This is the final tale of Portia's Caskets and the most confusing.
D. THE MYSTERY: WHAT WENT WRONG?
70. The fourth and final tale is the most baffling of all, and it illustrates a logical principle of basic importance. The suitor of the last story passed all three tests and happily claimed Portia III as his bride. They had many children, great-grandchildren, etc. Several generations later a descendant was born in America who looked so much like the ancestral portraits that she was named Portia Nth-henceforth to be referred to as "Portia." When this Portia grew to young womanhood she was both clever and beautiful-just like all the other Portias. In addition, she was highly vivacious and a bit on the mischievous side. She also decided to select her hus band by the casket method (which was somewhat of an anomaly in modern New York, but let that pass). The test she used appeared simple enough; she had only two caskets, silver and gold, in one of which was Portia's portrait. The lids bore the following inscriptions:
Gold
THE PORTRAIT IS NOT IN HERE
Silver
EXACTLY ONE OF THESE TWO STATEMENTS IS TRUE
Which casket would you choose? Well, the suitor reasoned as follows. If the statement on the silver casket is true, then it is the case that exactly one of the two statements is true. This means that the statement on the gold casket must be false. On the other hand, suppose the statement on the silver casket is false. Then it is not the case that exactly one of the statements is true; this means that the statements are either both true or both false. They can't both be true (under the assumption that the second is false), hence they are both false. Therefore again, the statement on the gold casket is false. So regardless of whether the statement on the silver casket is true or false, the statement on the gold casket must be false. Therefore the portrait must be in the gold casket. So the suitor triumphantly exclaimed, "The portrait must be in the gold casket" and opened the lid. To his utter horror the gold casket was empty! The suitor was stunned and claimed that Portia had deceived him. "I don't stoop to deceptions," laughed Portia, and with a haughty, trium phant, and disdainful air opened the silver casket. Sure enough, the portrait was there. Now, what on earth went wrong with the suitor's reasoning?
"Well, well!" said Portia, evidently enjoying the situation enormously, "so your reason didn't do you much good, did it? However, you seem like a very attractive young man, so I think I'll give you another chance. I really shouldn't do this, but I will! In fact, I'll forget the last test and give you a simpler one in which your chances of winning me will be two out of three rather than one out of two. It resembles one of the tests given by my ancestor Portia III. Now surely you should be able to pass this one!" So saying, she led the suitor into another room in which there were three caskets-gold, silver, and lead. Portia explained that one of them contained a dagger and the other two were empty. To win her, the suitor merely need choose one of the empty ones. The inscriptions on the caskets read as follows:
Gold
THE DAGGER IS IN THIS CASKET
Silver
THIS CASKET IS EMPTY
Lead
AT MOST ONE OF THESE THREE STATEMENTS IS TRUE
(Compare this problem with the first test of Portia III! Doesn't it seem to be exactly the same problem?)
Well, the suitor reasoned very carefully this time as follows: Suppose statement (3) is true. Then both other state;nents must be false-in particular (2) is false, so the dagger is then in the silver casket. On the other hand, if (3) is false, then there must be at least two true statements present, hence (1) must be one of them, so in this case the dagger is in the gold casket. In either case the lead casket is empty. So the suitor chose the lead casket, opened the lid, and to his horror, there was the dagger! Laughingly, Portia opened the other two caskets and they were empty! I'm sure the reader will be happy to hear that Portia married her suitor anyhow. (She had decided this long before the tests, and merely used the tests to tease him a little). But this still leaves unanswered the question: What was wrong with the suitor's reasoning?
This puzzle is so good, I'm going to post the answer tomorrow. Also, I'm going to try to work my head around the whole solution again, because I didn't really get it the first time.
Monday, 30 December 2013
Sunday, 29 December 2013
T2B31
Third Tale Third Puzzle
69c. The Third Test
If the suitor passed these two tests, he was led into another room containing a gold, silver, and lead casket. Again, each casket was fashioned by either Cellini or Bellini. Now in this test, the suitor's chances were one out of three (if he guessed blindly); Portia used a portrait of herself, and the portrait was in one of the caskets. To pass the test, the suitor had to (1) select the casket containing the portrait; (2) tell the maker of each casket. The three inscriptions read:
Gold
THE PORTRAIT IS NOT IN HERE
Silver
THE PORTRAIT IS NOT IN HERE
Lead
AT LEAST TWO OF THESE CASKETS WERE FASHIONED BY CELLINI
What is the solution?
The Answer:
We first show that the lead casket must be a Bellini. Sup pose it were a Cellini. Then the statement is false, which means that there must be at least two Bellinis, which must be silver and gold. This is impossible, since the portrait can't be in both the silver and gold caskets. Thus the lead casket is really a Bellini. Hence the statement on it is true, so there are at least two Cellinis. This means that the gold and silver are both Cellinis. Hence the statements on both of them are false, so the portrait is neither in the gold nor the silver caskets. Therefore the portrait is in the lead casket. Also, we have proved that the lead casket is a Bellini and the other two are Cellinis, which answers the second question.
69c. The Third Test
If the suitor passed these two tests, he was led into another room containing a gold, silver, and lead casket. Again, each casket was fashioned by either Cellini or Bellini. Now in this test, the suitor's chances were one out of three (if he guessed blindly); Portia used a portrait of herself, and the portrait was in one of the caskets. To pass the test, the suitor had to (1) select the casket containing the portrait; (2) tell the maker of each casket. The three inscriptions read:
Gold
THE PORTRAIT IS NOT IN HERE
Silver
THE PORTRAIT IS NOT IN HERE
Lead
AT LEAST TWO OF THESE CASKETS WERE FASHIONED BY CELLINI
What is the solution?
The Answer:
We first show that the lead casket must be a Bellini. Sup pose it were a Cellini. Then the statement is false, which means that there must be at least two Bellinis, which must be silver and gold. This is impossible, since the portrait can't be in both the silver and gold caskets. Thus the lead casket is really a Bellini. Hence the statement on it is true, so there are at least two Cellinis. This means that the gold and silver are both Cellinis. Hence the statements on both of them are false, so the portrait is neither in the gold nor the silver caskets. Therefore the portrait is in the lead casket. Also, we have proved that the lead casket is a Bellini and the other two are Cellinis, which answers the second question.
Saturday, 28 December 2013
T2B30
Third Tale Second Puzzle:
69b .. The Second Test. In this test, the suitor's chances (if he guessed blindly) were one out of two. Portia used only two caskets, gold and silver, and one of them contained her portrait (no dagger was used in this test). Again each casket was fashioned either by Cellini or Bellini. The caskets read:
Gold
THE PORTRAIT IS NOT IN HERE
Silver
EXACTLY ONE OF THESE TWO CASKETS WAS FASHIONED BY BELLINI
Which casket should the suitor choose in order to find the portrait?
Answer:
If the silver casket is a Bellini, then the statement is true, in which case the gold casket is a Cellini. Suppose the silver casket is a Cellini. Then it is not the case that exactly one of the caskets is a Bellini. This means that the gold is a Cellini (for if it were a Bellini, then it would be the case that exactly one is a Bellini!) Thus, whether the silver is Bellini or Cellini, the gold is surely a Cellini. Therefore the statement on the gold casket is false, so the portrait is in the gold casket.
69b .. The Second Test. In this test, the suitor's chances (if he guessed blindly) were one out of two. Portia used only two caskets, gold and silver, and one of them contained her portrait (no dagger was used in this test). Again each casket was fashioned either by Cellini or Bellini. The caskets read:
Gold
THE PORTRAIT IS NOT IN HERE
Silver
EXACTLY ONE OF THESE TWO CASKETS WAS FASHIONED BY BELLINI
Which casket should the suitor choose in order to find the portrait?
Answer:
If the silver casket is a Bellini, then the statement is true, in which case the gold casket is a Cellini. Suppose the silver casket is a Cellini. Then it is not the case that exactly one of the caskets is a Bellini. This means that the gold is a Cellini (for if it were a Bellini, then it would be the case that exactly one is a Bellini!) Thus, whether the silver is Bellini or Cellini, the gold is surely a Cellini. Therefore the statement on the gold casket is false, so the portrait is in the gold casket.
T2B29
C. INTRODUCING BELLINI AND CELLINI
The suitor of the last tale passed both tests and happily claimed Portia II as his bride. They lived happily ever after and had a lovely daughter Portia III-henceforth to be called "Portia." When she grew up to young womanhood, she was born smart and beautiful-just like her mommy and grandmommy. She also decided to choose her husband by the casket method. The suitor had to pass three tests in order to win her! The tests were quite ingenious. She went back to her grandmother's idea of having only one state ment inscribed on each casket rather than two. But she introduced the following new wrinkle: She explained to the suitor that each casket was fashioned by one of two famous Florentine craftsmen-Cellini or Bellini. Whenever Cellini fashioned a casket, he always put a false inscription on it, whereas Bellini put only true inscriptions on his caskets.
69a. The First Test. In this unusual test the suitor (if he guessed blindly) would have a two out of three rather than a one out of three chance. Instead of using a portrait, Portia used a dagger which was placed in one of the three caskets; the other two caskets were empty. If the suitor could avoid the casket with the dagger, then he could take the next test. The inscrip tions on the caskets were as follows:
Gold
THE DAGGER IS IN THIS CASKET
Silver
THIS CASKET IS EMPTY
Lead
AT MOST ONE OF THESE THREE CASKETS WAS FASHIONED BY BELLINI
The Answer:
Suppose the lead casket had been fashioned by Bellini. Then the statement would be true, hence the other caskets must have been fashioned by Cellini. This means that the other statements are both false-in particular the state ment on the silver casket is false, so the dagger is in the silver casket. Thus, if the lead casket is the work of Bellini, then the silver casket contains the dagger. Now, suppose the lead casket had been fashioned by Cellini. Then the statement is false, so at least two caskets were fashioned by Bellini. This means that both the gold and silver caskets are Bellini caskets (since the lead one is assumed Cellini). Then the statements on both the gold and silver are true. In particular, the one on the gold is true. So in this case, the dagger lies in the gold casket. In neither case can the dagger be in the lead casket, so the. suitor should choose the lead casket.
The suitor of the last tale passed both tests and happily claimed Portia II as his bride. They lived happily ever after and had a lovely daughter Portia III-henceforth to be called "Portia." When she grew up to young womanhood, she was born smart and beautiful-just like her mommy and grandmommy. She also decided to choose her husband by the casket method. The suitor had to pass three tests in order to win her! The tests were quite ingenious. She went back to her grandmother's idea of having only one state ment inscribed on each casket rather than two. But she introduced the following new wrinkle: She explained to the suitor that each casket was fashioned by one of two famous Florentine craftsmen-Cellini or Bellini. Whenever Cellini fashioned a casket, he always put a false inscription on it, whereas Bellini put only true inscriptions on his caskets.
69a. The First Test. In this unusual test the suitor (if he guessed blindly) would have a two out of three rather than a one out of three chance. Instead of using a portrait, Portia used a dagger which was placed in one of the three caskets; the other two caskets were empty. If the suitor could avoid the casket with the dagger, then he could take the next test. The inscrip tions on the caskets were as follows:
Gold
THE DAGGER IS IN THIS CASKET
Silver
THIS CASKET IS EMPTY
Lead
AT MOST ONE OF THESE THREE CASKETS WAS FASHIONED BY BELLINI
The Answer:
Suppose the lead casket had been fashioned by Bellini. Then the statement would be true, hence the other caskets must have been fashioned by Cellini. This means that the other statements are both false-in particular the state ment on the silver casket is false, so the dagger is in the silver casket. Thus, if the lead casket is the work of Bellini, then the silver casket contains the dagger. Now, suppose the lead casket had been fashioned by Cellini. Then the statement is false, so at least two caskets were fashioned by Bellini. This means that both the gold and silver caskets are Bellini caskets (since the lead one is assumed Cellini). Then the statements on both the gold and silver are true. In particular, the one on the gold is true. So in this case, the dagger lies in the gold casket. In neither case can the dagger be in the lead casket, so the. suitor should choose the lead casket.
T2B28
Second Tale Puzzle B:
If the suitor passed the first test, he was taken into another room in which there were three more caskets. Again each casket had two sentences inscribed on the lid. Portia ex plained that on one of the lids, both statements were true; on another, both statements were false; and on the third, one statement was true and one was false.
Gold
(1) THE PORTRAIT IS NOT IN THIS CASKET
(2) IT IS IN THE SILVER CASKET
Silver
(1) THE PORTRAIT IS NOT IN THE GOLD CASKET
(2) IT IS IN THE LEAD CASKET
Lead
(1) THE PORTRAIT IS NOT IN THIS CASKET
(2) IT IS IN THE GOLD CASKET
Which casket contains the portrait?
Answer:
If the portrait is in the gold casket, then the gold and silver casket lids each contain two false statements. If it is in the silver casket, then the silver and lead caskets each contain one true and one false statement. Therefore the portrait is in the lead casket (and the silver casket lid contains both true statements; the lead, both false; and the gold, one true and one false).
If the suitor passed the first test, he was taken into another room in which there were three more caskets. Again each casket had two sentences inscribed on the lid. Portia ex plained that on one of the lids, both statements were true; on another, both statements were false; and on the third, one statement was true and one was false.
Gold
(1) THE PORTRAIT IS NOT IN THIS CASKET
(2) IT IS IN THE SILVER CASKET
Silver
(1) THE PORTRAIT IS NOT IN THE GOLD CASKET
(2) IT IS IN THE LEAD CASKET
Lead
(1) THE PORTRAIT IS NOT IN THIS CASKET
(2) IT IS IN THE GOLD CASKET
Which casket contains the portrait?
Answer:
If the portrait is in the gold casket, then the gold and silver casket lids each contain two false statements. If it is in the silver casket, then the silver and lead caskets each contain one true and one false statement. Therefore the portrait is in the lead casket (and the silver casket lid contains both true statements; the lead, both false; and the gold, one true and one false).
Monday, 23 December 2013
T2B27
Portia and her husband did, as a matter of fact, live happily
ever after. They had a daughter Portia II-henceforth to be
called "Portia." When the young Portia grew to young
womanhood, she was both clever and beautiful, just like her mommy. She also decided to select her husband by the
casket method. The suitor had to pass two tests in order to
win her.
Second Tale, First Puzzle:
Second Tale, First Puzzle:
In this test each lid contained two statements, and Portia
explained that no lid contained more than one false
statement.
Which casket contains the portrait?
The Answer:
The Answer:
We can immediately rule out the lead casket, for if the
portrait were there, then both statements on the lead
casket would be false. So the portrait is in the gold or the
silver casket. Now, the first statements on the gold and
silver caskets agree, so they are both true or both false. If
they are both false, the second statements are both true
but they cannot be both true since they are contradictory.
Therefore the first statements are both true, so the portrait
cannot b e in the gold casket. This proves that the portrait is
in the silver casket.
T2B26
Second Puzzle:
Portia's suitor chose correctly, so they married and lived quite happily-at least for a while. Then, one day, Portia had the following thoughts: "Though my husband showed some intelligence in choosing the right casket, the problem wasn't really that difficult. Surely, I could have made the p r o b l e m h a r d e r a n d g o t t e n a r e a l ly c l e v e r h u s b a n d . " S o s h e forthwith divorced her husband and decided to get a clev erer one.
This time she had the following inscriptions put on the caskets:
Portia's suitor chose correctly, so they married and lived quite happily-at least for a while. Then, one day, Portia had the following thoughts: "Though my husband showed some intelligence in choosing the right casket, the problem wasn't really that difficult. Surely, I could have made the p r o b l e m h a r d e r a n d g o t t e n a r e a l ly c l e v e r h u s b a n d . " S o s h e forthwith divorced her husband and decided to get a clev erer one.
This time she had the following inscriptions put on the caskets:
Gold
THE PORTRAIT IS NOT IN THE SILVER CASKET
THE PORTRAIT IS NOT IN THE SILVER CASKET
Silver
THE PORTRAIT IS NOT IN THIS CASKET
THE PORTRAIT IS NOT IN THIS CASKET
Lead
THE PORTRAIT IS IN THIS CASKET
THE PORTRAIT IS IN THIS CASKET
Portia explained to the suitor that at least one of the three
statements was true and that at least one of them was false.
Which casket contains the portrait?
Epilogue _________________
As fate would have it, the first suitor turned out to be Portia's ex-husband. He was really quite hright enough to figure out this problem too. So they were remarried. The husband took Portia home, turned her over his knee, gave her a good sound spanking, and Portia never had any foolish ideas again.
Which casket contains the portrait?
Epilogue _________________
As fate would have it, the first suitor turned out to be Portia's ex-husband. He was really quite hright enough to figure out this problem too. So they were remarried. The husband took Portia home, turned her over his knee, gave her a good sound spanking, and Portia never had any foolish ideas again.
The Answer:
If the portrait were in the lead casket, then all three state
ments would be true, which is contrary to what is given. If
the portrait were in the silver casket, then all three state
ments would be false, which is again contrary to what is
given. Therefore the portrait must be in the gold casket
(and we have the first two statements true and the third one
false, which is consistent with what is given).
Saturday, 21 December 2013
T2B25
One of Raymond Smullyan's most famous puzzles includes Portia's Caskets. This set of riddles starts out fairly easy, and gets to the final and most difficult one.
I think that the last puzzle only keeps its difficulty if you do the entire series, so I'll start posting a puzzle a day until I get to the final one.
First Puzzle:
Tomorrow, puzzle number 2...
I think that the last puzzle only keeps its difficulty if you do the entire series, so I'll start posting a puzzle a day until I get to the final one.
First Puzzle:
The statements on the gold and lead caskets say the
opposite, hence one of them must be true. Since at most
one of the three statements is true, then the statement on
the silver casket is false, so the portrait is actually in the
silver casket.
This problem could be alternatively solved by the fol lowing method: If the portrait were in the gold casket, we would have two true statements (namely on the gold and lead caskets), which is contrary to what is given. If the portrait were in the lead casket, we would again have two true statements (this time on the lead and silver caskets). Therefore the portrait must be in the silver casket.
Both methods are correct, and this illustrates the fact that in many problems there can be several correct ways of arriving at the same conclusion.
First Answer:
Which casket should the suitor choose?
This problem could be alternatively solved by the fol lowing method: If the portrait were in the gold casket, we would have two true statements (namely on the gold and lead caskets), which is contrary to what is given. If the portrait were in the lead casket, we would again have two true statements (this time on the lead and silver caskets). Therefore the portrait must be in the silver casket.
Both methods are correct, and this illustrates the fact that in many problems there can be several correct ways of arriving at the same conclusion.
First Answer:
In Shakespeare's Merchant of Venice Portia had three
caskets-gold, silver, and lead-inside one of which was
Portia's portrait. The suitor was to choose one of the
caskets, and if he was lucky enough (or wise enough) to
choose the one with the portrait, then he could claim Portia
as his bride. On the lid of each casket was an inscription to
help the suitor choose wisely.
Now, suppose Portia wished to choose her husband not on the basis of virtue, but simply on the basis of in telligence. She had the following inscriptions put on the caskets.
Now, suppose Portia wished to choose her husband not on the basis of virtue, but simply on the basis of in telligence. She had the following inscriptions put on the caskets.
Gold
THE PORTRAIT IS IN THIS CASKET
THE PORTRAIT IS IN THIS CASKET
Silver
THE PORTRAIT IS NOT IN THIS CASKET
THE PORTRAIT IS NOT IN THIS CASKET
Lead
THE PORTRAIT IS NOT IN THE GOLD CASKET
THE PORTRAIT IS NOT IN THE GOLD CASKET
Portia explained to the suitor that of the three statements,
at most one was true.
Tomorrow, puzzle number 2...
Thursday, 19 December 2013
T2B24
3. The Joke Was on Me
A fellow graduate student of mine at the University of Chicago had two brothers, aged six and eight. I was a frequent visitor to their house and often did tricks for the children. One day 1 came and said, "1 have a trick in which I could turn you both into lions." To my surprise, one of them said, "Okay, turn us into lions." 1 replied, "Well, uh, really, uh, I shouldn't do that, because there is no way 1 could tum you back again." The little one said, "I don't care; 1 want you to tum us into lions anyway." 1 replied, "No, really, there's no way1 can tum you back." The older one shouted, "I want you to turn us into lions!" The little one then asked, "How do you tum us into lions?" I replied, "By saying the magic words." One ofthem asked, "What are the magic words?" 1 replied, "If I told you the magic words, I would be saying them, and so you would tum into lions." They thought about this for a while, and then one of them asked, " Aren't there any magic words which would bring us back?" 1 replied: "Yes, there are, but the trouble is this. If I said the first magic words, then not only you two but every body in the world-including myself-would tum into a lion. And lions can't talk, so there would be no one left to say the other magic words to bring us back." The older one then said, "Write them down!" The little one said, "But I can't read!" I replied, "No, no, writing them down is out of the question; even if they were written down rather than said, everyone in the world would still turn into a lion." They said, "Oh."
About a week later I met the eight-year-old, and he said, " Smullyan, there' s something I've been wanting to ask you; something which has been puzzling me." I replied, "Yes?" He said. "How did you ever learn the magic words?"
This is another one of Smullyan's riddles, and I wanted to point this one out because if you think about it several times, you realize that you don't really know what magic words he's talking about, the ones to turn people into lions or the one to turn them back.
Anyway. Cool puzzle.
A fellow graduate student of mine at the University of Chicago had two brothers, aged six and eight. I was a frequent visitor to their house and often did tricks for the children. One day 1 came and said, "1 have a trick in which I could turn you both into lions." To my surprise, one of them said, "Okay, turn us into lions." 1 replied, "Well, uh, really, uh, I shouldn't do that, because there is no way 1 could tum you back again." The little one said, "I don't care; 1 want you to tum us into lions anyway." 1 replied, "No, really, there's no way1 can tum you back." The older one shouted, "I want you to turn us into lions!" The little one then asked, "How do you tum us into lions?" I replied, "By saying the magic words." One ofthem asked, "What are the magic words?" 1 replied, "If I told you the magic words, I would be saying them, and so you would tum into lions." They thought about this for a while, and then one of them asked, " Aren't there any magic words which would bring us back?" 1 replied: "Yes, there are, but the trouble is this. If I said the first magic words, then not only you two but every body in the world-including myself-would tum into a lion. And lions can't talk, so there would be no one left to say the other magic words to bring us back." The older one then said, "Write them down!" The little one said, "But I can't read!" I replied, "No, no, writing them down is out of the question; even if they were written down rather than said, everyone in the world would still turn into a lion." They said, "Oh."
About a week later I met the eight-year-old, and he said, " Smullyan, there' s something I've been wanting to ask you; something which has been puzzling me." I replied, "Yes?" He said. "How did you ever learn the magic words?"
This is another one of Smullyan's riddles, and I wanted to point this one out because if you think about it several times, you realize that you don't really know what magic words he's talking about, the ones to turn people into lions or the one to turn them back.
Anyway. Cool puzzle.
Wednesday, 18 December 2013
T2B23
This is a story from http://weblog.raganwald.com/2008/05/few-easy-ones-from-raymond-smullyan.html:
Dr. Tarr is a psychologist with the Department of Health. Her job is to inspect asylums to determine whether they are in compliance with the law. Asylums have Doctors and Patients. In a compliant asylum, all the doctors are sane and all the patients are insane. Clearly, an asylum with an insane doctor or a sane patient is Not A Good Thing.
Sane persons are correct in all of their beliefs. Insane persons are incorrect in all of their beliefs. Both sane and insane persons are scrupulously honest: they always state what they believe to be the case. Unfortunately, the asylums are very modern and do not use identifying devices such as uniforms, ID tags, or other devices to show which persons are doctors and which are patients. Nor is it possible to know whether a person is sane or insane by any means other than questioning them.
One day, after inspecting a number of asylums, Dr. Tarr was having a drink and cigar with her good friend Professor Feather. The professor found her work interesting and asked her to recount some of her findings.
“Well,” said Dr. Tarr, “at the first asylum I visited, I met an inhabitant who made a single statement. I immediately took steps to have them released.”
“Wait,” interjected the professor, “so you’re saying this person was not an insane patient?”
“Of course,” replied Dr. Tarr.
Professor Feather thought for a moment, then asked “How is that possible? This sounds like the old Liar and Truth Teller puzzle. This person either told the truth or they lied. But there are four possibilities for any person in an asylum: Sane Doctor, Insane Patient, Insane Doctor, or Sane Patient.
“Even if you knew whether they were lying or telling the truth, that would only narrow the matter down to two possibilities. For example, if they told a truth such as ‘two plus two equals four’, you would know that they were Sane. But how would you know that they were a Patient, not a Doctor?”
Dr. Tarr replied with a chuckle “I agree that I could not have deduced what to do based on an inhabitant saying ‘two plus two equals four’. But in this case, the patient was quite intelligent and thought of a single statement which could establish the fact that only a Sane Patient could make that statement.
“I’m sure if you think about it, you could construct such a statement. Name a statement which could only be uttered by a Sane Patient.”
…
Dr. Tarr and Professor Feather shared a chuckle over that one. Then the professor took a more serious tone. “But have you ever had to remove a Doctor from an asylum?”
“Yes,” said Dr. Tarr sadly, “it has happened. Doctors do go insane once in a while. Recently I had just such a case. As it happened, I was visiting an asylum for the very first time and the first inhabitant I met made a single statement. I immediately had the inhabitant transferred to a special institution for former Doctors.”
“Don’t say it!” exclaimed the professor, “I want to work it out for myself…”
…
“Another time,” continued Dr. Tarr, “I was visiting an asylum which had been placed on probation for irregularities such as Insane Doctors and Sane Patients. I asked an inhabitant ‘Are you a patient’, and she said ‘yes’.”
“What did you do next?” asked Professor Feather. “Did you need to do any more investigating?”
…
“I’m glad you worked that out. Another asylum was on probation and I decided to ask the very same question of the first inhabitant I met. This time, when I asked ‘Are you a patient,’ he replied ‘I believe so…’.” Do you think I knew enough to close the asylum?
Professor Feather thought about this one for a very long time.
I did not find the answers to these, but I think they are put in an interesting format with a storylike sequence (I think many of Smullyan's puzzles are like this, they are based off the same basic problem, each one a more complex variation of the same problem, for a certain section/theme of problems). I think these are quite hard, although in the webpage, the author says that he put "easy" in the title to encourage people to try these puzzles.
Dr. Tarr is a psychologist with the Department of Health. Her job is to inspect asylums to determine whether they are in compliance with the law. Asylums have Doctors and Patients. In a compliant asylum, all the doctors are sane and all the patients are insane. Clearly, an asylum with an insane doctor or a sane patient is Not A Good Thing.
Sane persons are correct in all of their beliefs. Insane persons are incorrect in all of their beliefs. Both sane and insane persons are scrupulously honest: they always state what they believe to be the case. Unfortunately, the asylums are very modern and do not use identifying devices such as uniforms, ID tags, or other devices to show which persons are doctors and which are patients. Nor is it possible to know whether a person is sane or insane by any means other than questioning them.
One day, after inspecting a number of asylums, Dr. Tarr was having a drink and cigar with her good friend Professor Feather. The professor found her work interesting and asked her to recount some of her findings.
“Well,” said Dr. Tarr, “at the first asylum I visited, I met an inhabitant who made a single statement. I immediately took steps to have them released.”
“Wait,” interjected the professor, “so you’re saying this person was not an insane patient?”
“Of course,” replied Dr. Tarr.
Professor Feather thought for a moment, then asked “How is that possible? This sounds like the old Liar and Truth Teller puzzle. This person either told the truth or they lied. But there are four possibilities for any person in an asylum: Sane Doctor, Insane Patient, Insane Doctor, or Sane Patient.
“Even if you knew whether they were lying or telling the truth, that would only narrow the matter down to two possibilities. For example, if they told a truth such as ‘two plus two equals four’, you would know that they were Sane. But how would you know that they were a Patient, not a Doctor?”
Dr. Tarr replied with a chuckle “I agree that I could not have deduced what to do based on an inhabitant saying ‘two plus two equals four’. But in this case, the patient was quite intelligent and thought of a single statement which could establish the fact that only a Sane Patient could make that statement.
“I’m sure if you think about it, you could construct such a statement. Name a statement which could only be uttered by a Sane Patient.”
…
Dr. Tarr and Professor Feather shared a chuckle over that one. Then the professor took a more serious tone. “But have you ever had to remove a Doctor from an asylum?”
“Yes,” said Dr. Tarr sadly, “it has happened. Doctors do go insane once in a while. Recently I had just such a case. As it happened, I was visiting an asylum for the very first time and the first inhabitant I met made a single statement. I immediately had the inhabitant transferred to a special institution for former Doctors.”
“Don’t say it!” exclaimed the professor, “I want to work it out for myself…”
…
“Another time,” continued Dr. Tarr, “I was visiting an asylum which had been placed on probation for irregularities such as Insane Doctors and Sane Patients. I asked an inhabitant ‘Are you a patient’, and she said ‘yes’.”
“What did you do next?” asked Professor Feather. “Did you need to do any more investigating?”
…
“I’m glad you worked that out. Another asylum was on probation and I decided to ask the very same question of the first inhabitant I met. This time, when I asked ‘Are you a patient,’ he replied ‘I believe so…’.” Do you think I knew enough to close the asylum?
Professor Feather thought about this one for a very long time.
I did not find the answers to these, but I think they are put in an interesting format with a storylike sequence (I think many of Smullyan's puzzles are like this, they are based off the same basic problem, each one a more complex variation of the same problem, for a certain section/theme of problems). I think these are quite hard, although in the webpage, the author says that he put "easy" in the title to encourage people to try these puzzles.
Tuesday, 17 December 2013
T2B22
I find Raymond Smullyan's introduction to logic interesting. In the first part of his book, he puts forth this thought:
My introduction to logic was at the age of six. It happened
this way: On April 1, 1925, I was sick in bed with grippe, or
flu, or something. In the morning my brother Emile (ten
years my senior) came into my bedroom and said: "Well,
Raymond, today is April Fool's Day, and I will fool you as
youhaveneverbeenfooledbefore!" Iwaitedalldaylongfor
him to fool me, but he didn't. Late that night, my mother
asked me, "Why don't you go to sleep?" I replied, "I'm
waiting for Emile to fool me." My mother turned to Emile
and said, "Emile, will you please fool the child!" Emile then
turned to me, and the following dialogue ensued:
Emile | So, you expected me to fool you, didn't you?
Raymond | Yes.
Emile | But I didn't, did I?
Raymond | No.
Emile | But you expected me to, didn't you?
Raymond | Yes.
Emile | So I fooled you, didn't I!
Well, I recall lying in bed long after the lights were turned out wondering whether or not I had really been fooled. On the one hand, if I wasn't fooled, then I did not get what I expected, hence I was fooled. (This was Emile's argument.) But with equal reason it can be said that if I was fooled, then I did get what I expected, so then, in what sense was I fooled. So, was I fooled or wasn't I?
Raymond | Yes.
Emile | But I didn't, did I?
Raymond | No.
Emile | But you expected me to, didn't you?
Raymond | Yes.
Emile | So I fooled you, didn't I!
Well, I recall lying in bed long after the lights were turned out wondering whether or not I had really been fooled. On the one hand, if I wasn't fooled, then I did not get what I expected, hence I was fooled. (This was Emile's argument.) But with equal reason it can be said that if I was fooled, then I did get what I expected, so then, in what sense was I fooled. So, was I fooled or wasn't I?
I shall not answer this puzzle now; we shall return to it in one form or another several times in the course of this
book. It embodies a subtle principle which shall be one of
our major themes.
I thought the last paragraph was an EXCELLENT point. Also, the way he says it sounds smart :P. Never thought of saying it that way.
I thought the last paragraph was an EXCELLENT point. Also, the way he says it sounds smart :P. Never thought of saying it that way.
Monday, 16 December 2013
T2B21
I've realized that I am really bad at knights and knaves puzzles.
I don't really like to use the strategy of assuming different situations, I start more from the facts I know, then I use those to figure out the rest of the puzzle. I think it's harder to try a lot of different solutions, but I do know a lot of people who like to do so.
Saturday, 14 December 2013
Friday, 13 December 2013
T2B19
Death. Note. Is. So. Good. I. Love. This. Series.
The whole idea of justice is so DEEP, it's so much more than can be told with just words...
Beautiful.
Deathnote is now officially my favourite manga.
The whole idea of justice is so DEEP, it's so much more than can be told with just words...
Beautiful.
Deathnote is now officially my favourite manga.
Thursday, 12 December 2013
T2B18
This is a really weird riddle: A New York city hairdresser recently said that he would rather cut the hair of three Canadians than that of one New Yorker. Why?
Answer: He would earn three times as much money.
Answer: He would earn three times as much money.
Wednesday, 11 December 2013
T2B17
Did you know that if you boil an egg first, it'll spin longer?
This was in a riddle, but technically, it's not a riddle, it's trivia, therefore, I am not presenting it as a riddle.
I actually didn't know this before, but it's quite interesting.
Here is the scientific explanation from Planet Science:
Hard-boiled eggs are solid inside. In the raw egg, the liquid inside the egg slides about and stops the egg from spinning as fast.
When you stop the hard-boiled egg, it stops quickly. When you stop the raw egg, it keeps turning a little bit. You have only stopped the shell, not the liquid inside. The liquid is still moving, which causes the shell to keep turning.
Hard-boiled eggs will spin on their end if you spin them fast enough. The egg saves energy by spinning on its end and making a smaller circle.
Tuesday, 10 December 2013
T2B16
This excerpt is part of Raymond Smullyan's "What is the Name of this Book?" logic puzzle books. This is not actually a puzzle, it's a joke about mathematicians and physicists:
2 1 1 . And What About You? _______
Are you of the mathematician or physicist type? Well, there is the following delightful test to tell whether you are a mathematician or physicist.
You are in a country cabin in which there is an unlighted stove, a box of matches, a faucet with cold running water, and an empty pot. How would you get a pot of hot water? Doubtless you will answer, "I would fill the pot with cold water, light the stove, and then put the pot on until the water gets hot," To this I reply: "Good; so far, mathematicians and physicists are in complete agreement. Now, the next problem separates the cases."
In this problem, you are in a country cabin in which there is an unlighted stove, a box of matches, a faucet with cold running water, and a pot filled with cold water. How would you get a pot of hot water? Most people reply, "1 would light the stove and put the pot of cold water on it." I reply: "Then you are a physicist! The mathematician would pour out the water, reducing the case to the preceding prob lem, which has already been solved."
We could go a step further and consider the case of a pot of cold water already on a lighted stove. How do we get hot water? The physicist just waits for the water to get hot; the mathematician turns off the stove, dumps out the water, reducing the case to the first problem (or he mightjust turn off the stove, reducing the case to the second problem) . A still more dramatic variation goes as follows: A house is on fire. We have available a hydrant and a dis connected hose. How does one put out the fire? Obviously, by first connecting the hose to the hydrant and then squirt ing the building. Now, suppose you have a hydrant, a disconnected hose and a house not on fire. How do you put out the fire? The mathematician first sets fire to the house, reducing the problem to the preceding case.
Are you of the mathematician or physicist type? Well, there is the following delightful test to tell whether you are a mathematician or physicist.
You are in a country cabin in which there is an unlighted stove, a box of matches, a faucet with cold running water, and an empty pot. How would you get a pot of hot water? Doubtless you will answer, "I would fill the pot with cold water, light the stove, and then put the pot on until the water gets hot," To this I reply: "Good; so far, mathematicians and physicists are in complete agreement. Now, the next problem separates the cases."
In this problem, you are in a country cabin in which there is an unlighted stove, a box of matches, a faucet with cold running water, and a pot filled with cold water. How would you get a pot of hot water? Most people reply, "1 would light the stove and put the pot of cold water on it." I reply: "Then you are a physicist! The mathematician would pour out the water, reducing the case to the preceding prob lem, which has already been solved."
We could go a step further and consider the case of a pot of cold water already on a lighted stove. How do we get hot water? The physicist just waits for the water to get hot; the mathematician turns off the stove, dumps out the water, reducing the case to the first problem (or he mightjust turn off the stove, reducing the case to the second problem) . A still more dramatic variation goes as follows: A house is on fire. We have available a hydrant and a dis connected hose. How does one put out the fire? Obviously, by first connecting the hose to the hydrant and then squirt ing the building. Now, suppose you have a hydrant, a disconnected hose and a house not on fire. How do you put out the fire? The mathematician first sets fire to the house, reducing the problem to the preceding case.
Monday, 9 December 2013
T2B15
The Elegance of the Hedgehog, by Muriel Barbery.
Beautiful, beautiful book.
Deep.
Philosophical.
Like I said: Beautiful.
I'm only about 3/4 of the way through, and I borrowed it in like Term 1 for English IRP, but it is such a deep books about a hidden concierge and a super intelligent 12 year old girl (who by the way, has decided to suicide on her 13th birthday because she thinks the world is pointless). It's kind of mature subject matter at times, but it is SO. DANG. DEEP. It is just so... complex, it's just awesome.
Friday, 6 December 2013
T2B14
Toy Story 3
Dependence: The actors and director relying upon each other to produce the film as a cohesive whole, I think. Also, they must each do their job as well as they can to make an impressive end.
Independence: I guess that would be the thing where the director (I think it was the director) had to sketch out new characters/toys, I think, or something of that sort, and was waiting to show it to his peers, he was doing it independently (well, maybe he might not have been, considering someone else might have told him to do so, but it seemed independent).
Interdependence: I really liked the Day vs. Night short, because I thought it was interesting how Day and Night always wanted to impress the other with their scenes that happen within them. It always seems like the other person has it better, like the saying, "The grass is greener on the other side." But when they combined to become twilight and become the same, it united them. And they later turned to the other side, which I guess gave them what they wanted....
I really liked the black-and-white sketchy short, where they had a bonding thing to not shave/get a haircut, etc., because I thought that was really interesting and a bit crazy, but I didn't find that it had to do with my dependence/independence/interdependence thoughts.
Dependence: The actors and director relying upon each other to produce the film as a cohesive whole, I think. Also, they must each do their job as well as they can to make an impressive end.
Independence: I guess that would be the thing where the director (I think it was the director) had to sketch out new characters/toys, I think, or something of that sort, and was waiting to show it to his peers, he was doing it independently (well, maybe he might not have been, considering someone else might have told him to do so, but it seemed independent).
Interdependence: I really liked the Day vs. Night short, because I thought it was interesting how Day and Night always wanted to impress the other with their scenes that happen within them. It always seems like the other person has it better, like the saying, "The grass is greener on the other side." But when they combined to become twilight and become the same, it united them. And they later turned to the other side, which I guess gave them what they wanted....
I really liked the black-and-white sketchy short, where they had a bonding thing to not shave/get a haircut, etc., because I thought that was really interesting and a bit crazy, but I didn't find that it had to do with my dependence/independence/interdependence thoughts.
Thursday, 5 December 2013
T2B13
Dependence: Doctor to give me prescription for KCl
Independence: Me asking Ms. Smedley for a note for KCl and doing research and asking the pharmacy and doing all of that stuff before asking the doctor to give me a prescription for KCl
Interdependence: French skit practice time in class today? Like if one person gets distracted by accident (that would be me) the other people depend on this person to get back on track, and this person needs to depend on the other people to bring this person back on track?
Independence: Me asking Ms. Smedley for a note for KCl and doing research and asking the pharmacy and doing all of that stuff before asking the doctor to give me a prescription for KCl
Interdependence: French skit practice time in class today? Like if one person gets distracted by accident (that would be me) the other people depend on this person to get back on track, and this person needs to depend on the other people to bring this person back on track?
Wednesday, 4 December 2013
T2B12
Dependence: Borrowing beakers from the science lab for science fair. I needed to depend on Ms. Smedley to help me get beakers/sign me out.
Independence: Quote from Ms. Greskiw: "You don't teach a child independence by telling them what to do." We were talking about independence and being prepared, etc. in English today, and that came up.
Interdependence: For this I have absolutely no idea, unless you count French skits, where you have to depend on your partner and they have to depend on you...
Tuesday, 3 December 2013
T2B11
Dependence: Relying on mother to help with conversions for science
Independence: Figuring out math problems by myself
Interdependence: Relying on Sasha to a. do the assignment, and b. give the paper to me, and also she said I should write down the information on the assignment (science)
Independence: Figuring out math problems by myself
Interdependence: Relying on Sasha to a. do the assignment, and b. give the paper to me, and also she said I should write down the information on the assignment (science)
Monday, 2 December 2013
T2B10
Dependence: the state of relying on or needing someone or something for aid, support, or the like.
Independence: freedom from the control, influence, support, aid, or the like, of others.
Interdependence: mutually reliant on each other.
For example, today I was dependent on my mother for a ride. I was independent in getting to class. I was interdependent with Erica for editing English essays--she to edit mine, myself to edit hers.
It's really fascinating to stop and think about how much we rely on outside factors. If I am biking home today, I rely on the weather not to rain or snow. If I want to do research today, I have to rely on my Wi-fi working. If I want a note for science, I have to rely on the person giving me the note.
Well, okay, those examples may not exactly be dependence. But anyway, it's sometimes depressing to think about how little we are in control every day. Whether we like it or not, we can't be completely independent. This was a really big concept in MACC, and it was that autonomy is not complete independence, it's knowing when to be independent and when to be dependent--knowing when you can do something by yourself, and asking for help when you need it. Although, I suppose, that is a form of independence in itself.
Sunday, 1 December 2013
T2B9
Learning Outcome of completing math linear relations worksheet: To learn and review linear equations, and to provide questions and/or challenging puzzles to deepen your understanding of linear relations, even when given some crazy figures
I understand it pretty well, although they take some time for me to solve, so I'm going to use this opportunity to put in additional questions and ask my parents for help if I need it.
Also, math is fun. :)
I understand it pretty well, although they take some time for me to solve, so I'm going to use this opportunity to put in additional questions and ask my parents for help if I need it.
Also, math is fun. :)
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