Into The Solution Thicket?

Last week I explored some old puzzles that have survived the test of time to become minor parts of our culture. In particular, I was interested in puzzles that achieve their staying power in part because of some deliberate misdirection in the statement that leads us astray into a solution thicket rather than proceeding immediately to an “Ah-ha!” solution. The typical example I used was the St. Ives puzzle:

“As I was going to St. Ives,
I met a man with seven wives.
Each wife had seven sacks,
Each sack held seven cats,
Each cat had seven kits:
Kits, cats, sacks, and wives,
How many were going to St. Ives?”

This puzzle presents more than the obvious problem. Like, when was it written? To find a time in history when travelers through what is now Cornwall on their way to St. Ives could be open polygamists and not subject to burning at the stake or similar adventure would be an interesting challenge. However, a more bothersome response to this puzzle was the number of readers who challenged the answer by pointing out that the classical rendering of this puzzle contains as essential ambiguity: which way was the man with seven wives going? That is, taking a page from Clinton, what is the definition of “met?”

If “met” can include the concept of overtaking, then the man and his entourage could have been going to St. Ives, also. This is supremely logical since we can assume that a single traveler (another ambiguity, but we’ll pass of that for a moment) almost certainly makes better time than a huge caravan that is burdened with an inordinate number of pets.

In some sense, the best answer to the old puzzle is to give either solution equal weight and then average them. That would probably satisfy no one, but has precedents. For instance, what is the sum of the infinite series 1-1+1-1+1-1+1…? Arrange the terms by pairs like (1-1) + (1-1) +(1-1)… Each pair adds to zero, so the whole must be zero. Now arrange the terms by pairs starting with the second term like 1 + (-1+1) + (-1+1)… Since the terms in the parentheses add to zero, the total sum of the series must be 1. If both solutions seem correct, then the best answer we can give is the average or ½. Anyone have any difficulties with that?

Now consider “As I was going…” This simple statement is insufficient to assume, as most people do, that the traveler was alone. In a strict mathematical sense, if we assume “met” means the man with seven wives was coming from St. Ives, then the answer to how many were going to St. Ives is “Not less than one.” After all, even if the traveler is not directly associated with other pilgrims, we can easily assume the road to St. Ives has other people on it.

Consider the ambiguities and false conclusions that we can easily draw from the simple situation. Then think about how easily one can be led astray in trying to logically sort out the puzzles of everyday life where the ambiguities are rife, and we must make decisions quickly.

However, when all is said and done, I admitted last week that I have never been to St. Ives; that admission prompted the best response to the puzzle:

You really should look, as it is a lovely seaside town in Cornwall, England.



Being unable to pass up the challenge, I clicked on the site and was delighted to see that on the home page, St. Ives brags about being a wonderful place to get married! This gives credence to the interpretation that the man with seven wives had indeed stayed at St. Ives long enough to get married seven times and was headed home with his new wives and pets when he met the anonymous traveler. Case closed.

For those who wish to delve further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here.

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