Continued Fractions edits

src/Advanced/Continued Fractions/parts/01 part A.tex

@@ -78,7 +78,7 @@ An \textit{infinite continued fraction} is an expression of the form
 a_0 + \cfrac{1}{a_1+\cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4 + ...}}}}
 \]
 where $a_0, a_1, a_2, ...$ are in $\mathbb{Z}^+_0$.
-To prove that this expression actually makes sense and equals a finite number
+Showing that this expression converges to a finite number
 is beyond the scope of this worksheet, so we assume it for now.
 This is denoted $[a_0, a_1, a_2, ...]$.
 
@@ -133,7 +133,7 @@ A few examples are below. We denote the repeating sequence with a line.
 \problem{}
 \begin{itemize}
 	\item Show that $\sqrt{2} = [1, \overline{2}]$.
-	\item Show that $\sqrt{5} = [1, \overline{4}]$.
+	\item Show that $\sqrt{5} = [2, \overline{4}]$.
 \end{itemize}
 \hint{use the same strategy as \ref{irrational} but without a calculator.}
 
@@ -159,7 +159,7 @@ Express the following continued fractions in the form $\frac{a+\sqrt{b}}{c}$ whe
 
 \problem{Challenge II}
 Let $\alpha = [~a_0,~ ...,~ a_r,~ \overline{a_{r+1},~ ...,~ a_{r+p}}~]$ be any periodic continued fraction. \par
-Prove that $\alpha$ is of the form $\frac{a+\sqrt{b}}{c}$ for some integers $a,b,c$ where $b$ is not a perfect square.
+Show that $\alpha$ is of the form $\frac{a+\sqrt{b}}{c}$ for some integers $a,b,c$ where $b$ is not a perfect square.
 
 
 
@@ -168,7 +168,7 @@ Prove that $\alpha$ is of the form $\frac{a+\sqrt{b}}{c}$ for some integers $a,b
 
 
 \problem{Challenge III}
-Prove that any number of the form $\frac{a+\sqrt{b}}{c}$ where $a,b,c$ are integers
+Show that any number of the form $\frac{a+\sqrt{b}}{c}$ where $a,b,c$ are integers
 and $b$ is not a perfect square can be written as a periodic continued fraction.
 
 

src/Advanced/Continued Fractions/parts/02 part B.tex

@@ -65,7 +65,7 @@ Verify the recursive formula for $1\leq j\leq 3$ for the convergents $C_j$ of: \
 
 
 \problem{Challenge IV}<rec>
-Prove that $p_n = a_np_{n-1} + p_{n-2}$ and $q_n = a_nq_{n-1} + q_{n-2}$ by induction.
+Show that $p_n = a_np_{n-1} + p_{n-2}$ and $q_n = a_nq_{n-1} + q_{n-2}$ by induction.
 \begin{itemize}
 	\item As the base case, verify the recursive formulas for $n=1$ and $n=2$.
 	\item Assume the recursive formulas hold for $n\leq m$ and show the formulas hold for $m+1$.
@@ -97,7 +97,7 @@ we will show that $p_n q_{n-1} - p_{n-1}q_n = (-1)^{n-1}$.
 
 
 
-\problem{Challenge VI}
+\problem{Challenge V}
 Similarly derive the formula $p_nq_{n-2}-p_{n-2}q_n = (-1)^{n-2}a_n$.
 
 
@@ -141,7 +141,7 @@ We will show that $|\alpha-C_n|<\frac{1}{q_n^2}$.
 We are now ready to prove a fundamental result in the theory of rational approximation.
 \problem{Dirichlet's approximation theorem}
 Let $\alpha$ be any irrational number.
-Prove that there are infinitely many rational numbers $\frac{p}{q}$ such that $|\alpha - \frac{p}{q}| < \frac{1}{q^2}$.
+Show that there are infinitely many rational numbers $\frac{p}{q}$ such that $|\alpha - \frac{p}{q}| < \frac{1}{q^2}$.
 
 
 
@@ -154,8 +154,8 @@ Prove that there are infinitely many rational numbers $\frac{p}{q}$ such that $|
 
 
 
-\problem{Challenge VII}
+\problem{Challenge VI}
-Prove that if $\alpha$ is \emph{rational}, then there are only \emph{finitely} many rational numbers $\frac{p}{q}$
+Show that if $\alpha$ is \emph{rational}, then there are only \emph{finitely} many rational numbers $\frac{p}{q}$
 satisfying $|\alpha - \frac{p}{q} | < \frac{1}{q^2}$.
 
 
@@ -195,8 +195,8 @@ Let $\frac{a}{b}$ and $\frac{c}{d}$ be consecutive elements of the Farey sequenc
 
 
 
-\problem{Challenge VIII}<farey>
+\problem{Challenge VII}<farey>
-Prove that $bc-ad=1$ for $\frac{a}{b}$ and $\frac{c}{d}$ consecutive rational numbers in Farey sequence of order $n$.
+Show that $bc-ad=1$ for $\frac{a}{b}$ and $\frac{c}{d}$ consecutive rational numbers in Farey sequence of order $n$.
 
 \begin{itemize}[itemsep=2mm]
 	\item In the plane, draw the triangle with vertices (0,0), $(b,a)$, $(d,c)$.
@@ -237,7 +237,7 @@ $|\alpha - \frac ab| \geq |\alpha - \frac{p_n}{q_n}|$
 
 
 
-\problem{Challenge X}
+\problem{Challenge VIII}
 Prove the following strengthening of Dirichlet's approximation theorem.
 If $\alpha$ is irrational, then there are infinitely many rational numbers
 $\frac{p}{q}$ satisfying $|\alpha - \frac pq| < \frac{1}{2q^2}$.

src/Advanced/De Bruijn/parts/3 line.tex

@@ -49,7 +49,6 @@ Have an instructor check your solution.
 				(0) edge (2)
 				(2) edge (3)
 				(2) edge[bend left] (4)
-				(4) edge[bend left] (2)
 				(3) edge (1)
 				(4) edge (3)
 				(4) edge[loop right] (4)

src/Advanced/De Bruijn/parts/4 sturmian.tex

@@ -375,7 +375,7 @@ Construct a minimal Sturmain word of order 5.
 
 
 \problem{}
-Argue that the words we get by \ref{sturmanthm} are minimal Sturmain words. \par
+Show that the words we get by \ref{sturmanthm} are minimal Sturmain words. \par
 That is, the word $w$ has length $2n$ and $\mathcal{S}_m(w) = m + 1$ for all $m \leq n$.
 
 \begin{solution}

src/Advanced/Definable Sets/main.tex

@@ -2,7 +2,8 @@
 % use [solutions] flag to show solutions.
 \documentclass[
 	solutions,
-	singlenumbering
+	singlenumbering,
+	shortwarning
 ]{../../../lib/tex/handout}
 \usepackage{../../../lib/tex/macros}
 

src/Advanced/Definable Sets/parts/0 logic.tex

@@ -158,7 +158,7 @@ the same $A$ and $B$. \par
 \vfill
 
 \problem{}
-Write an expression equivalent to $A \lor B$ using only $\lnot$, $\rightarrow$, and $()$?
+Write an expression equivalent to $A \lor B$ using only $\lnot$, $\rightarrow$, and $()$.
 
 \begin{solution}
 	$((\lnot A) \rightarrow B)$

src/Advanced/Retrograde Analysis/parts/02 easy.tex

@@ -252,7 +252,7 @@ It is Black's move. Can Black castle? \par
 \difficulty{2}{5}
 
 Neither Black nor White captured a piece on their last move. \par
-It is Black's move. Can he castle? \par
+It is Black's move. Show that he cannot castle? \par
 \hint{What was White's last move? Check the cases.}
 
 \manyboards{
@@ -348,7 +348,7 @@ It is White's move. What was the previous move?
 	Bc8,
 	pg6,
 	Pg5,kh5,
-	Pd4,Qg4,Bh4,
+	Pd4,qg4,bh4,
 	pd3,
 	Pd2,Be2,Bg2,
 	Nc1,rd1,Ne1,Kf1,Qg1,Rh1