Videos and Additional Comments
Day 1: January 16, 2015
- Video 1.1:
http://youtu.be/6Velp4mNLK4
- Number Bonds:
- Start with how many ways to write 10 as a sum of non-negative
integers. Called number bonds in my son's class. Make sure problem is
clear (summands cannot be negative, view 6+4 as different than 4+6). Talk
about gathering data and deducing patterns.
- Saw the answer was clear when had two summands. With three summands
eventually saw the
triangle numbers
emerge. What do we think we get in general? Any thoughts on the degree of
the polynomial? How do we get a formula for the triangle numbers (can look
at adding dots in a right triangle, double and have a square but need to
add the diagonal twice; could also write the numbers backwards and add).
- I call the general case the cookie problem, though others
call it
stars and bars. I have a very detailed solution of how to derive this
and use this as a module in schools on my
math riddles page (email me
for the password to the student / teacher corner).
- Proof by Example doesn't exist:
- 16/64 = 1/4, 19/95 = 1/5, 49/98 = 1/2, but 12/24 doesn't equal 1/4;
for the first three it looks like the way you divide two digit numbers
with the same value on the anti-main diagonal is by cancelling; a pattern
doesn't always hold up and there is no such thing as proof by example
(unless you go through all possible cases).
- Here is a link to some comments on types of proofs:
http://web.williams.edu/Mathematics/sjmiller/public_html/341/handouts/proofs.html
- Tic-Tac-Toe:
-
We talked about tic-tac-toe today
as a counting problem: how many `distinct' games are there. We are willing
to consider games that are the same under rotation or reflection as the
same game; see http://www.btinternet.com/~se16/hgb/tictactoe.htm for
a nice analysis, or see the image
here for optimal strategy.
-
Probably the most famous movie occurrence of tic-tac-toe
is from Wargames; the clip
is here (the entire movie
is online here,
start around 1:44:17; this was a classic movie from my childhood).
-
Develin and
Payne: bidding tic-tac-toe analysis
-
Interesting variants:
- We talked about chirality and mirror images in biology; there are a
lot of great articles on this. Look at http://www.rowland.harvard.edu/rjf/fischer/background.php .
- A great exercise is to ask students to design generalized tic-tac-toe
games. Maybe need n in a row on a bigger board, and to help the second
person put some O's already on the board. See very quickly it's easy to
get 3 in a row on a big board if you go first. Try to give students the
freedom to ask questions and explore
- Do dogs know calculus:
-
A common theme that surfaces as we do more and more mathematical modeling
is that simple models very quickly lead to very hard equations to solve.
The drowning swimmer problem is actually the same as Snell's
law, for how light travels / bends in going from one medium to another.
If you write down the equations for the drowning swimmer, you quickly find
a quartic to solve.
-
What
is locally the best direction to move might not be the fastest way to
reach the global extremum; there's the possibility of getting trapped in a
local extremum. Thus doing what is locally best is not always globally
best. A great example of this
are the two papers on whether or not dogs (or one dog in particular,
Elvis) know calculus. Looking at the the
path the dog takes to get the stick in the water, if the dog is in the
water close to the stick the optimal choice is to swim straight for the
stick; however, if the stick is far away it's best to swim to water, run
on water, then jump back in. In other words, if you only can go for a few
seconds the closest you can get to the stick is to swim for it; however,
if you're in it for the long haul it's worthwhile to do something which is
not locally optimal but which will lead to optimal. For interesting
articles related to this, see the two papers below by Pennings on whether
or not dogs know calculus. Click
here for
a picture of his dog, Elvis, who does know
calculus.
- Won 3 of 4:
- Just a cute example on how easy it is to mislead with statistics. If
you've won 3 of 4 it seems like you're doing well, but a little thought
reveals you lost 5 games ago, won 4 games ago, and have 1 loss in the past
3, thus you're really 3 of 5 or 2 of 3!
- Great example from Yes, Prime Minister on how to manipulate people
through questions:
https://www.youtube.com/watch?v=G0ZZJXw4MTA
- Video 1.2:
http://youtu.be/3WarJhQjT9w
- Fractions: 2/3 = 4/6:
- Talked about how to prove two fractions are equal. For 2/3 = 4/6 can
have a pizza split into thirds and then divide each piece in half, and go
from 2/3 to 4/6, but really have units in teh background. It's really 2
big / 3 big = 4 small / 6 small, and the units cancel. Could also do by
adding another pie where again have 2/3 going to one person, and now have
2 big / 3 big si the same as 4 big / 6 big, but need to argue that the
percentage is the same when we have two pizzas with the same split.
- I find a lot of problems are easier if we have units. The scientists
(bio, chem, physics) are often very good with this. This is called
factor
labeling.
- Dimensional analysis and the pendulum:
- Pythagorean Theorem:
-
There are many
proofs of the Pythagorean theorem, including James
Garfield's
(Williams, Class of 1856) proof, and then the dimensional analysis proof.
We then discussed other dimensional analysis problems, such as the period
of a pendulum (see above). Starts with the need for proof by giving an
example of something that works for awhile, but eventually fails.
- James' log-5 method:
-
Sabermetrics is
the `science' of applying math/stats reasoning to baseball. The formula I
mentioned in class is what's known as the log-5
method;
it's simple, it's easy to use, it does a good job, and there is a lot that
can be done with it. My favorite is using it to talk about looking at
extreme cases to get a feel for what shape a formula should take.
-
A better formula is the Pythagorean
Won - Loss formula (someone
linked my
paper deriving this from a reasonable model to
the wikipedia page). ESPN, MLB.com and all sites like this use the
Pythagorean win expectation in their expanded series.
-
Talk of mine on the subject:
- How to do algebra: telescoping series:
- Telescoping sums:
http://en.wikipedia.org/wiki/Telescoping_series
- Lots of examples of how important it is to write algebra in a
good way -- different ways highlight different aspects of the problem.
Sadly, sometimes one way works well for one type and another way works for
another!
- Another great example of arranging
calculations efficiently is
Horner's
algorithm.
Application to fractal geometry and iterating polynomials. A great example
is the Mandelbrot set (see
this video);
one of the first instances of fractals being used in movies was the
Genesis torpedo in Star Trek II: The Wrath of Khan (click
here for the video).
- If you know linear algebra, there's also Strassen's
algorithm for
matrix multiplication.
- Video 1.3:
http://youtu.be/MVqcTLObVps
- M&M game:
- Some talks and papers / slides on the M&M game:
- For me, the most important lesson of the M&M game is the importance of
encouraging questions, and then exploring. There are lots of great ways to
attack a problem, and often there are multiple paths to a solution. We
discussed how to get a recurrence relation in two indices; this is a nice
generalization of the Fibonacci numbers (we'll talk more of these in the
second class).
- Geometric Series from Hoops:
-
The proof we gave today of the geometric series formula (by shooting
baskets) uses many great techniques in mathematics. It is thus well worth
it to study and ponder the proof.
-
Memoryless process: once both
people miss, it is as if we've just started the game fresh.
- There is a real need to have good variable names. It's easy to get
confused and forget that the probability of winning depends on the
probabilities of both people and
who starts with the ball!
- These arguments are given in the M&M slides and video above.
- Babylonian mathematics and look-up tables:
- We talked about how the Babylonians did
mathematics. In addition to the horrors of base
60 (which Wikipedia tells
me is due to the Sumerians,
which must be true if they've posted it), they did give us the look-up
table. The point is to reduce long, painful calculations to
pre-computed quantities, with perhaps some (hopefully linear)
interpolation as
you can't pre-compute everything. In base 60, one would need to have
tables for about 3600/2 = 1800 multiplications to be able to do xy; the
Babylonians noticed xy = ((x+y)^2 - x^2 - y^2) / 2; this reduces the
problem to knowing just squares (only 60 entries needed) and the ability
to subtract and divide by 2. It's much better to do these simpler problems
than the original harder one.
- Click here for more on
Babylonian
mathematics: Click
here for more on base 60.
- Another great example of arranging
calculations efficiently is
Horner's
algorithm.
- ISBN numbers, Check Digits, ....:
- Hamming codes and Hat game:
- Error Detection, Correction and Target:
Day 2: January 25, 2015
- Video 2.1: Lines Of
Symmetry:
http://youtu.be/DtI0dshSNGg
- Lines of Symmetry:
- Talked about lines of symmetry. Main takeaway: need a way to
methodically go through possibilities. It's very easy to get overwhelmed;
break a problem into smaller problems and work on those. Build intuition.
Applications range from how planes are built (and the addition of the
wingtips now) to analyzing tic-tac-toe (from Day 1).
- Be careful about extrapolating too much from small examples. At first
looks like lines of symmetry go through a vertex, but see that's not
always the case. Make conjectures. Try to generalize from patterns. We
found a few nice ones: always go through a vertex or a midpoint, split the
region into two of equal area.
- Try to generalize: what about 3-dimensions and planes of symmetry?
What would we expect? Encourage people to ask questions and create.
- Patterns can be misleading. The OEIS
(online encyclopedia of integer sequences) is a great place to
start. If you search on 1, 2, 4, 7 there are 993 results, though the first
is what I was thinking of, and there is a wealth of additional
information: https://oeis.org/A000124.
What do you think the next number is? How do you find it? Is there a
greedy algorithm lurking here like in other places?
- Math riddle: pentagram:
- The riddle is posted here:
http://mathriddles.williams.edu/?p=67
- Again, one of the issues is figuring out how to go through the
infinitely many possibilities. Break into cases. Maybe the two lines meet
in a point, or they don't. Maybe that point is inside the region, maybe it
isn't. Think about assumptions you might be making (you get triangles by
dividing the region you already have, but that ignores creating new
triangles using some of the exteriors). Try doing what is globally best:
draw a line to give you as many triangles as possible (we can do a line
that gives us three, though it does temporarily cost us some others). This
is somewhat related to the greedy algorithm: doing what is best at a given
moment often leads to a great strategy.
- Video 2.2: \(e^\pi\) vs
\(\pi^e\) and bases: http://youtu.be/yeJKZJ1qMSM
- \(e^\pi\) vs \(\pi^e\) and bases:
- Saw a great application of calculus: calculus allows us to find maxima
and minima by looking for critical points. This is one of the key concepts
of the subject.
- For the comparison, we found it was a little easier to look not at
\(e^x\) and \(x^e\) but rather the logarithms of each. This is a common
theme: try to make the algebra easier, and logarithms convert products to
sums!
- We then moved to the Farmer Brown problem (enclosing the maximum
rectangular pen for a given perimeter); while this can be solved by
calculus, we solved it by seeing the possibilities fell on a parabola.
Interestingly, the Farmer Bob problem (where we get one side for free as
we border water) can be solved without calculus by appealing to symmetry:
imagine Aquaman is building a fence for his fish at the same time and
mirrors what Bob does! This ties in nicely with the first part of the day,
and gives another example of symmetry.
- Partitioning a number to maximize a product:
- We continued with one of my favorite problems, given \(S = a_1
+ \cdots + a_n\) with each \(a_i\) a positive integer and the goal to
maximize the product of the \(a_i\). We quickly see the optimal is when each
\(a_i\) is 2 or 3, and since \(2\ast 2 \ast 2 < 3 \ast 3\) we want \(3\)'s
over \(2\)'s. We converted to a real problem and assumed there were \(n\)
summands, each a real number. We got a function defined on the integers to
maximize, replaced it with a function defined on the reals so calculus would
be applicable. We then curve sketched and saw the function was increasing to
its maximum and decreasing past it, so the optimal integer soln was either
to the left or right of the optimal real soln (here optimal soln is
referring to the number of summands). It's unusual to be this fortunate.
- We had to maximize \(a_1 \ast \cdots \ast a_n\) given \(a_1
+ \cdots + a_n = S\) and each \(a_i > 0\). We can do this with Lagrange
multipliers, or since each \(a_i\) is in \([1, S]\) we can appeal to the
\(n=2\) case because a real
continuous function on a compact set attains its max and min. What is
nice is that this existence result from real analysis improves to being
constructive; if we were at the optimal point and all coordinates were not
equal, we could simply replace two of them with the average and improve the
product. Thus repeated applications of the Farmer Brown problem give us all
the \(a_i\)'s are equal.
- A nice application of this problem is that for disk storage
(see radix
economy), base
3 has advantages over base
2, though base 2 has the very fast binary
search. Another nice example of base 3 occurs with the Cantor
set.
- Great paper on the power of base 3:
http://www.americanscientist.org/issues/pub/third-base
- We talked a lot about how to use logarithms to make an
analysis easier, or to exponentiate. For example, \((S/x)^x = \exp(x \log(S/x))\).
- When I teach calculus I end with a warning that you may have been fooled into
believing you learned certain derivatives when you hadn't. For example:
- \(f(x) = x^n\) has derivative \(n x^{n-1}\). This follows
from the definition of the derivative and the binomial theorem to expand \((x+h)^n\)
when \(n\) is a positive integer.
- \(f(x) = x^{p/q}\) has derivative \(\frac{p}{q} x^{p/q-1}\).
This follows by setting \(g(x) = f(x)^q = x^p\) and then differentiating,
which gives \(g'(x) = q f(x)^{q-1} f'(x) = p x^{p-1}\), and then
substituting and solving for \(f'(x)\). We cannot get it the same was as the
derivative of \(x^n\), as that would require knowing the binomial theorem
for non-integral exponents.
- \(f(x) = x^{\sqrt{2}}\) has derivative \(\sqrt{2}
x^{\sqrt{2}-1}\). This follows from using the exponential function and the
chain rule: \(x^{\sqrt{2}} = \exp(\sqrt{2} \log x)).
- Thus, \(x^r\) does have derivative \(r x^{r-1}\), but the
proof for general \(r\) goes through the exponential function.
- Video 2.3: Fibonacci:
Part 1:
http://youtu.be/9ERNg3WiLqo
- Basic Properties:
- Rough size: we started with a simple argument: \(2 F_{n-2} \le F_n \le
2 F_{n-1}\), so \(\sqrt{2}^n \le F_n \le 2^n\) and thus we have a rough
sense of the growth rate of the Fibonacci numbers (they grow exponentially
fast). This suggests the divine inspiration guess of \(F_n = r^n\)
later.
- Here is a nice video on the Fibonacci numbers in nature: http://www.youtube.com/watch?v=J7VOA8NxhWY
- Binet's formula:
There are many ways to prove Binet's
formula for
an explicit, closed form expression for the n-th Fibonacci number. One is
through divine inspiration, the second through generating
functions and partial
fractions.
Generating functions occur in a variety of problems; there are many
applications near and dear to me in number theory (such as attacking the Goldbach or Twin
Prime Problem via
the Circle
Method).
The great utility of Binet's formula is we can jump to any Fibonacci
number without having to compute all the intermediate ones. Even though it
might be hard to work with such large numbers, we can jump to the
trillionth (and if we take logarithms then we can specify it quite well).
- Applications: Leslie matrices: we gave a great application of
difference equations to modeling population growth:
http://en.wikipedia.org/wiki/Leslie_matrix
- Applications: Double plus one: here's the video I made with OIT at
Williams: https://www.youtube.com/watch?v=Esa2TYwDmwA
- Note in the double plus one application that we see it's easier to
write and solve the recurrence when we look for the probability we do
not have 5 consecutive blacks in n spins; this continues a common
theme that sometimes one problem leads to easier algebra than another.
- Video 2.4:
Fibonacci: Part 2:
http://youtu.be/o-z4rL67vVY
- Interesting relations:
- Fibonacci Quilt: It's really called the Golden Spiral:
http://en.wikipedia.org/wiki/Golden_spiral
- Talked about the beautiful relation \(F_1^2 + F_2^2 + \cdots + F_n^2 =
F_n \cdot F_{n+1}\). We saw how this is reasonable as each term on the
right is an area and this suggests something like the golden spiral, so we
expect it to equal something involving products of Fibonacci numbers. This
is a nice connection to our earlier work on dimensional analysis. Do we
expect a similar formula to exist for sums of cubes of Fibonacci numbers?
You can ask the students to conjecture, to try to find something to fit
the data, .... The danger, of course, is that we are summing numbers and
there is nothing that forces us to assign units to them, so a relationship
if it exists might not involve products of three terms. Thus, we need to
be careful. (In fact there is a relation and it involves only first
powers; see
https://www.math.hmc.edu/~benjamin/papers/sumfibocubes.pdf ).
- Nice TED talk on this by Arthur Benjamin of Harvey Mudd:
http://www.ted.com/talks/arthur_benjamin_the_magic_of_fibonacci_numbers
- Log cabin quilt: YouTube tutorial:
https://www.youtube.com/watch?v=nniiZsdpnx0
- Golden ratio:
http://en.wikipedia.org/wiki/Golden_ratio
- Silver ratio:
http://en.wikipedia.org/wiki/Silver_ratio
- If you're interested in new sequences related to the Fibonacci Quilt,
email me and I'll share work in progress.
- There is a lot of great research on tiling the plane with various
figures. You can talk about regular tilings (equilateral triangles,
squares, regular hexagons) and ask students what shapes they think can
work. You can talk about tiling 3-D or higher. You can talk about tilings
by irregular figures, or multiple figures (octogon and square). Another
option is Platonic solids. It all comes down to trying to find ways to put
things together nicely. Homework can be to look at tilings in their house
/ school / .... Lots of nice jumping points.
- Video 2.5: Fibonacci:
Part 3:
http://youtu.be/hUk5lFbLvT4
- Birthday Problem:
- As a bonus we discussed the
Birthday Problem (Wikipedia
gives the Taylor expansion argument from taking logarithms,
which is also in my book if anyone wants additional comments). You should
have a Pavlovian response and always always always think `take a
logarithm' when you see a product. What is particularly nice about this
problem is you can see the parameter dependence. This problem is `simple'
in that there's only one parameter, the number of days in the year (or two
if you consider the percentage variable as well). The argument generalizes
easily; in the book I give the general case. This is but one of many
possible generalizations. You should get into the habit of asking what
else can you ask. What are other good questions? What if we ask for how
many people we need to have at least a 50% chance that at least three will
share a birthday? Or that there will be at least two pairs of people
sharing birthdays? Questions like these are great extra credit / challenge
problems: if you're interested, just let me know.
-
There are practical applications of the birthday problem. One of these
is in the birthday
attack, which arises in cryptography
(and thus connects to the next unit!).
-
Implicit in our analysis was the pigeon-hole
principle, a powerful method and one worth knowing.
-
We also needed the sum of the integers up to n-1; there's an industry in
computing these values. One great way is via mathematical
induction, but this has the drawback of requiring you to know the
answer ahead of time. The sums of integers are the triagonal
numbers; there are formulas for sums of powers (see here and here).
- Alternate definition:
- We then
returned to Fibonacci numbers, and talked about an alternative definition of
the Fibonacci numbers as the unique sequence of positive integers such that
every number can be represented uniquely as a sum of non-adjacent terms of the
sequence. We proved the equivalence of the two using the
Greedy algorithim
(this is a wonderful and powerful technique; what's nice is that no thought is
required, but that's also a drawback and sometimes it leads to suboptimal
results). It's interesting to see what kinds of generalizations are possible,
namely you can take a sequence you've studied all your life and find new
twists. Below are some generalizations I've done with students / colleagues;
some of these projects were done with high school students and I have research
projects here available. This is just a subset -- email me if you want more
information.
- Video 2.6: Cryptography:
http://youtu.be/9p2ZMoDGaGM
- General Information:
- There is a wealth of information on cryptography. Here are a
collection of links from a winter study I taught on the subject a few
years back:
http://web.williams.edu/Mathematics/sjmiller/public_html/crypto/additionalcomments.htm
- Here is Wikipedia's page on RSA:
http://en.wikipedia.org/wiki/RSA_%28cryptosystem%29
- One of the key facts we needed was that if an integer \(a\) is
relatively prime to a prime \(p\) then \(a^p \equiv a \bmod p\). This is
normally proved in abstract algebra / group theory as a consequence of
Lagrange's theorem, but it is possible to prove elementarily. The key
facts are the
binomial theorem and that if \(\left({p \atop k}\right)\) is the
binomial coefficient with \(p\) prime then it is a multiple of \(p\) if
\(1 \le k \le p-1\) (it's an integer and the denominator isn't a multiple
of \(p\), but the numerator is).
- Here's the proof. We proceed by induction. It's clearly true
when \(a=1\), and now we show that if it is true for \(a-1\) then it is
true for \(a\). (We need a slightly stronger statement, namely that the
result also holds if \(a\) is a multiple of \(p\), but the proof is
trivial in that case as both sides are multiples of \(p\) and hence equal
modulo \(p\).) By assumption we know \((a-1)^p \equiv a-1 \bmod p\). Then
by the Binomial Theorem \(a^p = ((a-1)+1)^p = \sum_{k=0}^p \left({p \atop
k}\right) (a-1)^k 1^{p-k}\); as all the terms but the first and last are a
multiple of \(p\), if we look modulo \(p\) we find \(a^p \equiv (a-1)^p +
1^p \bmod p\). Now we use the inductive assumption that \(a-1)^p \equiv
a-1 \bmod p\), and we get \(a^p \equiv a-1 + 1 \bmod p\), completing the
proof.
- Note the above is not a full proof of RSA, but it does give us one of
the key steps.
- We talked about fast multiplication; you can find that (as well as a
description of the cookie problem and solution) in chapter one of another
book I wrote, available online here:
http://press.princeton.edu/chapters/s8220.pdf
- In addition to fast ways to multiply numbers, there are also
significantly faster ways to multiply matrices! See the Strassen
algorithm.
- Below are some old drafts of chapters from a cryptography book I
wrote. Please don't distribute these further, but they give some sense of
what is out there.
-
Classical cryptography: This is the core chapter. It begins with one
of the oldest ciphers, and continues through subsequent improvements.
-
Enigma and Ultra: The Germans believed their Enigma code was secure;
fortunately for us the Allies were able to crack it. In this short chapter
we'll discuss some of the mathematics behind Enigma, and see why the
Germans reasonably felt that it would be impossible to break.
-
Errror detecting and correcting: This
chapter here deals with transmitting information in such a way that, not
only can we detect when we've made certain errors, but we can also correct
them! The mathematics is motivated through some riddles. These riddles can
be used to excite and interest students, and naturally lead to the more
advanced material. You've seen
these ideas all your life with UPC
codes.
- Some cryptography links:
- Here are some books recommended by the class a few years ago
