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\noindent {\large \bf Homework 1 [40 pts] \hfill 21-127 Concepts of Math \hfill due 07.06.2012}\\
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\noindent{\bf\large Notes:} All of these problems are designed to be solvable by you {\it now}, using just your thinking caps. I encourage you to get started on this {\it early} and come to office hours to ask questions. Also, since we haven't discussed much about what constitutes a {\it proof}, just keep the following guidelines in mind: your answers should be well-written (both grammatical and legible) and understandable by both me and your fellow classmates. We just want to get you thinking and trying to {\it explain} your thoughts as best you can!

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{\large\bf Explain}:
\begin{enumerate}
\item {\bf ``Wean''-ing into the course:} Find my office in Wean 7104 sometime this week while I am there and tell me your name. It doesn't count unless I'm there!\\
{\bf [2 pts]}
\item {\bf Two Negs Make a Pos:} Explain why $(-1)(-1)=1$. Pretend you are writing for a fellow classmate of similar intelligence who is skeptical of this fact and needs to be convinced.\\
{\bf [2 pts]}
\item {\bf Proof or Spoof?} Consider the following ``proof'' that $1=2$. What is wrong with it, if anything?

``Let $x=1$ and $y=1$. Thus, $x=y$. Multiply both sides to get $x^2=xy$. Add $x^2$ to both sides to get $2x^2=xy+x^2$. Subtract $2xy$ from both sides to get $2x^2-2xy=x^2-xy$. Factor and write $2(x^2-xy)=1(x^2-xy)$. Divide both sides and conclude that $2=1$.''

Be as {\it specific} as possible; identify {\it where} the errors are and {\it why} they're errors.\\
{\bf [3 pts]}
\item {\bf Proof or Spoof 2? The Return of the Spoof:} Let $a,b,c$ be real numbers, with $a\neq 0$. What is wrong with the following ``proof'' that $-\frac{b}{2a}$ is a solution to the equation $ax^2+bx+c=0$, if anything?

``Let $x$ and $y$ be solutions to the equation. Subtracting $ay^2+by+c=0$ from $ax^2+bx+c=0$ yields $a(x+y)(x-y)+b(x-y)=0$. Hence, $a(x+y)+b=0$, and so $x+y=-\frac{b}{a}$. Since $x$ and $y$ were {\it any} solutions, we may repeat this computation with $x=y$. Thus, $2x=-\frac{b}{a}$, and therefore $x=-\frac{b}{2a}$ is a solution.''

Again, be as {\it specific} as possible about the error(s).\\
{\bf [4 pts]}
\end{enumerate}
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\noindent\large{\bf Apply}:\normalsize
\begin{enumerate}[resume]
\item {\bf Rooting Out Roots:} Solve $\sqrt{x-1}=x-3$ for all real values of $x$ that satisfy the equation. Explain your work, and try to indicate why your answer/s is/are the {\it only} answer/s.\\
{\bf [3 pts]}
\item {\bf The First Rule Of Logic Club Is $\dots$:} To join Logic Club, you must decide to {\it always} tell the truth or {\it always} lie. Members of Logic Club know who lies and who soothsays. I do not belong to Logic Club, but I encounter three members on the street who make the following statements:
\begin{itemize}
\item Jack: ``All three of us are liars.''
\item Tyler: ``Exactly two of us are liars.''
\item Chuck: ``Jack and Tyler are liars.''
\end{itemize}
Who should I believe, if anyone?\\
{\bf[3 pts]}
\item {\bf Road Trip:} Let's say we take a class field trip to UCLA to visit renowned mathematician {\color{blue}\link{Terence Tao}{http://en.wikipedia.org/wiki/Terence\_Tao}}. In a feat of super-human attention, I drive the bus the entire way there, averaging 90 kilometers per hour. We spend a few days in the sun and, again, I drive all the way back, this time averaging 100 kilometers per hour because we're late for class. What was our average speed for the entire trip, in terms of kilometers per hour?\\
{\bf [3 pts]}
\item {\bf Triominoes on a Chessboard:} Consider a standard $8\times 8$ chessboard, with two adjacent squares removed from each corner. (See picture.) Can you tile this board with T-shaped triominoes? (See picture.) If so, how? If not, why not?\\
{\bf [4 pts]}
\begin{center} \includegraphics[scale=0.55]{triomino-chess.png} \end{center}
\end{enumerate}
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\noindent\large{\bf Explore}:\normalsize
\begin{enumerate}[resume]
\item {\bf Use the Fuse:} You have two strings of fuse. Each will burn for exactly one hour. The fuses are not necessarily identical, though, and do not burn at a constant rate. All you have with you is a lighter and these two fuses. Can you measure exactly 45 minutes? If so, explain how. If not, explain why.\\
\noindent{\bf[3 pts]}
\item {\bf Integral Solutions:} Consider the variable equation $6x+15y=93$. We want to find some {\it integral} solutions; that is, we want to find values of $x$ and $y$ that are both {\it integers} (natural numbers, zero, and negative natural numbers) that satisfy the equation.
\begin{enumerate}
\item Find a solution where both $x$ and $y$ are positive integers. Decribe, with a few sentences, how you came up with this solution.\\
{\bf [2 pt]}
\item Find a solution where one of the values, $x$ or $y$, is positive and the other is negative. Again, describe how you came up with this solution.\\
{\bf [2 pts]}
\item How many solutions do you think there are? Try to write down a characterization of all possible solutions, or describe how you might find them all.\\
{\bf [2 pts]}
\end{enumerate}
\item {\bf Not The Loneliest Number:} How many whole numbers less than or equal to 1000 have a 1 as at least one of their digits? For instance, we want to count 1 and 12 and 511 once each.\\
{\bf [3 pts]}
\item {\bf The Cutest Infestation Problem Ever:} We have several piles of koala bears. In an attempt to disperse them, we remove one koala bear from each pile and place them into a new pile. For example, if we started with koala piles of sizes 1, 4, and 4, we would then end up with koala piles of sizes 3, 3, and 3; or, if we started with 3 and 4, we would end up with 2 and 2 and 3.

Is it possible that we do this operation and end up with the exact same pile sizes as we started with? Identify all the collections of piles that have this property and explain why they are the only ones.\\
{\bf [4 pts]}
\item Read pages 1-7 in {\it A Mathematician's Lament} by Paul Lockhart (available on the website).\\
{\bf [0 pts]} but highly recommended reading!
\end{enumerate}
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