According to the previous post called Probability and Actuarial Science, actuaries are specially-trained professionals who use their knowledge of probability, statistics, and specific models to evaluate and price risks, e.g. risks of living too long, the risks of dying too soon, the risks of property damages, the risks of becoming sick and a whole host of other risks that are quantifiable. To be an actuary, a candidate must pass a series of exams.

It is fitting that the first actuarial exam, called Exam P, is on probability. How hard is Exam P? What is the pass rate? How much time should a student spend in studying for Exam P?

Exam P is administered by the Society of Actuaries (SOA). It is a 3-hour multiple choice exam consisting of 30 questions. There is a rule of thumb that says that the number of hours required for studying an actuarial exam is the length of the exam times 100. This would mean Exam P may requires 300 hours of studying and preparation.

The 300-hour requirement is not far fetched. Typically a student should take an undergraduate course in probability and statistics (calculus based) prior to taking Exam P. The class time for such as course can certainly be a part of the 300 hours.

Let’s say the class time is around 90 hours (spread over two 15-week semesters with 3 hours of class time each week). That leaves 210 hours for a period of intense studying prior to taking Exam P, sometime after taking the probability and stats course.

In fact, the most effective form of studying is working practice problems. In these 210 hours, the student should review concepts for sure. However, the best way to review is to apply what he or she has learned in the probability and stats course through problem solving.

Intense problem solving is not optional as the pass rate for Exam P and other actuarial exams tend to be around 40%. If a student spends 15 hours a week studying, the 210 hours of studying would take about 3 and a half months. Passing actuarial exams requires dedication.

We give another plug for our Exam P site. It has over 100 realistic Exam P practice problems. We know of students who passed Exam P as a result of studying our practice problems.

We want to talk about treasure hunt in mathematics. But first, focus on a treasure hunt that involves a treasure chest that is similar to the one shown above.

An ornate Romanesque 10-inch by 10-inch treasure chest weighing about 40 pounds that is filled with gold coins and other gems is “hidden in the Rocky Mountains, somewhere between Santa Fe and the Canadian border at an elevation above 5,000 feet. It’s not in a mine, a graveyard or near a structure,” according to Forrest Fenn, a millionaire art dealer in his 80s who lives in Sante Fe, New Mexico. His self-published memoir included a cryptic poem that describes the location of the treasure box.

Nobody knows for sure if this is not just a hoax. Eight years after the publication of the poem, tens of thousands of treasure hunters have flocked to locations in the vast Rocky Mountain region looking for Fenn’s treasure box, which is thought to be worth over $1 million. Four of these treasure hunters had died while searching. The most recent death occurred in the summer of 2017.

In fact, the wife of another treasure hunter who died while searching for Fenn’s hidden box told NBC News that the hunt was “ludicrous” and “should be stopped.” Here’s another piece from NPR that profiles Forrest Fenn and his treasure box.

The payout of this treasure hunt, if successful, is only in the one to two million dollars range. The Rocky Mountains stretch more than 3,000 miles (4,800 km) from the northernmost part of British Columbia, in western Canada, to New Mexico, in the Southwestern United States. The odds of finding the treasure seem long. How would that compare with the odds of winning the Powerball jackpot, which are one in 292 millions? The size of the Powerball jackpot winning is usually in the range of hundreds of million dollars. It seems that buying a Powerball ticket may be a better bet – much larger payout, virtually no chance of dying as a result of buying the ticket.

We do not advocate for buying Powerball tickets or other lottery tickets. The comparison serves to illustrate the ludicrousness of searching for Fenn’s box. To some, like the wife of one of the hunters who died, the hunt may seem ludicrous to the point that buying a Powerball ticket makes more sense.

From a financial standpoint and safety standpoint, hunting for the hidden box of Fenn makes no sense. Of course, there is another way to look at the hunt. For some, it is all about the thrill of the chase, as the title of Forrest Fenn’s book suggests.

In mathematics, the most famous chase may well be the one set off by Pierre de Fermat. Pierre de Fermat claimed in 1637 that no positive integers , and satisfies the equation for any integer greater than 2 (this statement was later called Fermat’s Last Theorem). He scribbled in the margin of his own copy of the Greek text Arithmetica by Diophantus commenting that a proof of this statement was too large to fit in the margin. The following image shows the page from the 1670 edition of Diophantus’ Arithmetica.

The above image is from an edition that came after the death of Fermat in 1665. So it is not from the book with Fermat’s scribbles. This 1670 edition did incorporate the commentary by Fermat (see the part that says Observatio Domini Petri de Fermat).

The claim made by Fermat set off a search for a proof that lasted 358 years. The first successful proof was presented by Andrew Wiles in 1994 and formally published in 1995. With the fact that the solution took three and a half centuries to emerge, it would be obvious that this is a very difficult mathematical problem. Indeed, the Guinness Book of World Records made the obvious official by declaring Fermat’s last theorem as the “most difficult mathematical problem”, another reason being that it had the largest number of unsuccessful proofs.

The mathematical treasure hunts, unlike the one for Forrest Fenn’s box, are not in the physical realm but in the realm of ideas. The payout is not in the form of gold coins or gold nuggets. A more appropriate metaphor is the climbing of Mount Everest. Why climb a tall mountain? Because it is there and it is tall. The mathematicians work on hard problems because they are hard and no one else had solved them before.

Andrew Wiles’ proof was built on the work of all the giants that came before (and all the unsuccessful proofs that came before). Fermat’s last theorem, as an unsolved problem throughout the centuries, stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. The mathematical treasure hunts tend to blaze new trails.

Some mathematical treasure hunts even have $1 million prize attached. The Millennium Problems are sponsored by the Clay Institute, a list of 6 unsolved problems with a $1 million prize money each (it used to be a list of 7 problems). Also see here. Needless to say, these problems are difficult problems. Riemann hypothesis, one of the problems, was formulated in 1859! Any progress in these problems will expand our views of the mathematical world and the physical world.

The hunt for Fenn’s treasure box could very well take years or even decades. Stepping on a loose stone in a steep hill in the wilderness may be fatal. This hunt will end, if successful, in the retrieval of the treasure box and nothing more. The treasure hunt of the mathematical kind can lead to new vista in the mathematical world as well as new ways of looking at the physical world. The mathematical treasure hunt can be done in the comfort of one’s home or office, or in any other relaxed environment such as a park. The mathematical treasure hunts, in some cases, can also end in a million dollar payout.