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).
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