PUZZLES
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Doubles Tournament
Ping Pong…now there’s a game!
Puzzle 1:
Solution 1: The actuaries are
able to select actuary A to be
the “calculator.” He/she uses
one of the methods to determine the average salary of all, as
determined in the previous “Average Difficulty” puzzle of Nov/Dec
2003.
Thus, for example, suppose A is elected
the “calculator.” A whispers a dummy number to B, who adds his salary and whispers
to C—similarly C to D, D to E, E to F, F
to G, and G back to A . Then A deducts
dummy number and thus determines total
salaries of all remaining 6 actuaries.
A keeps this total number to himself.
Now—remaining seated actuaries B, C,
D, E, F, G decide upon an order—in the
following rounds exactly one of them will
drop out in succession—and agree this
order will never be revealed to A. Thus,
in the next round A does same calculation—but with a new dummy number
and with one less of the group participating. In the following round an additional
person drops out, and the same calculation occurs. At the end, Actuary A knows
the sums for the 6 actuaries, then 5, etc.
After the 6 rounds, A trivially solves for
individual salaries, and after including his
own salary announces the seven salaries
randomly to the others.
Solution 2: As in previous Nov/Dec
puzzle solution 2—have all seven split
salaries in two and report in succession
to A and B. A and B decide together a random remix of the order of all 7 applicable
to both, but do not disclose to others. A
and B whisper their sums in succession to
C—C calculates the 7 sums, and then remixes (do you see why?) and announces
the 7 salaries.
Among the submitted solutions
were the following ideas.
Puzzle 2: Same situation, but of the seven actuaries it is known that one earns
significantly more than the others. The
problem here is to determine solely what
the highest salary is without anybody determining any other salaries or any other
salary rating information.
1. The Ping Pong “doubles” tournament.
Suppose three equally matched opponents—Andy, Bob, and Charlie—decide
to have a ping pong tournament. Two people will play each game, with the winner
of that game staying on and playing the
person that sat out. The games will continue until the winner is declared—that
person being the first to win two games
in a row.
Using the well-known childhood actuarial randomizer known as “One potato, Two potato”—it is determined that the
first game will be played between Andy
and Bob, with Charlie sitting out.
a. Just as the first ping pong game was
about to begin, Philip, an astute mathematician with no access to a computer,
quickly told each of the players their odds
of winning the tournament. Assuming
you have no computer, similarly determine the precise odds of each player winning the tournament?
b. Charlie, not an actuary, was surprised
when Philip subsequently mentioned
the mathematical expectation that he
had quickly calculated, of the number of
games played in the tournament. How did
Philip calculate this, what was the number, and why was Charlie surprised?
Please submit your solutions via email to [email protected] (that’s with 3
z’s!) or by mail to PUZZLES, 17 Ravine
Rd., Great Neck, NY 11023. Please submit answers as soon as possible to make
the solvers list. And please send any ideas
or any favorites for consideration for future issues to the same addresses.
LAST ISSUE’S PUZZLES
In year 2054, seven actuaries
with computer-like brain capacity are
seated. They wish to determine their seven
individual salaries without anyone knowing any other’s salary or salary rating, and
have no access to pencil or paper.
Answer:
74
Co n t i n g e n c i e s
May/June 2004
Answer: A few solutions were submitted—here is one that works well.
Solution: (Taking advantage of the actuarial computer brain implant.) The actuaries each multiply their own salaries to the
100th power. A makes up a dummy number and whispers to B. As in above they
go around back to A. A deducts dummy
and adds back his (to the 100th power)
salary. Thereafter A takes the 1/100 power
of the sum. This should yield the highest salary. Procedure can be duplicated at
higher power to verify accuracy. When A
sees results do not vary, he announces the
salary—which is the highest salary.
Note: Yan Fridman pointed out that
by essentially voting 1 for yes and 0 for
no and using the same technique as solution 1 above (i.e., whereby first actuary
whispers a dummy number to second,
etc.) the actuaries can basically ask questions and determine how many of the
group anonymously answer affirmatively.
This method could theoretically be used
(though perhaps somewhat time consuming) to solve both puzzles. Perhaps
more noteworthy, it becomes a general
method to get anonymous feedback on
any subject, even in a paperless office.
January/February Solvers: D. Beers,
C. Chacosky, M, Cook, J.M.Crooks, M.
Danburg, A. Dean, D. DeKeiser, D.
Doddridge, L. Dyrland, G. Hansen, E. Klis,
R. Kwok, D. Martin, S. Mu, S. Penn , K.
Thompson. (Note: due to an e-mail glitch,
a few solutions sent 1/20-1/21 may not be
reflected; apologies to anyone omitted.º)
March/April Solvers: R. Bartholomew,
R. Ellerbruch, D. Engelmayer, Y. Fridman,
C. Kwok, M. Lee, A. Spooner, Z. Wadia, H.
Wang , W. Wong
B ONO TOM S TUDIO
This Issue’s Puzzles