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The first contest was purely aesthetic: Use Maple or another graphing
utility to graph the most creative or beautiful curve.
Congratulations to
Christos Tsonis
for a graph which looked something like a hairy carneval mask:
> plot[sin(260*t)^2 + cos(2*t),t,t=-Pi..Pi];
- A professor had ten different pairs of socks, but a malicious dryer
ate six (individual) socks. An optimist would say
that with luck, there will still be
7 matching pairs, while a pessimist would say that there will more probably be
only 4 matching pairs. Who is more likely to be right, and by how
much? Give a quantitative comparison of the probabilities.
Congratulations to Karthik Balakrishnan and Jay Underwood for
correctly showing that the pessimist is 112 times more likely to
be right than the optimist.
-
Over in Athens, Georgia, many people -
far, far, too many - believe that (f g)' = f' g'.
This is known as the Ugamath formula. Once in a while it is even right,
for example when f and g are constants,
or f(x) = g(x) = exp(2 x). How many examples can you find
where (f g)' = f' g' but NOT = 0?
Congratulations to Karthik Balakrishnan for the largest number of
solutions.
-
As you know from a class in Euclidean geometry, two triangles are
congruent if they share two angles and a side (ASA) or two
sides and the angle between them (SAS), but not necessarily if, for example
they share two sides and a different angle (SSA). The relative positions
of the 6 possible elements (3 S's and 3 A's) need to be given
for these theorems.
This does not
exclude the possibility that two triangles might share 5 of the 6
elements but fail to be congruent, if we don't
say which angles are adjacent to which sides. How many examples can
you find of such pairs of triangles with integer sides?
Congratulations to Karthik, again, for many solutions.!
-
Define a sequence of polynomials by P0 := 1, P1 := x+1,
and Pn+1 := Pn + x Pn-1. Show that
all roots of Pn are real, for all n.
Sadly for the polynomials, this one was not solved by the extended deadline.
Keep trying for an exciting prize, though no points!
-
Arithmetic codes.
There is a category of arithmetic puzzles, in which each digit is encoded
as a letter, and the arithmetic message spells something meaningful. The
classic one is a letter sent home from college, which reads
Dear Father,
S E N D
+ M O R E
_________
M O N E Y
This is a good puzzle because there is exactly one solution.
Our contest has two parts. The first is to solve for the arithmetic encoded by
AB*CD=EEE
E*CD-AB=CC
in order to find A*B*D . This much may get you an exciting prize.
The second part is more creative. Make up a puzzle, preferably something
like the SEND + MORE = MONEY code by using concepts from this class.
The winning puzzle will have one and only one solution, and will be judged
on the basis of beauty.
P.S., the father's response to the letter above was:
U S E
+ L E S S
_________
S O N N Y
Congratulations to Tim Lee for two sums, including
I
+ AGREE
+ ITS
_________
TOUGH
- Three-way duel, if "duel" is the proper word.
Three duelists stand at the corners of an equilateral
triangle. The rules of the duel are that they shoot in order, and they draw
straws to see who begins. At each turn a given duelist gets one shot, and
the shooting continues until there is only one left standing.
Now, they all know that Dead-Eye Dana is a great sharpshooter who will
hit the target 100% of the time, while Average Alex hits the target 80% of
the target 80% of the time and So-So Sal hits the target 50% of the time.
Although the duel began over some passionate irrationality, now their
minds are focused, and they each act rationally to maximize the chances
of their survival. The contest is to figure out the probability each of
the three has to survive, and to explain your answer clearly.
Congratulations to Steven Paul for showing (quantitatively) that the
worst shot is the most likely to win!
-
The infidelity contest.
Note: This logic puzzle is the creation of
John Allen Paulos,
and was recounted in his book Once upon a Number,
which was chosen by the
Los Angeles Times
as one of the
best nonfiction books of 1998. The innovation of alien
genders is, however, my own.
A nominally monogamous species on the planet Thrae consists of two
genders, "wen" and "momen," the singulars being "wan" and
"moman." Now, wen are very intelligent and logical, and each wan
knows that all the other wen are equally so. They have the
following well-known habits. 1. If a wan has proof that the moman
to whom the wan is married has been unfaithful, the wan will kill
the moman within one day. 2. Wen's intuition always allows them
to know when another wan's spouse has been unfaithful. 3. Wen
never tell any wan else about infidelity. 4. For some reason, wen's
intuition fails them as to their own spouses. 5. Every wan takes
as proved whatever is printed in the Equinchet, a local oracle.
In the village of Chetag there are forty married couples,
and all the momen have been unfaithful. Of course, every wan
lives in the belief of being the one wan with a faithful spouse.
Well, maybe. One day the Equinchet prints the statement that
it has evidence that at least one moman in Chetag has been
unfaithful.
Does any wan learn anything not already known? Do any killings
ensue? If so, how many and when? To win this one, you have to
have the correct answer and the correct logical reasons for it.
(Hint: One wins when one wonders well when one wan wonders whether
one wan's one wanders.)
Ah, well, no correct solutions to this one by the deadline
-
The literature contest.
Who can find the best examples of realmath in
real literature? I am aware of some examples in work by
Poe, Calvino, Borges, Stephenson, and some others. If you find a
good example our panel of judges was unaware of,
and back it up with the evidence, you
may win a few points.
A split win this time, between Karthik Balakrishnan and Eric Rahm!
Karthik provided a really nice
website on math and fiction, while Eric came up with the
movie
;
check out the
movie's web site.
Honorable mention to Steven Paul for finding Flatland on the
Web.
- Three politicians are having a debate around a round table on the planet
Thrae. To the left of A sits B, to the left of B sits C, and to the left of
C sits A. Now, there are three political parties, the Truthtellers, who always
tell the truth, the Liars, who always lie, and the Wafflers, who
lie and tell the truth in strict alternation. In the course of the debate they
report the following facts (in order) about their opponents:
speaker The person to my left is The person to my right is I am
A a Liar a Waffler a Truthteller
B a Waffler a Waffler a Truthteller
C a Liar a Truthteller a Truthteller
Congratulations to Tim Lee and, minutes later, Casey, Zach, Steven, and
Karthik, for showing that two of the politicians are Wafflers and one a
Truthteller.
-
Eigendoggerel. Write some rhymed couplets about eigenvectors
and eigenvalues. Your couplet must contain
at least one true mathematical statement
(and no false mathematical statements), and must be original by you.
I.e., not downloaded from the Web, where there is no doubt a vast literature
of wonderful poetry about linear algebra. The winner will be judged on the
basis of wit. Your entries are due on Monday, 27 November.
Congratulations to Rebecca Fink, whose poem was voted the best, though
there were also awe-inspiring eigenpoems by Tim Lee and Zach Eaton.