It's a Puzzlement

Guessing May Not be the Answer

by John P. Robertson

One quarter of the point value of the question will be subtracted for each incorrect answer. No points will be added or subtracted for responses left blank." — Excerpt from the CAS Exam 6 Instructions

This surprising puzzle is from Tom Struppeck.

A student taking Part 6 knows exactly what he knows, and knows what he doesn't know. He has answered every question for which he knows the answer, and he knows that he has scored exactly 53 points so far. He also knows that he needs 54 points to pass (we won't ask how he knows this). There are 20 one-point, multiple-choice questions left, each with five possible answers. He can't do any of these, and can't even eliminate any answers. He wants to guess randomly at these remaining 20 in the hopes of getting his final score up to 54 or more. That is, he wants to maximize the probability of scoring at least one point in total on these 20 one-point questions, by random guessing. How many questions should he guess at, and how many should he leave blank?

For "extra credit," suppose there are n one-point questions left and you want to maximize the probability of getting one point or more by guessing. Can you give a rule that tells how many you should guess at, and how many you should leave blank? Can you prove that rule? Can you make any general statements about maximizing the probability of getting r or more points (proof not required)?

## Puzzling Dissection

The diagram at right shows the solution to this very difficult puzzle. Robert Ballmer, Bob Conger, and Walter Fransen submitted solutions.

## Palindrome Challenge

We had many fine entries to the actuarial palindrome contest, which makes ranking the results very difficult. With help from Mark Saltveit of The Palindromist Magazine (www.realchange.org/pal/index2.htm), Walt Wright, and Pete Lindquist, three "first place" winners have been, somewhat arbitrarily, selected:

Bret Shroyer—"NAIC, it's I, Tat," said I, a statistician.

Bob Spitzer—Palindromic actuary's business card: "STATS WONK" I Know Stats

Paul Ivanovskis—Assess a risk, sir: assess ä.

Space does not permit printing all entries, but a few of the others submitted were:

Paul Ivanovskis—Dr. Occam, axe dud exam? Accord!

Sums are Part 1 Trap, Erasmus.

"Eh—ports at a crisis?" "Si, sir—catastrophe!"

Mike Ziniti—A disgruntled exam taker confronts CAS President Bob Conger on the fact that he just failed: Yo, B., revel! CAS exam axes a clever boy!

Jay Hall—Look! Si! Risk sir is kool!

Bob Gardner (double)—Part one—no trap?

NO! (ital.) Exam ten—I net max elation!

Todd Hubal—Data, et al up in a min. I manipulate a tad.

Jonathan Evans—Benefits, trends, development for adjusted premiums and losses with ratemaking actuaries—All most useful!—making useful most all actuaries ratemaking with losses and premiums, adjusted for development, trends, benefits.

Louis Doray, George M. Levine, Charlie Orlowicz, David Uhland, and Glenn Walker also submitted palindromes. All entries are available by clicking here or by sending an e-mail to me at jrobertson@platinumre.com (specify Word or PDF). Read them all, and pick your own favorites!