T2B32

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.