Solutions to Logic and Math Puzzle 2

2. Just4Fun 2 - (English)  Reasoning Skill:  Three ladies and Five Hats 

 Problem Statement:

Miss Swank, Miss Social, and miss Highhat descended upon the hatter and demanded the latest creations in chapeaux.  The hatter asked the three ladies to take three chairs.  He had, however, arranged the chairs sin such a manner that Miss Highhat could see the hats that Miss Swank and Miss Social tried on, and Miss Swank could see Miss Social's hat, but Miss Social could not see the other two ladies' hats.  The hatter showed the ladies five hats, three of which were green and two of which were pink.  He mixed the hats and from the rear placed a hat upon each ladies; head.  He asked Miss Highhat the color of her hat.  She relied that she did not know.  He asked Miss Swank the color of her hat.  She replied she did not know.  When Miss Social was asked the color of her hat, she could tell the hatter the correct color.  What was the color of Miss Social's hat?

Problem source:

"General College Mathematics", W. L. Ayres, Cleota G. Fry, H. F. S. Jonah, McGraw Hill Book Company, 1952  p.35

 Solutions: 

 Sivakumaran Venkataraman:

A can find out the color of her hat only if the other two, B & C have pink ones. Even if one of them is green she can't guess. Since she says 'no' B & C guess that one of them at least must be having a green hat. B can guess that she has a green hat only if the C's hat is pink. Since B also says no, C easily guesses that hers is Green.

Vasudevan Tirumurti

Green. Miss.Highhat can guess right only if the two hats she sees are pink. - there are only two pink hats and if she sees two pink she has to be wearing green. So at least one hat is green. Both could be Green too. Given this Miss Swank can guess that her hat is green only if she sees Miss. Social wearing pink hat. If Miss. Social is wearing green it wont be clear if Miss.Highhat saw two greens or one. She could not guess. Ergo Miss.Social is wearing Green.
 

Rahim (using his own "reformed" spelling)

Social was wearing green and she said so.

When she heard Hihat telling that she did not no the kolor of the hat she herself was wearing Soshial konluded that (i)  Both myself and Swank r wearing green (ii) I am having green and Swank pink (iii) vice versa.

But when Swank also said "I do not no" Social was sure she she was not wearing pink. Had she been wearing pink, Swank wud have said "I am having green"  

Prasad:

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About Me Muthu Muthusubramanyam Jack of many trades!  Have tried radio assembling (valve and transistor) photography (including printing and developing), Vedic and Western Astrology, palmistry (Cheiro) besides going to school until pretty old age! My blogs muththuvin puthirkaL சொல் விளையாட்டுக்கள் - மேல்நிலைப் புதிர்கள் குழு குறுக்கெழுத்து Word Puzzles in Tamil for Kids About M Muthusubramanyam My philosophy: Work is worship  (செய்யும் தொழிலே தெய்வம் - பட்டுக்கோட்டை கல்யாணசுந்தரம்:  ceyyum tozilE theivam - PattukkOttai KalyANasuntharam) Occupation Online Puzzles  I nterests blogging, puzzles solving and making, surfing the web, astrology, spiritual enlightenment, p(rov)oking nephews and nieces in Facebook Favorite Movies பலே பாண்டியா, சபாஷ் மீனா, கல்யாண பரிசு, Aah (Hindi), Shri 420 (Hindi), "The Good, the bad and the ugly", Hatari, Psycho Favorite Books Three Men in a Boat (Not to say of the Dog) by Jerome k. Jerome, பொன்னியின் செல்வன் View my complete profile

Just4Fun 2 - Tamil

மோஹன், சேகர், கணேஷ் மூவரும் குல்லாய் வாங்கச் சென்றனர். குல்லாய்க் கடைக்காரர் அவர்கள் மூவரையும் மூன்று நாற்காலிகளில் ஒருவர் பின் ஒருவராக உட்கார வைத்தார். மோஹனால் சேகர், கணேஷ் இருவரின் குல்லாயையும் பார்க்க முடியும்.  சேகரால் கணேஷ் குல்லாயை மட்டும் பார்க்க முடியும்.  கணேஷ் மற்ற இருவரின் குல்லாயையும் பார்க்க முடியாது.  கடைக்காரர் தன்னிடம் இருந்த மூன்று பச்சைக் குல்லாய்களையும் இரண்டு சிவப்புக் குல்லாய்களையும் காட்டினார்.  பின் மூவர் தலையிலும் குல்லாய் அணிவித்தார். மோஹனிடம் அவர் குல்லாய் என்ன நிறம் என்று கேட்டார். அவர் தெரியாது என்றார். சேகரும் அவ்வாறே சொன்னார். கணேஷிடம் அவரது தொப்பி நிறம் கேட்டதற்கு, அவர் தன் குல்லாய் நிறத்தைச் சரியாகச் சொல்லி விட்டார். கணேஷின் குல்லாய் என்ன நிறம்? அவர் எவ்வாறு அதைக் கண்டுபிடித்தார்?

Please send your answers through commentts (preferred) or by email to inamutham@gmail.com


மூலம்: 

"பொது கல்லூரி கணிதம்", W. L.  அய்ரஸ், க்ளியோடா G. ஃப்ரை, H. F. S ஜோனா, மெக்ரா ஹில் புத்தக கம்பெனி, 1952 p.35

Problem source:

"General College Mathematics", W. L. Ayres, Cleota G. Fry, H. F. S. Jonah, McGraw Hill Book Company, 1952  p.35

Just4Fun 2 - English

Reasoning Skill:  Three ladies and Five Hats

Miss Swank, Miss Social, and miss Highhat descended upon the hatter and demanded the latest creations in chapeaux.  The hatter asked the three ladies to take three chairs.  He had, however, arranged the chairs sin such a manner that Miss Highhat could see the hats that Miss Swank and Miss Social tried on, and Miss Swank could see Miss Social's hat, but Miss Social could not see the other two ladies' hats.  The hatter showed the ladies five hats, three of which were green and two of which were pink.  He mixed the hats and from the rear placed a hat upon each ladies; head.  He asked Miss Highhat the color of her hat.  She relied that she did not know.  He asked Miss Swank the color of her hat.  She replied she did not know.  When Miss Social was asked the color of her hat, she could tell the hatter the correct color.  What was the color of Miss Social's hat?

Problem source:

"General College Mathematics", W. L. Ayres, Cleota G. Fry, H. F. S. Jonah, McGraw Hill Book Company, 1952  p.35

Solutions to Logic and Math Puzzles

1.  Three wise men and their reasoning skills:

 Problem Statement: 

Three wise men were tested for their reasoning skill by being blindfolded and having a finger rubbed on each of their foreheads, after being told that one or more would have 

smudges on their foreheads.  Actually, all three were given smudges.  They were told to tap once if they saw one or more spots on removing the blindfold and to tap twice 

if they decided that there was a spot on their own forehead.  When the blindfolds were removed, all three tapped once.  After a pause, one man tapped twice.  What was his reasoning?

 Solutions:

1. Prasad

Best Solution (clear logic and best presentation) - Thanks to Prasad

Let us name the wise men as A, B and C with all possible scenarios as shown below:
X indicates the presence of the smudge in the forehead of the wise man on top of the column

	A	B	C
Scenario 1	x
Scenario  2		x
Scenario  3			x
Scenario  4	x	x
Scenario  5		x	x
Scenario  6	x		x
Scenario  7	x	x	x

Case 1: (Rows 1, 2 and 3)
If only one of them received the smudge, then the one who received the smudge cannot notice any smudge and would not tap. Since all of them tapped once, they all realize that there must be at least two smudges.

Case 2: (Rows 4, 5 and 6)

If only two of them received smudges, those with the smudges would notice no smudge on the third and realize that they themselves must have a smudge. They both would have tapped a second time. This cannot occur if the third also has a smudge.

Case 3: (Row 7)

Since Case 2 did not occur, one of them realizes that it is because all three have received smudges. So he goes ahead and taps twice after a pause.

2. Anthony:

A man taps second only when he knows for sure that he is having a spot on forehead. Lets put him as A. If only one of them were given, then it is easier to find out by the fact that only two people will tap first and the one who didnt tap will tap after a pause. If two of them were given, (Say A not given) all three will tap first. But the persons who were given (B/C) sees that the other person (C / B) also taps in first round and finds out that they themselves are given. A, who quickly strikes out these two scenarios knows that all three were given, since B and C couldn't decide only becuase A himself was given. But why the other two B & C couldnt come to the same conclusion at the same time is a very good question, but thats not part of our concern if we are to explain A's logic only.

3. T. V. Sivakumaran

The most intelligent guy who who tapped twice , let us say A should have thought as follows.

If he didin't have the smudge in his  forehead, B  would have easily found out that he must have it on his own forehead, since both A and C has tapped once looking at the smudge on his  (B's) forehead. He would have immediately tapped twice.

Ditto for C.

4. Rahim

Let A, B and C be the 3 wise men.

Let A be the one who found out: his reasoning : "I must be having the spot because, if not, B would have tapped twice as C also has a spot but tapped only once !" A's argument : If I had no spot, then B, who is wise, would have tapped twice thinking that he was having a spot because C had tapped looking only his spot. Since B was smarter than C, say.   Since I also had a spot, B was confused. May be, B thought, C had tapped looking at my, (A's) spot. B would have thought he may or may not have a spot.   If A and B both no spot, then C wud not have tapped at all !                    
The beauty in this puzzle is, any one else, that is B or C,  would have given a similar reason. But A was a little smarter than the other 2.  A quickly decides that he too must be having the smudge and taps twice. 

2. Just4Fun 2 - (English)  Reasoning Skill:  Three ladies and Five Hats 

 Problem Statement:

Miss Swank, Miss Social, and miss Highhat descended upon the hatter and demanded the latest creations in chapeaux.  The hatter asked the three ladies to take three chairs.  He had, however, arranged the chairs sin such a manner that Miss Highhat could see the hats that Miss Swank and Miss Social tried on, and Miss Swank could see Miss Social's hat, but Miss Social could not see the other two ladies' hats.  The hatter showed the ladies five hats, three of which were green and two of which were pink.  He mixed the hats and from the rear placed a hat upon each ladies; head.  He asked Miss Highhat the color of her hat.  She relied that she did not know.  He asked Miss Swank the color of her hat.  She replied she did not know.  When Miss Social was asked the color of her hat, she could tell the hatter the correct color.  What was the color of Miss Social's hat?

Problem source:

"General College Mathematics", W. L. Ayres, Cleota G. Fry, H. F. S. Jonah, McGraw Hill Book Company, 1952  p.35

 Solutions: 

 Sivakumaran Venkataraman:

A can find out the color of her hat only if the other two, B & C have pink ones. Even if one of them is green she can't guess. Since she says 'no' B & C guess that one of them at least must be having a green hat. B can guess that she has a green hat only if the C's hat is pink. Since B also says no, C easily guesses that hers is Green.

Vasudevan Tirumurti

Green. Miss.Highhat can guess right only if the two hats she sees are pink. - there are only two pink hats and if she sees two pink she has to be wearing green. So at least one hat is green. Both could be Green too. Given this Miss Swank can guess that her hat is green only if she sees Miss. Social wearing pink hat. If Miss. Social is wearing green it wont be clear if Miss.Highhat saw two greens or one. She could not guess. Ergo Miss.Social is wearing Green.
 

Rahim (using his own "reformed" spelling)

Social was wearing green and she said so.

When she heard Hihat telling that she did not no the kolor of the hat she herself was wearing Soshial konluded that (i)  Both myself and Swank r wearing green (ii) I am having green and Swank pink (iii) vice versa.

But when Swank also said "I do not no" Social was sure she she was not wearing pink. Had she been wearing pink, Swank wud have said "I am having green"  

Prasad:

வேடிக்கைக் கணக்கு/புதிர்கள்

பின்னணி:

நான் அரசினர் கலைக் கல்லூரி சென்னை -அப்போது மதராஸ், (முன்னாள் மதராஸ் முகமதியன் கல்லூரி;  இப்போது அரசினர் மகளிர் கலைக் கல்லூரி),  மாணவனாக இருந்த போது ,  அமெரிக்க நூலகம் (அக்காலத்தில் USIS நூலகம் என்று அழைக்கப்பட்டது) கல்லூரியிலிருந்து ஆயிரம் விளக்கு திசையில் மவுண்ட் ரோடு வழியாக நடந்து செல்லும் தூரத்தில் இருந்தது. நான் நூலகத்தில் உறுப்பினராகச் சேர்ந்து, இடைவேளைகளில் அங்கு சென்று புத்தகங்களை கடன் வாங்கி வருவதுண்டு.

அவற்றில்  Ayres, Fry, Jonah என்ற மூவர் எழுதிய "General College Mathematics" புத்தகமும் ஒன்று.

பல ஆண்டுகளுக்கு பிறகு நான் என் பட்டமேற்படிப்புத்  தொடர West LaFayette, இந்தியானாவில் உள்ள பர்ட்யூ பல்கலைக் கழகம் சென்றேன். அந்தக் கால கட்டத்தில், ஒரு பழைய புத்தகக் கடையில் அதே புத்தகம் விற்பனைக்கு இருக்கக் கண்டேன். அதைத் திறந்து பார்த்த போது, அந்த ஆசிரியர்கள் Ayres, ஃப்ரை மற்றும் ஜோனா பர்ட்யூ பல்கலைக் கழக (முன்னாள்)  பேராசிரியர்கள் என்று அறிந்தேன். இந்த செய்தி எனக்கு அந்தப் புத்தகத்தின் மேல் நாட்டம் இன்னும் அதிகரிகச் செய்யவே, அந்தப் புத்தகத்தை வாங்கி விட்டேன்.

பிறகு நான் நாட்டின் பல்வேறு பகுதிகளுக்குச் சென்று வாழவேண்டி இருக்க, கடந்த 35 ஆண்டுகளில், நான் ஒவ்வொரு முறை ஊர் மாறும் போதும் பல புத்தகங்களை விட்டுவிட்டுச் செல்ல நேர்ந்தது.  ஆனால் ஒரு சிலவற்றை மட்டும் தங்க வைத்துக் கொண்டேன். அவற்றில் ஒன்று இந்த "பொது கல்லூரி
கணிதம்" .

இந்த புத்தகத்தில், ஒவ்வொரு அத்தியாய முடிலும் ஒரு "வேடிக்கைக் கணக்கு" கொடுக்கப்பட்டிருக்கிறது.  இந்த கணக்கு/புதிர்கள் கணிதம் (இயற்கணிதம்), பகுத்தறிவு அல்லது இரண்டும் கொண்டு தீர்க்கக் கூடியவை. நான் கல்லூரியில் ஒரு புகுமுக மாணவனாக இருந்த போதே சிலவற்றிற்கு விடை கண்டு பிடிக்க முடிந்தது.

பல எனக்கு மிகவும் கடினமாக இருந்தன. அந்தப் புத்தகத்தில் புதிர்களுக்கு விடை கொடுக்கப் படவில்லை.  அந்தப் புதிர்களில் பலவற்றிற்கு இன்றும் கூட என்னால் விடை கண்டு பிடிக்க முடியவில்லை! 

நான் அவற்றை உங்களிடம் பகிர்ந்து விடைகளும் அறிய விரும்புகிறேன்.

முதலில் ஒரு பகுத்தறிவுப் (solvable using logic and reasoning) புதிருடன் தொடங்குவோம்:


புதிர்:

மூன்று அறிஞர்களின் ஆராய்ந்தறியும் திறன் பரிசோதனை செய்யப் பட்டது. அவர்கள் கண்களைக் கட்டிவிட்டு, நெற்றியில் விரலால் தீத்தப் பட்டது. சிலர் முகத்தில் மை தீத்தப் பட்டிருக்கும் என்றும் சொல்லப் பட்டது. உண்மையில், மூவர் நெற்றியிலும் மை தீத்தப்பட்டது. பிறகு, கண்கட்டவிழ்த்துவிட்டு, ஒன்று அல்லது அதிகம் மைக் குறிகள் பார்த்தவர்கள் ஒரு கையைத் தூக்கவும், தன் நெற்றியில் குறி இருப்பதாகத் தீர்மானித்தால் இரண்டு கைகளையும் தூக்குமாறும் கேட்டுக்கொள்ளப் பட்டது. 
முதலில் மூவரும் ஒரு கை தூக்கினர்; சில நொடிகளில், ஒருவர் இரண்டாவது கையையும் தூக்கினார்.  அவர் அவ்வாறு ஏன் செய்தார்?

ProblemsJust4Fun - 1

Background:

When I was a student of the Government Arts College, Chennai (then Madras) (former Madras Mohammedan College and now Govt. Arts College for Women),  the American Library (then known 

as the USIS  library) was within walking distance from the college along Mount Road in the direction towards Thousand Lights.  I became a member of the library and used to walk over there during breaks and borrow books.

One of the books I remember was  titled "General College Mathematics" by Ayres, Fry and Jonah.

Several years later I went to Purdue University in West Lafayette, Indiana to pursue my Doctorate.  At that time, 

I found the same book for sale in an old books store.  

When I opened, I found the authors Ayres, Fry and 

Jonah were faculties of Purdue University.  This added 

to my interest and I went ahead and bought that book.  

As I have moved to several different parts of the country 

in the past 35 years, I got rid of many books at each move 

but retained a few.  This book, "General College 

Mathematics" is one of them.

In this book, every chapter also has a "Problem Just 

for Fun".  These problems were solvable by simple 

mathematics (algebra), reasoning or both.  I could 

solve some of the problems even as a freshman in college 

back then; many were too much for me.  The book does 

not give answers.

Many of those problems are not solvable by me even today! 

I want to share these with you all and get the solutions.

Let me start with one of the logic problems from this book:

Problem Statement:

Three wise men were tested for their reasoning skill by being blindfolded and having a finger rubbed on each of their 

foreheads, after being told that one or more would have 

smudges on their foreheads.  Actually, all three were given smudges.  They were told to tap once if they saw one 

or more spots on removing the blindfold and to tap twice 

if they decided that there was a spot on their own forehead.  

When the blindfolds were removed, all three tapped once.  

After a pause, one man tapped twice.  What was his reasoning?