The Three Caskets Problem

A Shakespearean Puzzler

A version of this article appears in TWISTED LOGIC: Puzzles, Paradoxes, and Big Questions, by Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.

THE THREE CASKETS PROBLEM

The narrative of William Shakespeare’s ‘Merchant of Venice’ contains intrigue around the character of the young heiress Portia. Amid the various plot developments, one of the more fascinating elements of the story lies in a puzzle set for anyone seeking her hand in marriage. Three caskets made of gold, silver, and lead each contain a different item. Only one holds the prize, a miniature portrait of Portia which symbolises the route to her heart. Portia alone knows that the portrait’s true location is in the lead casket.

SUITORS AND THE CRYPTIC CASKETS: UNRAVELLING THE PECULIAR TEST

As the story unfolds, we learn that to claim Portia in holy wedlock a suitor must choose the casket that houses her portrait. Each casket comes engraved with a cryptic inscription, adding a layer of interest and sophistication to the task.

THE ALLURING GOLD: THE FIRST SUITOR’S TEST

The Prince of Morocco steps forward to face this intriguing test. He is confronted with the inscriptions on the caskets, each one at least as cryptic as the others. Drawn by the promise of desire inscribed on the gold casket, ‘Who chooseth me shall gain what many men desire’, he chooses it, hoping to find ‘an angel in a golden bed’. His dreams are shattered when he finds a skull and a cryptic scroll, instead of the image of Portia. The message on the scroll serves as a harsh reminder, ‘All that glisters is not gold’. With a heavy heart, he retreats, leaving Portia with a sigh of relief, uttering, ‘A gentle riddance’.

SILVER’S DECEPTION: THE SECOND SUITOR’S TURN

Emboldened by his self-worth, and unaware of which casket his predecessor had chosen, the Prince of Arragon interprets the inscription on the silver casket, ‘Who chooseth me shall get as much as he deserves’, as a validation of his worthiness. He selects this casket.

ADDING COMPLEXITY: THE PUZZLE TAKES A TWIST

Now let’s indulge in a thought experiment by introducing an intriguing layer to this complex puzzle. Suppose that, after Arragon’s selection of the silver casket, Portia must open one of the remaining caskets without revealing the portrait’s location. She must, therefore, open the gold casket, which she knows does not contain her likeness. This presents Arragon with the opportunity to hold to his initial choice, the silver casket, or switch to the remaining, unopened casket made of lead.

WEIGHING THE ODDS: ARRAGON’S PROBABILITY PARADOX

If Arragon believes that Portia’s knowledge of the caskets is equal to his, should he stick with his initial choice or take a chance on the unopened lead casket? His decision is far from straightforward, hinging on his interpretation of the cryptic inscriptions, his understanding of the shifting probabilities, and his perception of Portia’s actions.

THE PROBABILITY PUZZLE: DECIPHERING THE GAME OF CHANCE

To understand the implications of the new development, we must first delve into the realm of probability. At the outset, Arragon’s initial choice, the silver casket, had a one-third chance of being correct, assuming he has no other information. There is, therefore, a two-thirds probability that the portrait lay in one of the other two caskets.

Portia’s revelation that the gold casket doesn’t contain the portrait effectively shifts these odds if we can assume that she knows which of the caskets contains her portrait, and must not reveal it. The two-thirds chance, which was initially split between the gold and lead caskets, now converges entirely on the lead casket. Consequently, if Arragon changes his choice from the silver casket to the lead one, his probability of finding Portia’s portrait doubles from one-third to two-thirds, other things being equal.

FATEFUL DECISION: TO SWITCH OR NOT TO SWITCH

If he dismisses the inscriptions as mere distractions and recognises the probability shift in favour of the lead casket, then switching seems like the most rational move. However, if he believes that he has deciphered the true meaning of the inscriptions, he might decide to stick with his original choice.

ARAGON’S DECISION: TO OPEN THE SILVER CASKET

Arragon is either unaware of the true probabilities or else is swayed by the cryptic clues. He chooses the silver casket. However, it only harbours disappointment. Instead of Portia’s portrait, he discovers an image of a fool and a note mocking his decision, ‘With one fool’s head I came to woo, But I go away with two’. His self-confidence leads to his downfall, leaving him more foolish than when he first arrived.

THE POWER OF THE INSCRIPTIONS: GUIDE OR DISTRACTION?

The inscriptions on the caskets add an extra layer of uncertainty and complexity to Arragon’s decision-making process. They could be seen as guides leading the suitors to the correct choice, or they could be deceptive distractions meant to confuse and mislead. The inscription on the lead casket, ‘Who chooseth me must give and hazard all he hath’, could be perceived as a warning of the risks involved or as a subtle hint about the potential rewards of choosing what appears to be the least valuable casket.

CONCLUSION: THE POWER OF INFORMATION

In this thought experiment, the key element is the new information introduced by Portia when she opens the gold casket. After all, she knows where the portrait is. This single action has the potential to increase significantly Arragon’s chance of success. If Arragon understands and acts upon this new information, he can potentially improve his chances of selecting the correct casket from one in three to two in three. However, this seemingly simple shift in probability is complicated by the presence of other potentially influential factors, such as the cryptic inscriptions on the caskets. This makes the problem different from the basic Monty Hall decision.

He might also believe that Portia has no idea which casket contains the portrait. In that case, by opening the gold casket, she would be adding no information to what Arragon already has. He may as well be guided by any additional information he thinks he might pick up from the cryptic inscriptions. Either way, he faces a lonely but life-altering decision.