puzzles | m a r k d a n b u r g - W y l d
Take the Money and Run
This Issue’s Puzzle
90 Contingencies | SEP/OCT.08
thieves, or one thief plus some amount of
treasure. Further, at no point could any
subset of the gang of thieves be on the
homeward side of the river in possession
of more than their agreed-upon share of
the treasure. (Having some thieves on the
homeward side of the river with less than
their share of the treasure is fine.) Finally,
since they are in a bit of a hurry the thieves
need to minimize the total number of trips
across the river. What’s the minimum
number of trips required, and who and/
or what is on the boat for each trip?
previous Issue’s puzzle
Coin Tricks
I spent the weekend visiting an old friend,
Mr. Maxwell Chance. He’s an eccentric
sort, mostly harmless, but he had recently
developed a numismatic obsession and
so playing with coins took up most of
our time. This lead to the following three
“quickie” puzzles:
1. The Scales: Max set up a coin slot that
led down to a pair of balance scales. The
arms of the scales were slanted so that a
coin might roll either to the right or left,
then fall into a bucket. He took $100 worth
of dimes and set one dime in each bucket.
He explained that the angle of the scale’s
arms was such that the probability of a
coin rolling right or left was proportional
to the number of coins already in each
bucket. For example, if the right bucket
had 13 coins and the left bucket had four,
the probability of the next coin rolling to
the right was 13/17.
To start off, Max placed a single dime in
each bucket (to prevent the first roll from completely deciding the remaining outcome).
“What would you give me now in advance,” he asked, “for the right to the lighter
bucket after I roll the remaining 998 dimes
into the scales?”
Dimes spend as well as any other cash,
so what is a fair bid?
2. The Chessboard: Max set out quarters
down the diagonal of a chessboard, then
played a (somewhat tedious) game of laying
out additional coins based on the rule that if
an empty square shared two or more sides
with a filled square, it should also be filled.
The board quickly filled in.
“The diagonal is kind of a special case,”
he explained. “If I lay them out at random, I
usually need nine or 10 to be sure of filling
in the board. I think eight is the minimum,
but I haven’t been able to prove it.”
Can you?
3. The Table: Max took $2.82 in nickels and
pennies, and threw them onto a table that he
had covered with a large sheet of paper. After
carefully checking that no three coins fell onto
a single line, he challenged me to connect the
coins by straight line segments, such that each
line connected a nickel to a penny and no two
lines crossed.
Easy enough—but your puzzle is to
find an algorithm that always works.
Solution
All three puzzles are ones that can be dealt
with quickly, but also not so quickly. Proofs
by induction seemed to come to mind for a
shutterstock
Once upon a time, there were four thieves
who successfully trekked across a desert,
rowed across a river, traversed a forest,
tunneled under the wall, and made off with
18 sacks of the Sultan’s jewels and gold.
The names of these thieves were Antone,
Bruno, Cristoph, and Dan. Because Antone
had organized the raid, they all agreed his
share should be the largest, so he got eight
sacks. Bruno was the second-in-command,
so he got five sacks. Cristoph got three and
Dan got the last two. Everyone agreed to
this division, and swore an oath to abide
by it.
They seemed to be set for life, because
even two sacks of gold and jewels will
go pretty far in these inflationary times.
They made it back through the tunnel and
crossed the forest without any difficulty
but ran into some trouble at the river’s
edge.
You see, as is common for these types
of stories the boat was only large enough
to carry across two people at a time. Since
the thieves were traveling with heavy
sacks of gold and jewels, they were further restricted because they couldn’t fit
two people and even a single sack of gold
and jewels into the boat at one time. They
could either carry up to two people, or one
person plus some of the Sultan’s treasure.
Although they had all agreed to the
division of treasure and all trusted each
other enough to undertake this huge expedition, they also had a fair understanding
of human psychology. They knew that if,
at any point, any members of the gang
were on the homeward side of the river
with more than their share of the treasure,
they would likely succumb to the temptation to abandon the rest of the gang and
run off. None of them wanted to place his
companions in such a predicament.
The gang needed a way to get all of
themselves and all the treasure across the
river, so that each trip took one thief, two
lot of solvers. The quick solutions are:
1. While not obvious, the split of coins between the two buckets is uniform—there’s
an equal chance of the lighter bucket
having anywhere from one to 999 coins.
Thus, the fair value of the lighter bucket
is $25.025
2. Note that the sum of the perimeters of
the covered squares stays constant. That
is, each time a new square is covered, two
edges that were previously outer edges become interior and two new exterior edges
are created. Thus, to eventually get the outer perimeter of the entire board, you need
to cover at least eight squares. Moreover,
those eight squares cannot have any edges
in common. Of course, this doesn’t show
that any set of eight squares will eventually
cover the board, but it does show that the
ADV E R T I S E R
INDEX
minimum is eight.
3. Although I didn’t make this clear
enough, my intention was that there would
be an equal number—47—of pennies and
nickels. To attach each nickel to a penny
without any lines crossing, it’s sufficient
to minimize the total length of the edges.
If two lines did cross, say vx and yz, they
then would be cross diagonals of a convex
quadrilateral. So, by the triangle inequality,
the sides vz and yx would be short.
●
Solver List
Solvers of all three puzzles—Robert
Bartholomew, Bob Byrne, Bill
Cross, Mark Evans, Rui Guo, John
Hubenschmidt, Lee Michelson, Geoff
Moak, David Promislow, Noam Segal, Al
Spooner, Kevin Trapp
Solutions may be e-mailed to [email protected] or mailed to Puzzles, 65 W. 35th
Place, Eugene, Ore. 97405.
In order to make the solver list,
please make sure that your answers
and solutions are received by
Sept. 30, 2008. Depending on the
response volume, solver lists may contain only the names of
people who solved puzzles on the
first attempt.
Solvers of at least 1 puzzle—Jason
Choi, John Cook, Mark Fowler, Philip
Silverman, Barry Zurbuchen
To add your company’s name to this list, call Mohanna & Associates at
800-800-0341 or e-mail [email protected].
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Contingencies | SEP/OCT.08 91