Freshman Seminar 24i applicants' Essay 2 problems (Fall 2010)
(in roughly the order received, with some further comments):

Euler's celebrated identity
e^{iπ}+1=0.
[This is the key to De Morgan's quote
Imagine a person with a gift of ridicule ... [He might say]
First that a negative quantity has no logarithm; secondly that
a negative quantity has no square root; thirdly that the first
nonexistent is to the second as the circumference of a circle
is to the diameter. (A Budget of Paradoxes, 319—320)
which unwraps to log(sqrt(1))/sqrt(1)=π.]
Euler's identity is proved as a special case of
Euler's formula
e^{ix} = cos(x) + i sin(x)
[a.k.a. "cis(x)"], by comparing Taylor series
or differential equations. This link
between the exponential and trigonometric functions is a key
building block of the mathematical universe, and we shall see that
it also gives some very useful problemsolving tools  for starters
it encapsulates most of the important trig identities (e.g. compare
real and imaginary parts in
e^{ix} e^{iy} =
e^{ix+iy} =
e^{i(x+y)}
[a.k.a. cis(x) cis(y) = cis(x+y)]
to recover the formulas for
cos(x+y) and sin(x+y)
respectively).

Connnect a 3x3 array of 9 dots with an uninterrupted path of 4
straight line segments.
[A famous example of literally “thinking outside the box”;
I've also seen creative approaches to reducing the count of segments
even below 4, by folding the paper, exploiting the thickness of
physical dots and pen(cil)points to make a Zshaped path of
notquiteparallel line segments, and even using a writing implement
with a really thick point to reduce the count to 1!
Ignoring such cheats, how few segments does it take for an
mbyn array?]

Find a positive integer which doubles when its last digit is moved in
front. [For example, if the problem said
“quintuples” instead of “doubles” you could use
142857. Can you
find more such numbers, whether for doubling, quintupling, or
other multiples? Can you describe all the possibilities?]

Using the derivative of a function to find extrema, inflection points,
etc. [we'll also see in the seminar how to find some extrema without
calculus; these include some standard calculus examples as well as
some that would be very hard to do with calculus techniques.
To be sure there are also many max/min problems for which calculus
provides the simplest if not the only solutions.]

Surface area integrals in Calculus BC

Mathematical problems arising in AP physics

The problem of the
Bridges of Königsberg, which led Euler
to inaugurate combinatorial graph theory.
[Note that this first example of a graph has multiple edges,
which many modern treatments of graph theory regard as unorthodox.]

Cauchy's
theorem in group theory [this is rather abstract compared with
the kind of math we'll usually work with in 24i, but the technique
is useful also in many more downtoearth problem, such as one
standard proof of (p1)!≡1 mod p
for all primes p (the harder part of
Wilson's theorem in elementary number theory).]

An international society has 1978 members from six different countries.
The members are numbered 1, 2, 3, …, 1978. Prove that there is
at least one member whose number is the sum of the numbers of two
(not necessarily distinct) members from his own country.
[a tricky
IMO problem that turns out to be part of
Schur's theorem in Ramsey theory;
this we will be able to do in 24i as an example of the
“pigeonhole principle” and/or induction.]

A Calculus BC project: compute (without resorting to scales,
graduated cylinders, and the like) the volume of an irregularly shaped
Bundt cake.

A lemma on continued fractions: of two consecutive
rational approximations
p_{n}/q_{n}
(n=i, i+1)
to any real number x, at least one is within
1/(2(q_{n})^{2})
of x.

A few of the problems that lead to the The remarkable Catalan numbers
(2n)! / (n!(n+1)!) = 1, 2, 5, 14, 42, 132, 429, 1430, 4862, ...
— see for instance
Richard Stanley's list of lots of
“Exercises on Catalan and Related Numbers”
from his magnum opus Enumerative Combinatorics
(that page also links to the solutions of these exercises,
and a “Catalan addendum” that as of July brings the
number of combinatorial interpretations to 166).

Evaluate the continued fraction
1+1/(1+1/(1+1/(1+…))).
[This yields the unreasonably ubiqutious
golden ratio
φ = (1+sqrt(5))/2 = 1.61803…;
it arises in one explanation of why φ is the
“most irrational” number, in the sense of being
farthest in a certain precise sense from any rational approximation
— cf. the other continuedfraction problem above.]

Simplify this scary expression
(which turns out to be an example of a “telescoping sum”).

The traveling salesman problem [besides modeling a
realworld problem, this one is essentially equivalent to
one of the milliondollar problems of the
Clay Math Institute
headquartered just outside Harvard Yard.

A memorable puzzler from 7th grade math:
a hoop girding the Earth at its Equator expands by 10 feet,
so it is now a constant height above the Equator (assumed to be
a perfect circle — never mind that Quito is almost 3 kilometers
above sea level, etc.). What is this height? [Surprisingly one
doesn't need to know the size of the Earth for this. A followup
that may be even more surprising: pull on a belt around the same
idealized equator, again lengthening it by 10 feet, but this time
concentrated at some point P on the Equtor, so that the belt now
consists of two line segments AP, PB tangent to the Earth at the
Equator, together with the long arc BA around the Equator.
How far above the Equator is P? Here you will have to know
the size of the Earth (the Equator is very nearly 40000 kilometers
or 4⋅10^{7} meters long,
while 10 feet is about 3 meters), as well as some calculus at
the level of Harvard's Math 1b.]

The Futurama mindswapping theorem
(about generating arbitrary permutations by certain simple
transpositions).

Of the repeating decimal periods of the fractions
1/13=.076923…,
2/13=.153846…,
3/13=.230768…, etc. till
12/13=.923076…,, the six cyclic permutations
of 1/13 occur precisely for
n/13 where n is one of the
“quadratic residues” 1, 3, 4, 9, 10, 12 of 13
(that is, one of the numbers that arises as a nonzero remainder
mod 13 of a perfect square)!
[This is somewhat related with the earlier comment regarding 142857.]

Equations like x  ¼ = x that irritatingly
have no solutions. [Such equations are a part of the mathematical
universe, and it's not usually worth being irritated by them,
though on occasion important advances have been made by famously
expanding that universe to include solutions of equations like
x+5=3, 2x=1, and
x^{2}+1=0 that previously had
no solution; even for x¼=x
it is sometimes right to say x=∞
is the (unique) solution.]

Zeno's “paradox of dichotomy”,
which leads to the modern notions of limits and convergence

The "Monty Hall problem"; curiously it seems that
Monty Hall himself was not aware either of the controversy or of
the fact that switching offered a clear advantage,
until a statistician asked him about it only a few years ago!
[I also just learned that he shares my birthday, albeit
45 years earlier; NB that's not an example of the
"birthday paradox" — do you see why? How long a list of random
birthdays must I go through before I have a better than 50% chance
of finding somebody else with the same birthday as me?]

Systems of simultaneously linear equations [a kind of problem which
goes back literally millennia and which leads to the
remarkably fruitful ideas of modern
linear algebra]

Historical and still productive theorems, as of Euler
(in calculus and elementary number theory) and Lagrange;
the use of L'Hôpital's rule to evaluate limits

The “100 blueeyed islanders” riddle
[again an example of mathematical induction;
see also the similar puzzle of
the lions and the lamb,
and this example of the family of
“pirates' gold” puzzles]

Find two “primitive Pythagorean triples” (solutions of
a^{2}+b^{2}=c^{2}
in relatively prime natural numbers a,b,c)
beyond the familiar (3,4,5)
and the possibly less familiar (5,12,13).
[A classical “Diophantine equation” (equation to be solved
in integers). Can you generate infinitely many Pythagorean triples?
All of them?]

Given 12 balls, all but one of the same weight, and an unmarked
balance scale, determine in only three weighings which ball is
different and whether it is lighter or heavier than the rest.
[Can you prove that 12 is the maximal number for which this task
can be accomplished with only three weighings? What if you can
perform four, five, … n weighings?]

A recent problem from the
AIME: In a parallelogram ABCD, point M is on AB
such that AM/AB = 17/1000 and
point N is on AD so that AN/AD = 17/2009.
Let P be the point of intersection of AC and MN. Find AC/AP.
[A good example of a problem in plane geometry that can be most easily
solved with coordinates or vectors]

Trigonometry problems, and their reconsideration in calculus
as with the limit of sin(x)/x
as sin(x)→0
[which in effect goes back to the Greek method of approximating π
using areas of inscribed and circumscribed regular polygons]

The “missing dollar” riddle