You can search for articles in back issues of Contingencies from July/August 2000 to March/April 2009 using the search box to the right. Simply type in subject words, author's name, or article title and click search. To search for articles from May/June 2009 to the present, go to the current digital issue of the magazine and use the search function on the left of the top navigation bar.

Google Custom Search

Tax Season Hiring

by Yan Fridman

PDF version


Let’s finish the year with a relatively simple math puzzle. It was offered a long time ago at a math competition.

It’s November, and an online tax preparation firm,, starts to recruit CPAs for the next year as the tax filing season approaches rapidly. The company isn’t large, and Mr. Rouge, the manager, plans to finish the hiring process by the end of November.

Mr. Rouge is known to be a very peculiar man, and so is his hiring pattern. During the first day, he plans to fill one position and then one-seventeenth of the remaining positions. (Mr. Rouge knows in advance the number of openings.)

During the second day, he will hire two more people and then fill one-seventeenth of the remaining positions. The pattern continues, and during the final day of hiring, he will employ n people and then fill one-seventeenth of the remaining positions. Please answer the following questions:

1. How many business days will it take Mr. Rouge to complete hiring?
2. How many people will Mr. Rouge hire?
3. How peculiar is the hiring pattern?
4. Is there more than one answer to the puzzle?



3. No such q exists.
4. Chances for the Police in question 1 and for the Army in question 2 are the same (50 percent) regardless of the probability p. Police wins the tournament with 25 percent probability regardless of the probability q in question 3.

The Solution

Miners and Bankers are able to create some inequity in the chances to win the tournament solely due to the asymmetrical scoring system; a combination of a win and a loss is better than two ties. Why in this situation these two teams would tie is unknown. Perhaps in the real world it would never happen. However, it makes an interesting math puzzle. The actual solution could be quite long, so I will try to squeeze the main points within the limits of the puzzles page.

1. There are 243 different scoring combinations. They can be split based on the number of ties (excluding the tie between the Miners and the Bankers) as follows: If the number of ties is t, the number of possible combinations is or 5, inequities are not explored and the probability of making the final match for any team is 50 percent. However, if t = 2 or 3, then the Police team gets an advantage in just a few cases and the expectation of making finals increases to 42 2/3 (compared to 40) out of 80 (t=2) and 22 (compared to 20) out of 40 (t=3). The cases could be counted with some work and are left to the readers as an exercise. (Hint: Police will be at an advantage, for example, if one of the cheating teams ties all games.) Total probability of the Police team making the final game is

By setting to zero, you can find the answer.

2. This is very similar to the first case. Inequities arise when t = 3 or 4, where t is the total number of ties. Expectation of making finals decreases to 18²⁄³ (compared to 20) out of 40 (t=3) and 4 (compared to 5) out of 10 (t=4). Again, the cases are left as an exercise. (Hint: Army will be at a disadvantage if it ties all three games; this will be offset against the cases in which Police ties all three games.) Total probability of the Police team making the final game is

By setting to zero, you can find the answer.

3. Police wins the championship with 50 percent probability if it advances to the final game with certainty. The probability of the Police team’s making the finals equals . Since p is assumed to be uniformly distributed on [0, 1], the following must take place:1

By solving the integral equation, you find q > 1, which is impossible. A more interesting question: What probability q triggers the expectation of the Police advancing to the final game to be equal to one-half? The same logic applies as we need to solve the equation.0.
This equation solves for q = 8/13.

4. The old scoring system was fair in that it prevents inequities from occur- ring. Therefore, the probability of Army’s or Police’s winning the group stage is 50 percent, regardless of the Miners-Bankers strategy.

Chess Puzzle. White to move and mate in three.

Initial position: White—Ka7, Rf1, Ba2, Bc5, pawns b4, c6, e2. Black—Ke4, pawns c7, d7, e5.


Case A: 1. Bg8! dxc; 2. Rf7 Kd5; 3. Rf4# Case B: 1. Bg8! d6; 2. Bg1 d5; 3. Bh7#


Due to an administrative deadline, names of only those people who submitted cor- rect solutions by Sept. 30, 2005, are shown on the lists.

Soccer Puzzle: Bob Byrne, Mark Evans, Chi Kwok, Lee Michelson, Brian Miller, Al Spooner, Jeanette Woodhall

Chess Puzzle: Mike Crooks, Lee Michelson


White to Move and Mate in Three.
I selected the easiest chess puzzle so far, since the holiday season starts soon and readers may not want to spend a lot of time solving puzzles.

Solutions may be e-mailed to or mailed to Puzzles, 25 Sparrow Walk, Newtown, Pa. 18940.

In order to make the solver lists (separately maintained for the regular and chess puzzles), please submit your answers and solutions by Nov. 30, 2005.

Depending on the response volume, solver lists may contain only the names of people who solved puzzles on the first attempt.


Contingencies (ISSN 1048-9851) is published by the American Academy of Actuaries, 1100 17th St. NW, 7th floor, Washington, DC 20036. The basic annual subscription rate is included in Academy dues. The nonmember rate is $24. Periodicals postage paid at Washington, DC, and at additional mailing offices. BPA circulation audited.

This article may not be reproduced in whole or in part without written permission of the publisher. Opinions expressed in signed articles are those of the author and do not necessarily reflect official policy of the American Academy of Actuaries.

November/December 2005

Enterprise Risk Management for Insurers: Actuarial Theory in Practice

Operational Risk: The New Frontier

Fundamental Changes

Social Security Reform: What's the Best Fix?

Inside Track:
Tough Love


Is One Euro of Actuaries Worth the Same as One Euro of Financial Economists?

Policy Briefing:
Mr. Smith Comes to the Academy

Turning the Tables: Mortality Tables Should Reflect Improving Mortality

Understanding Insurance, Part I: Comparing Written Premium With Paid Losses

Tax Season Hiring

A Tribute to My Father

Past Issues

Contact us

American Academy of Actuaries