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[edit] Current Puzzles
There is no current puzzle, so that you can find the parrot.
[edit] Announcements & news
12/14/06: Hi guys, well class is over, *sob*, but I was checking out the portal on wikipedia, and the weeks article is on fermat's last theorem. you might want to check it out. Peace, have a great holiday - - Avi [1]
11/20/2006: Incase anyone has free time over the break and would like to tackle a different kind of problem, here is a puzzle from my work for the final:
Create a chess match where the king is only moved as a result of forcing moves (i.e. a check, to prevent the loss of material, etc.) and the final position on the board is white king on a1 and black king on h8 (these are the only two pieces on the board).
As a tip, I wouldn't worry about making the moves plausible (they do still have to be legal however) and as I haven't come to a solution yet, I don't know that the moves in the match would be plausible.
--Alex.
11/15/2006: Inspiration! "Take chances, make mistakes, get messy." --Miss Frizzle, Magic School Bus --Reid
11/14/06: Just for fun that I got from an e-mail forward. Word Games --Reid
11/12/06: Firstly - no problem Jason. Secondly, sorry, but CDFGJKNPQTVWXYZ is not acceptable for PYX is a word, so I guess that brings us down to 14. Thirdly, no worries Reid; www.wineverygame.com ---- Avi
oy. --jrz
A fun little website: http://www.onelook.com/ By the way, "x-ray" has both "x" and "y." Does that count as a word? --Reid
10/10/06: Thanks a lot, Avi. Now I have to upload another edition of "Say the Word." A trip to the ol' dictionary reveals that an ai is a kind of sloth, and that qi is an acceptable alternative spelling of the traditional chinese concept of bodily energy (usually spelled chi). The qi spelling is normally capitalized, but American Heritage also lists lower-case q as an option.
So maybe instead of using the example AQZI as a silly string, it could be AQZE. At a glance, I don't see any allowable words in there, but it would be worth a systematic check before I upload another edition.
If possible, I would like the example to be short and to have two vowels. Otherwise it looks too much like the eventual answer.
--jrz
Two things:
1. (2)(3)(5)(7)(11)(13) + 1 = (59)(509). That settles the prime number issue.
2. I really liked Avi's question about whether a calculator could have an "inverse factorial" button! This leads to two thoughts. First, given a randomly chosen integer Q, what would be an efficient algorithm for finding out whether Q is the factorial of some integer? Interesting to think about.
Second, there is actually a continuous function that allows one to compute the "factorial" of a decimal number such as 1.73. I'll call this function F(x). It is defined as follows: F(x) = the integral of t^x e^(-t), from t=0 to t=infinity. You can verify that if x is an integer, then the integral evaluates to x!. For values of x lying between two integers n1 and n2, the integral evaluates to something intermediate between n1! and n2!. As an example, F(3) = 6 and F(4) = 24, and F(3.32) = 9.096....
So the question of the inverse-factorial button is really the question of making an F-inverse button. I don't know of a calculator that has an F-inverse button (not even Mathematica seems to have it). But one could presumably be designed.
Somewhere on the web I saw a suggestion for the symbol "?" to be used to denote the inverse factorial. This is cute, because then you get things like 24? = 4, 3 = 3!?, etc.
By the way, our F(x) is basically Euler's "Gamma function", but shifted by 1. That is to say, F(x) is Gamma(x+1).
[edit] Current Assignment
Fall 2006-Bennington College-jrz
1. Be prepared on Monday to talk for 5 minutes (this is a long time) about your final project ideas.
2. We are on a 50-page-a-day pace with The Making of the Atomic Bomb. Don't lag! I did my 50 yesterday, did you?
[edit] Interesting stuff to go look at
Lately we've been looking at transcendental equations. One of the most famous and important transcendental equations is "Kepler's Equation," which must be solved in order to determine the angle of an orbiting planet as a function of time. MathWorld is a good starting point for learning more about Kepler's Equation.[2].
[edit] Class topics
Hi Everyone! Over the summer I worked for the Grassroot Institute of Hawaii. They recently published two articles on Global Warming, and given our recent conversation, I thought you might enjoy them:
Brazilian Ethanol Fantasies
Man-Made Global Warming Continues to Unravel
--Reid
[edit] Archives
[edit] Assignment 6
Due Thursday 10/26:
1. Problem 13(c) from Hildebrand.
2. Classify, and find the general solution y(x): (dy/dx)^2 -9y + 18 = 0.
3. Classify, and find the general solution v(t): (d/dt)[ v(1-v^2)^(-1/2)] = 4.
[edit] Assignment 5
Assigned Monday, October 9
Due Monday, October 18
Mainly a calculus review to warm up for differential equations, but see the last two problems, which are pretty cool.
\begin{enumerate}
\item A function $f$ is defined for values of $x$ in the range $[0, {9\over 4}]$, according to the rule $f(x) = 2x^3 - 9x^2 + 12x$. Find the maximum and minimum values of this function over the range of definition.
\item A triangle has two sides of length 1. The third side has length $c$. The angle opposite $c$ is denoted $\theta$.
Imagine steadily opening up the triangle, so that the angle $\theta$ increases with time. This will cause length $c$ to increase with time as well.
At a moment in time when $\theta$ is $\pi\over 2$ and $\theta$ is increasing at a rate of 0.5 rad/sec, how fast is $c$ changing?
\item The speed of a whiffle ball is given as a function of time by $v(t) = {1\over t}$. Find the average speed of the whiffle ball over the interval from $t=1$ to $t=5$.
\item Differentiate the following functions:
\begin{enumerate} %parts
\item $f(x) = 8$
\item $g(t) = -2t^2 + 3t - 6$
\item $f(x) = {1\over x^8}$
\item $f(x) = x^2 - {1\over 2}\cos{x}$
\item $f(x) = 3x(6x-5x^2)$
\item $g(r) = {r-1\over r+1}$
\item $a(z) = z^3\sin{z}$
\item $g(t) = (t^3 + 8)^{100}$
\item $f(x) = (\cos{x})^2$
\item $f(x) = \sin{\!\left(x+x^{1/3}\right)}$
\item $s(\theta) = \tan\theta$
\item $g(x) = \sqrt{\tan{x}}$
\item $k(x) = x^x$
\end{enumerate} %parts
\item Evaluate the following integrals:
\begin{enumerate} % parts
\item $\int_1^3{(x+2)\,dx}$
\item $\int_{-2}^2{\sqrt{4-x^2}\,dx}$
\item $\int{(x^3 + 6)\,dx}$
\item $\int{4\cos{5x}\,dx}$
\item $\int{x\sqrt{2x-1}\,dx}$
\item $\int{(x^2+x)(2x+1)\,dx}$
\item $\int{(\sin{x})^3\cos{x}\,dx}$
\item $\int{(x^2+1)^2\,dx}$
\item $\int{3x^3\sin{x^4}\,dx}$
\item $\int_2^5{(y-1)e^{-y^2}\,dy}$
\end{enumerate} % parts
\item In this problem, we are going to use integration by parts to evaluate $\int{e^x\sin{x}\,dx}$.
\begin{enumerate} %parts
\item Approach the integral using integration by parts, with $u = \sin{x}$ and $v^\prime = e^x$. The resulting integral will not be any easier than the one we started with. However\dots
\item Use integration by parts again, with $u = \cos{x}$ and $v^\prime = e^x$. You should end up with an equation that can be solved {\it algebraically} for the original integral. (Suggestion: replace $\int{e^x\sin{x}\,dx}\,$ in your equation with the symbol $I$, and solve for $I$.)
\item Once you have found $\int{e^x\sin{x}\,dx}\,$, differentiate your answer to make sure that you get what you ought to get.
\end{enumerate} %parts
\item Given that \begin{equation} \int_{-\infty}^\infty{e^{-x^2}\,dx} = \sqrt{\pi}\, \end{equation} show using integration by parts that \begin{equation} \int_{-\infty}^\infty{x^2\,e^{-x^2}\,dx} = {\sqrt{\pi}\over 2}\,. \end{equation} What about $\int_{-\infty}^\infty{x^4\,e^{-x^2}\,dx}$, $\int_{-\infty}^\infty{x^6\,e^{-x^2}\,dx}$, and so on?
\item Verify that $\int{xe^x\,dx} = xe^x - e^x + C$.
\item Find the value of $b$ so that the area under the graph of $f(x) = 1/x^2$ between $x=1$ and $x=b$ is equal to 0.5.
\item A spaceship has a speed $v(t)$ that depends on time as \begin{equation} v(t) = {3t\over \sqrt{1 + 9t^2}} \end{equation} Time in this problem runs from $t=0$ to $t=\infty$.
\begin{enumerate}%parts
\item Show that there are no stationary points for $v(t)$.
\item Show that the spaceship is always speeding up, but at a decreasing rate.
\item Sketch $v(t)$ for values of $t$ in $[0,\infty]$.
\item How far does the spaceship travel between $t=0$ and $t=1$? How far does the spaceship travel between $t=1$ and $t=2$? How about between $t=2$ and $t=3$? Do these answers make sense given your sketch of $v(t)$? Can you sketch the antiderivative of $v(t)$?
\item Evaluate the definite integral $\int_0^{t_{\rm f}}{v(t)\,dt}$. Call the answer $D(t_{\rm f})$.
\item Find a simpler expression for $D(t_{\rm f})$ that is approximately valid for early times ($t_{\rm f} \ll 1$).
\end{enumerate}%parts
\item {\it From the 2001 \lq\lq Green Chicken Contest," held yearly at Williams College.} Consider the functions $f(x) = x^2 + 2bx + 1$ and $g(x) = 2a(x+b)$, where the variable is $x$ and the constants $a$ and $b$ are fixed real numbers. Let $S$ be the set of all pairs $(a,b)$ such that the graphs of $y=f(x)$ and $y=g(x)$ do not intersect in the $x$-$y$ plane. What is the area of $S$ in the $a$-$b$ plane?
\item {\it From the 1996 \lq\lq Green Chicken Contest," held yearly at Williams College.} A hallway of width $a$ turns through 90 degrees into a hallway of width $b$. A ladder is to be passed around the corner. If the movement is within the horizontal plane, what is the maximum length of the ladder?
\end{enumerate}
[edit] Assignment 4
"Fermat's second challenge to the mathematicians" (handed out in class).
I found these sites: [3] , [4] --jrz
And this good document: [5] --jrz
Liz has figured out my error in applying the continued-fraction algorithm. I was truncating the representation too early.
By Thursday, let's be fluent with solving Pell's equation for simple cases such as d=3,5,6,7,8,10, etc.
The continued-fraction algorithm seems to be very specific to the equation x^2 - d y^2 = 1, specifically with a 1 on the right-hand side. I wonder how a person would go about solving x^2 - d y^2 = 2 ? -in class we talked about how to generate additional solutions from a given solution and a solution to the unit form.
[edit] Assignment 3
Assigned: Thursday, September 14
Due: Monday, September 25
1. Find a solution to the equation e^x = cos{x} to two significant figures (besides x=0!).
2. For what values of a does e^x = asin{x} have a positive solution? Find an approximate form of the smallest positive solution, as a function of the parameter a.
3. Still due on 9/28, but sooner would make more sense given the way things turned out: A two- to three-page paper describing your attempts to solve the tower-of-sevens problem---the approaches taken, the false starts, and how you found the path to success. Also include and discuss subjective facts, such as your state of mind at various times during the process.
[edit] Assignment 2.5 The Cock-Up Puzzle
The most mathematically-flavored puzzle from Jason's book "Sentence of Death: More Puzzles for the Seriously Smart."
Cock-up
In British slang, a “cock-up” is a screwed-up situation. Call a collection of sentences a cock-up if the sentences are logically impossible when taken altogether. For example, the following two sentences comprise a very famous cock-up:
The next sentence is true.
The previous sentence is false.
Why is this a cock-up? Well, if the first sentence were true, then based on what it says, we would have to regard the second sentence as true; but then in that case we would have to regard the first sentence as false — contrary to our beginning supposition that the first sentence was true!But perhaps all this means is that the first sentence must be regarded as false…. Well, let’s see. If the first sentence were false, then we would have to regard the second sentence as false; but then we would have to conclude that the first sentence is true, contrary to the beginning supposition that the first sentence was false!Thus, there is no consistent way to assign truth values to both of these sentences. That’s a cock-up.
Here’s another example of a cock-up, this one consisting of three sentences:
1. Sentence 2 is true.
2. Sentence 3 is false.
3. Sentence 1 is true.
Do you see how that’s a cock-up?Now for the puzzle:
1. Sentence 2 is true.
2. Sentence 3 is false.
3. Sentence 4 is false.
4. Sentence 5 is true.
5. Sentence 6 is false.
6. Sentence 7 is true.
7. Sentence 1 is _____.
What should Sentence 7 say in order to make this collection of seven sentences a cock-up?
[edit] Assignment 2.25 The Twelve-Coin Puzzle
When given twelve visually-identical coins, you are told one is counterfeit. All the coins have the same weight, except for the counterfeit. Given a two-sided balance, how can you identify with certainty which coin is the counterfeit and whether it is lighter or heavier than the other coins in three weighings?
[edit] Assignment 2
1. Find all real x such that both hold simultaneously:
- (sinx - cosx)^3 > 0
- sinx > -0.1.
2. Still due on 9/28, but sooner would make more sense given the way things turned out: A two- to three-page paper describing your attempts to solve the tower-of-sevens problem---the approaches taken, the false starts, and how you found the path to success. Also include and discuss subjective facts, such as your state of mind at various times during the process.
[edit] Assignment 1
1. Consider an exponential tower consisting of three thousand 7's:
(7^(7^(7^( . . . )^ 7)))
What is the remainder when you divide the tower by 11?
I expect you to work together on this problem. Submit your final answer as a group.
All I want is the final answer, a single integer in the range 0-10. Write this integer on a piece of paper, and sign all your names underneath it.
class meetings will resume when I receive the correct answer, or on 9/28 if you have not found the correct answer before then.
If you submit an incorrect answer, then you must wait a week before submitting another answer.
Use any means necessary. I will not assist you.
2. Due on 9/28: A two- to three-page paper describing your attempts to solve this problem--the approaches taken, the false starts, and how you found the path to success. Also include and discuss subjective facts, such as your state of mind at various times during the process.
