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}
-Prove that if $\alpha$ is \emph{rational}, then there are only \emph{finitely} many rational numbers $\frac{p}{q}$
+\problem{Challenge VI}
+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>
-Prove that $bc-ad=1$ for $\frac{a}{b}$ and $\frac{c}{d}$ consecutive rational numbers in Farey sequence of order $n$.
+\problem{Challenge VII}<farey>
+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/0 intro.tex

@@ -21,7 +21,7 @@ Unlock this lock with only 5 keypresses.
 \end{solution}
 \vfill
 
-Now, consider the same lock, now set with a three-digit binary code.
+Now consider the same lock, but configured with a three-digit binary code.
 \problem{}
 How many codes are possible?
 \vfill

src/Advanced/De Bruijn/parts/1 words.tex

@@ -20,7 +20,11 @@ We say $v$ is a \textit{subword} of $w$ if $v$ is contained in $w$. \par
 For example, \texttt{11} is a subword of \texttt{011}, but \texttt{00} is not.
 
 \definition{}
-Recall \ref{lockproblem}. Let's generalize this to the \textit{$n$-subword problem}: \par
+Recall the lock problem from the previous page.
+Let's generalize this to the \textit{$n$-subword problem}:
+
+\vspace{1mm}
+
 Given an alphabet $A$ and a positive integer $n$,
 we want a word over $A$ that contains all possible length-$n$ subwords.
 The shortest word that solves a given $n$-subword problem is called the \textit{optimal solution}.
@@ -67,7 +71,7 @@ Find the following:
 
 \problem{}<sbounds>
 Let $w$ be a word over an alphabet of size $k$. \par
-Prove the following:
+Show that all of the following are true:
 \begin{itemize}
 	\item $\mathcal{S}_n(w) \leq k^n$
 	\item $\mathcal{S}_n(w) \geq \mathcal{S}_{n-1}(w) - 1$
@@ -103,7 +107,7 @@ Prove the following:
 
 \definition{}
 Let $v$ and $w$ be words over the same alphabet. \par
-The word $vw$ is the word formed by writing $v$ after $w$. \par
+The word $vw$ is the word formed by writing $w$ after $v$. \par
 For example, if $v = \texttt{1001}$ and $w = \texttt{10}$, $vw$ is $\texttt{100110}$.
 
 \problem{}
@@ -116,7 +120,6 @@ We'll call this the \textit{Fibonacci word} of order $k$.
 	\item What are $F_3$, $F_4$, and $F_5$?
 	\item Compute $\mathcal{S}_0$ through $\mathcal{S}_5$ for $F_5$.
 	\item Show that the length of $F_k$ is the $(k + 2)^\text{th}$ Fibonacci number. \par
-	\hint{Induction.}
 \end{itemize}
 
 \begin{solution}

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

@@ -127,10 +127,6 @@ Mate the king in one move. \par
 \pagebreak
 
 
-
-
-
-
 % Sherlock, a question of survival
 \problem{An empty board}
 \difficulty{2}{5}
@@ -161,42 +157,6 @@ There is one more piece on the board, which isn't shown. What color square does
 \pagebreak
 
 
-% Sherlock, another monochromatic
-\problem{The knight's grave}
-\difficulty{3}{5}
-
-In the game below, no pieces have moved from a black square to a white square, or from a white square to a black square.
-The white king has made less than fourteen moves. \par
-Use this information to show that a pawn was promoted. \par
-
-% spell:off
-\manyboards{
-	ke8,
-	Pb2,Pd2,
-	Ke1
-}
-% spell:on
-
-\begin{solution}
-	Knights always move to a different colored square, so all four missing knights must have been captured on their home square.
-	What pieces captured them?
-
-	\vspace{2mm}
-
-	We can easily account for the white knights and the black knight on G8, but who could've captured the knight from B8?
-	The only white pieces that can move to black squares are pawns, the Bishop (which is trapped on C1), the rook (which is stuck on column A and row 1), or the king (which would need at least 14 moves to do so).
-
-	\vspace{2mm}
-
-	If this knight was captured by a pawn, that pawn would be immediately promoted. If it was captured by a piece that wasn't a pawn, that piece must be a promoted pawn.
-\end{solution}
-
-\vfill
-\pagebreak
-
-
-
-
 
 
 % Arabian Knights, intro (given with solution)
@@ -292,7 +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{
@@ -372,4 +332,122 @@ Which bishop was it, and what did it capture? \par
 \end{solution}
 
 \vfill
-\pagebreak
+\pagebreak
+
+
+% Sherlock, appendix
+\problem{Moriarty's first}
+\difficulty{3}{5}
+
+
+No captures have been made in the last four moves. \par
+It is White's move. What was the previous move?
+
+% spell:off
+\manyboards{
+	Bc8,
+	pg6,
+	Pg5,kh5,
+	Pd4,qg4,bh4,
+	pd3,
+	Pd2,Be2,Bg2,
+	Nc1,rd1,Ne1,Kf1,Qg1,Rh1
+}
+% spell:on
+
+\begin{solution}
+	To see what the position was four moves ago,
+	move the Black queen to E4, the knight on E1 to F3,
+	the Black bishop to E1, and the White bishop on C8 to H3.
+
+	The following sequence of moves brought the game to the present position:
+
+	\begin{itemize}
+		\item bishop to c8, check
+		\item bishop to h4, check
+		\item knight to e1, check
+		\item queen to g4.
+	\end{itemize}
+
+	This is the only way the present position could have arisen,
+	so Black's last move was with the queen from E4 to G4.
+
+
+	Try any other last move, and you will find it impossible to play back three more moves.
+\end{solution}
+
+\vfill
+\pagebreak
+
+
+% Sherlock, appendix
+\problem{Moriarty's second}
+\difficulty{3}{5}
+
+
+Neither the White king nor queen has moved
+during the last five moves, nor has any piece
+been captured during that time.
+What was the last move?
+
+% spell:off
+\manyboards{
+	kh8,
+	Kg6,Bh6,
+	pa4,
+	Qa2
+}
+% spell:on
+
+\begin{solution}
+	Put the Black pawn on A7, the Black king on G8, remove the
+	White bishop, and put a White pawn on d5; this was the position
+	five moves ago. The following sequence of moves brought the
+	game to its present position:
+
+	\begin{itemize}
+		\item White: P-d6
+		\item Black: K-h8
+		\item White: P-d7
+		\item Black: P-a6
+		\item White: P-d8 = B
+		\item Black: P-a5
+		\item White: B-g5
+		\item Black: P-a4
+		\item White: B-h6
+	\end{itemize}
+\end{solution}
+
+\vfill
+\pagebreak
+
+
+% Sherlock, appendix
+\problem{Moriarty's third}
+\difficulty{3}{5}
+
+
+No pawn has moved, nor has any piece been
+captured in the last five moves. \par
+The Black king has been accidentally
+knocked off the board. \par
+On what square should he stand?
+
+% spell:off
+\manyboards{
+	rh8,
+	pa7,pb7,pc7,pd7,pe7,Kf7,pg7,Ph7,
+	Pg6,
+	na2
+}
+% spell:on
+
+\begin{solution}
+	The only way to avoid a retrograde stalemate for White is by
+	placing the Black king on C8. Black's last move was with
+	the rook from D8, White's move before that was with his
+	king from G8, and Black's move before that was to castle.
+\end{solution}
+
+\vfill
+\pagebreak

src/Advanced/Retrograde Analysis/parts/03 medium.tex

@@ -169,16 +169,156 @@ White to move. Which side of the board did each color start on? \par
 
 
 
+% Sherlock, another monochromatic
+\problem{Monochromatic}
+\difficulty{4}{5}
+
+In the game below, no pieces have moved from a black square to a white square or from a white square to a black square.
+The white king has made fewer than fourteen moves. \par
+Use this information to show that a pawn was promoted. \par
+
+% spell:off
+\manyboards{
+	ke8,
+	Pb2,Pd2,
+	Ke1
+}
+% spell:on
+
+\begin{solution}
+	Knights always move to a different colored square, so all four missing knights must have been captured on their home square.
+	What pieces captured them?
+
+	\vspace{2mm}
+
+	We can easily account for the white knights and the black knight on G8, but who could've captured the knight from B8?
+	The only white pieces that can move to black squares are pawns, the Bishop (which is trapped on C1), the rook (which is stuck on column A and row 1), or the king (which would need at least 14 moves to do so).
+
+	\vspace{2mm}
+
+	If this knight was captured by a pawn, that pawn would be immediately promoted. If it was captured by a piece that wasn't a pawn, that piece must be a promoted pawn.
+\end{solution}
+
+\vfill
+\pagebreak
 
 
 
 
+% Sherlock, another question of location
+\problem{Superposition}
+\difficulty{4}{5}
+
+A white pawn is missing; it is either on F2 or G2. \par
+Where is it?
+
+% spell:off
+\manyboards{
+	ke8,rh8,
+	pa7,pf7,pg7,
+	pa6,pb6,
+	pb5,
+	Pa4,Pb4,Pc4,
+	pa3,
+	Pa2,Pb2,
+	Ke1
+}
+% spell:on
+
+\vfill
+\pagebreak
+
+% Sherlock, another question of location
+\problem{Possibility}
+\difficulty{4}{5}
+
+Show that black can castle to either side. \par
+We know the following:
+
+\begin{itemize}
+	\item White started the game missing one rook.
+	\item White has not moved either knight
+	\item No promotions have been made
+	\item White's last move was from E2 to E4.
+\end{itemize}
+
+% spell:off
+\manyboards{
+	ra8,ke8,rh8,
+	pa7,bb7,pc7,pd7,pf7,pg7,ph7,
+	nc6,nh6,
+	pe5,qg5,
+	bb4,Pe4,
+	Pb2,Pc2,Pd2,Pf2,Pg2,Ph2,
+	Nb1,Bc1,Qd1,Ke1,Bf1,Ng1,Rh1
+}
+% spell:on
+
+
+\vfill
+\pagebreak
+
+
+
+
+
+% Sherlock, little exercise 2
+\problem{Kidnapping}
+\difficulty{4}{5}
+
+On which square was the White queen captured?. \par
+
+% spell:off
+\manyboards{
+	ra8,qd8,ke8,ng8,rh8,
+	pa7,pb7,pc7,pe7,pf7,ph7,
+	nc6,pe6,ph6,
+	Pb3,
+	Na2,Pb2,Pc2,Pd2,Pe2,Pf2,Pg2,Ph2,
+	Ra1,Ke1,Rh1
+}
+% spell:on
+
+\begin{solution}
+	White is missing a queen, both bishops, and one knight. \par
+	The black pawns on E6 and H6 account for two captures.
+
+	\vspace{2mm}
+
+	Neither white bishop could've been captured by these pawns,
+	since both are trapped by their pawns. Thus, these black pawns must have captured a queen and a knight.
+
+	\vspace{4mm}
+
+	The white pawn on B3 must have captured a black bishop. \par
+	The white queen got onto the board through A2. \par
+	Therefore, the pawn on B3 made its capture before the queen escaped,
+	and the black bishop was captured before the white queen.
+
+	\vspace{4mm}
+
+	Similarly, the bishop from C8 must have been
+	captured on B3 after the capture on E6, since it
+	got on the board through D7.
+
+
+	\vspace{4mm}
+
+	The capture on E6 was made before the capture on B3 (black bishop),
+	which was made before the white queen was captured.
+	Therefore, the white queen was not captured on E6, and must
+	have been lost on H6.
+\end{solution}
+
+\vfill
+\pagebreak
+
 
 
 
 % Arabian Knights 4
 \problem{A missing piece}
-\difficulty{4}{5}
+\difficulty{6}{8}
 
 
 There is a piece at G4, marked with a $\odot$. \par

src/Advanced/Retrograde Analysis/parts/04 hard.tex

@@ -2,7 +2,7 @@
 
 % Arabian Knights 5
 \problem{The hidden castle}
-\difficulty{7}{7}
+\difficulty{8}{8}
 
 There is a white castle hidden on this board. Where is it? \par
 None of the royalty has moved or been under attack. \par
@@ -30,7 +30,7 @@ None of the royalty has moved or been under attack. \par
 
 % Arabian Knights 6
 \problem{Who moved last?}
-\difficulty{7}{7}
+\difficulty{8}{8}
 
 After many moves of chess, the board looks as follows. \par
 Who moved last? \par
@@ -58,7 +58,7 @@ Who moved last? \par
 
 % Arabian Knights 3
 \problem{The king in disguise}<kingdisguise>
-\difficulty{7}{7}
+\difficulty{8}{8}
 
 The white king is exploring his kingdom under a disguise. He could look like any piece of any color.\par
 Show that he must be on C7.
@@ -119,7 +119,7 @@ Show that he must be on C7.
 
 % Arabian Knights 3
 \problem{The king in disguise once more}
-\difficulty{5}{7}
+\difficulty{5}{8}
 
 The white king is again exploring his kingdom, now under a different disguise. Where is he? \par
 \hint{\say{different disguise} implies that the white king looks like a different piece!}

src/Warm-Ups/Painting/main.typ

@@ -2,9 +2,8 @@
 #import "@preview/cetz:0.4.2"
 
 #show: handout.with(
-  title: [Warm-Up: What's an AST?],
+  title: [Warm-Up: The Painting],
   by: "Mark",
-  subtitle: "Based on a true story.",
 )
 
 #problem()
@@ -13,7 +12,8 @@ Hang the painting on two nails so that if either is removed, the painting falls.
 
 #v(2mm)
 You may detach the string as you hang the painting, but it must be re-attached once you're done. \
-#hint[The solution to this problem isn't a "think outside the box" trick, it's a clever wrapping of the string.]
+
+The solution to this problem isn't a trick, it's a clever wrapping of the string.
 
 
 #v(2mm)