Jan. 2005Logical ReasoningSlide 1 Logical Reasoning A Lesson in the “Math + Fun!” Series.

Jan. 2005 Logical Reasoning Slide 1

Logical ReasoningA Lesson in the “Math + Fun!” Series


About This Presentation

Edition Released Revised Revised

First Jan. 2005

This presentation is part of the “Math + Fun!” series devised by Behrooz Parhami, Professor of Computer Engineering at University of California, Santa Barbara. It was first prepared for special lessons in mathematics at Goleta Family School during the 2003-04 and 2004-05 school years. The slides can be used freely in teaching and in other educational settings. Unauthorized uses are strictly prohibited. © Behrooz Parhami


Aging Three Years in One Year?Amy says that she was 10 years old the day before yesterday and that she will turn 13 next year. How is this possible?(Think for a few minutes before you advance to the next slide.)

2004 2005 2006

12 13

TodayYesterday

The day before yesterday

10 11

This year Next year

Age:

Jan. 1

Dec. 31 Dec. 31


Mislabeled BagsWe have three bags, each holding two balls.One bag has two black balls. One bag has two white balls.One bag has one black and one white ball.The bags are labeled black, white, and mixed.None of the labels is correct, however.Correct the labeling by drawing a ball from only one bag.

B W M B M WW B MW M BM W BM B W

Extra challenge:

The number of balls in bagsis not known, but they are all black, all white, or mixed.

How do you solve the puzzleIn this case?

Minesweeper: Exampleof a reasoning game


Making Change for a Dollar

Can you have more than one dollar in change and still not be able to give exact change for one dollar?

Extra challenge: What is the most amount of change that you can have and still not be able to give exact change for one dollar?


River-Crossing Problems

Example: Person, fox, duck, corn must crossBoat can carry the person plus one animal/thingFox and duck should not be left togetherDuck and corn should not be left together

P F D C

There are people, animals, and things on one side of a riverThey must be transported to the other sideThere is a boat that cannot carry all of them at onceThere are restrictions on who can be left alone with whom or what

FC

PD

P FC

D CPFD

FPDC

PCF

DCF

PD

P FDC

PDCF


Graphs to Solve River Crossings

Example: Person, fox, duck, corn must crossBoat can carry the person plus one animal/thingFox and duck should not be left togetherDuck and corn should not be left together

P F D C

C/PFD

F/PCD

D/PCF

PDC/F

PDF/C

PD/CF -/PDCFFC/PD

FC

PD

PFC/D

P FC

D

PFDC/-

P FDC

From each state, there are a number of legal “moves”

Our previous solution

A different solution


Activity 1 – River CrossingThree soldiers get to a river they must cross; there’s no bridge in sight.They spot two boys playing in a tiny boat.The soldiers ask the boys if they can use the boat to cross the river.The boat is so tiny that it can carry only one soldier or the two boys. How can the three soldiers cross the river?

S1S2S3B1B2


Activity 2 – Bridge CrossingFour people must cross a bridge in 16 minutes or less.Individually, they can cross in 1, 2, 5, and 9 minutes.It is dark, so a flashlight is needed.They have only one flashlight.At most two can cross together.They must walk at the pace of the slower person.No other equipment is available.

P1P2P5P9

Flashlight

1 min

2 min

5 min9 min


Three Heads and Five HatsAmy, Bert, and Cindy are arrested in Fantasyland and taken to an ogre.They are shown five hats, two white and three black.They are lined up, Amy in front, Bert in the middle, and Cindy last.They are blindfolded and one of the five hats is placed on each one’s head.The blindfolds are removed, but each one can see only those in front of him/her.The ogre says that if any of them correctly guesses the color of his or her hat in less than one minute, all three will be set free; otherwise, . . .No one says anything for 58 secondsOn the 59th second, Amy shouts “black”, and they are set free. How did she know?

AmyBert

Cindy

? ??

Black!Black!

Black!


ChuckieAngelica

Bob

Activity 3: Blue or Red Spot?Angelica, Bob, and Chuckie go to a job interview.The manager says that she will put a red or blue spot on each forehead.Each will see the spots on the other foreheads, but not the one on his/her own.There are no mirrors in the room.The manager puts red spots on all three foreheads. She tells them that if they see a red spot, they should raise their right hand; as soon as a person knows the color of his/her spot, he/she should lower his/her handAfter a minute or so, Angelica lowers her hand and says “I must be red.” Why?

Red!


Which Switch Turns the Light on?

There are three light switches in one room and a lightbulb in an adjacent room.You know that the lightbulb is off now.You are allowed to go or look into the other room only once.How can you tell which light switch turns the lightbulb on?

Flipping one switch and checking on the lightbulb may not provide the answer.

Flipping two switches and checking on the lightbulb may not provide the answer.

Solution:Flip the first switch; wait for a few minutes; flip the first switch again.Flip the second switch and go into the next room.If the lightbulb is off, but warm to the touch, the first switch controls it; if it is on, the second switch controls it; if it is off, but cold, the third switch controls it.

Hint: How can you tell a lightbulb that has just been turned on from one that has been on for a while?


Truth Tellers and LiarsTwo tribes live on the LiaTru island. Members of one tribe always tell the truth; members of the other tribe always lie. For example, if you ask a boy on the LiaTru island “Are you a boy?” he will answer “Yes” if he is a truth teller and “No” if he is a liar. If a girl on the LiaTru island is 10 years old and you ask her “Are you 9 years old?” she will answer “No” if she is a truth teller and “Yes” if she is a liar.Members of the two tribes can recognize each other, but you can’t tell them apart.

Which way to GFS?

That way!

Puzzle 1: On your way to GFS, you arrive at a fork in the road.Someone appears, but you don’t know whether he is a truth teller or a liar. How can you find out which way leads to GFS?

Puzzle 2: How can you discover whether a person you meet is a truth teller or a liar?


Activity 4: Truth Tellers and Liars

Puzzle 2: You meet two people, C and D. C says, “Exactly one of us is from the liars tribe.” Which tribe is D from?

Puzzle 3: You meet two people, E and F. F says, “It is not the case that both of us are from the truth tellers tribe.” Which tribe is E from? What about F?

Puzzle 4: You meet two people, G and H. Each of the two makes a statement: G: “We are from different tribes.”H: “G is from the liars tribe.”Which tribes are G and H from?

Puzzle 1: You meet two people, A and B. A says, “Both of us are from the liars tribe.” Which tribe is A from? What about B?

BAYou


Truth Tellers, Liars, and RandomsThree tribes live on the LiaRanTru island.

Members of one tribe always tell the truth; members of the second tribe always lie; members of the third tribe answer completely at random.For example, if you ask a boy on the LiaRanTru island “Are you a boy?” he will answer “Yes” if he is a truth teller, “No” if he is a liar, and “Yes” or “No” (unpredictable) if he is a random. Members of the tribes can recognize each other, but you can’t tell them apart.

Puzzle : Three people from the LiaRanTru island, one representing each tribe, come to town. You must identify who is from which tribe by asking only three yes/no questions. Each question must be directed to only one person, but you can ask the same person multiple questions. P1

Hint : There are 6 possibilities for P1, P2, P3: LRT, LTR, RLT, RTL, TLR, TRL

P3P2


Activity 5: Other Logical Puzzles

Puzzle 2: Three travelers are told that their hotel room costs $30, so they pay the clerk $10 each. The clerk later realizes that he made a mistake and should have only charged them $25. He gives the bellboy $5 to return to the travelers. The bellboy is dishonest and gives each of them only $1, keeping $2 for himself. So the travelers actually spent $27 and the bellboy pocketed $2. What happened to the other dollar of the original $30?

Puzzle 3: Two mathematicians are walking down the street. The first says to the second, "I know you have three children. What are their ages?" The second replies, “The product of their ages is 36." The first says, "I can't tell their ages from that." The second says, “Well, the sum of their ages is the same as that address across the street." The first says, "I still can't tell." The second says, "The oldest is visiting his grandfather today." The first says, "Now I know their ages." Can you tell their ages?

Puzzle 1: Twelve black socks and 12 white socks are mixed up in a drawer. How many socks must you pull out, blindly, to get a matching pair for sure?


Next LessonThursday, February 3, 2005

A problem to think about: How many squares can you find that have dots in their corners?

Jan. 2005Logical ReasoningSlide 1 Logical Reasoning A Lesson in the “Math + Fun!” Series.

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