was quiet for three consecutive days; but yet, no pair of them made noise on more than one day during Homer's vacation. On which day of his holiday was Homer actually able to sleep late?

Solution by Group 2

1. Character Symbols C= Construction Workers D = Neighbor's dog S = Salesman

2. Diagram Confirming the Days Homer was Awoken :

Sat: S

Sun: D

Mon: S+C

Tues: ?

Wed: S+D

Thurs: ?

Fri: ?

Sat: C

Sun: D

3. Key Information to keep in mind. a) Not one of the 3 noisemakers were quiet for 3 consecutive days

b)No pair of the noisemakers made noise on more than 1 day during Homer's holiday, which means that

there could be a pair of the dog+construction workers on a certain day.

c)The construction worker has to annoy Homer on Thursday.

d) From the information listed above, the salesman has to come by on Friday, since the last stated day he came was on Wednesday. He could not come by on Thursday.

e) The last time the dog makes a sound is on Sunday so the dog has to bark on a day prior to Thursday. But due to conflicting information, the dog has to bark on Thursday. It should work as the construction workers and dog did not combine their forces for any time in the given information.

4. Chart:

Sat: S

Sun: D

Mon: S+C

Tues: Rest

Wed: S+D

Thurs: C+D

Fri: S

Sat: C

Sun: D

5. Conclusion The vacation day where Homer gets to sleep has to be Tuesday.