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116
Advanced/Retrograde Analysis/main.tex
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@ -13,6 +13,47 @@
\usepackage{afterpage}
\usepackage[hang]{footmisc}
% A row of stars, for showing difficulty
% first arg: filled stars
% second arg: total stars
\def\liststars#1#2{%
\def\oncolor{\color{ogrape}}%
\def\offcolor{\color{white}}%
%
\count255=1%
\count254=#2%
\advance\count254 by -1%
\ifnum #1 > 0
% The $$ are required around \bigstar.
% the special \odot chess piece breaks
% star sizing if they are ommited.
\loop
{\oncolor $\bigstar$}%
\ifnum\count255 < #1
\advance\count255 by 1
\repeat%
\else%
{\oncolor $\bigstar$}%
\fi%
%
\ifnum \count255 < #2%
\loop
{\offcolor $\bigstar$}%
\ifnum\count255 < \count254
\advance\count255 by 1
\repeat%
\fi%
}
\def\startimes#1{
{\color{ogrape} $\bigstar \times #1$}
}
\def\starvalue#1{
\stars{#1}{#1} \par
\vspace{1mm}
}
\def\difficulty#1#2{
\textbf{Difficulty:} \stars{#1}{#2} \par
@ -60,8 +101,20 @@ $\odot$}}
\chessboard[setpieces = {#1}]
\fi
}
\makeatother
\@makeORMCbox{hintlist}
{Available Hints}
{black!12!white}
{black!80!white}
\long\def\hintcontent#1#2#3{
\if@solutions
\liststars{#1}{#2} \par #3
\else
\liststars{#1}{#2}
\fi\par
}
\makeatother
@ -83,18 +136,71 @@ $\odot$}}
}
\section{Rules}
This handout contains a set of problems. \par
Each is marked with a few gold stars, which you collect by presenting a solution.
\vspace{2mm}
Below each problem is a list of hints, written next to their cost. \par
To get a hint, you must pay the marked number of stars. \par
Hints are usually provided in order, and are the same for all groups. \par
\vspace{2mm}
There is one exception: instructors may offer certain hints for a reduced fee,
judging by your explanation of what you have so far.
\begin{instructornote}
This exception is for groups that are stuck on the last bit of a problem. \par
If they've solved most of it, give them the information they already know
for a steep discount (or even for free).
\end{instructornote}
An example problem is below.
\vspace{4mm}
\generic{Problem $0$:}
\starvalue{4}
This is a problem that's worth four points. \par
Two hints are available. The first costs two stars (of the four available). \par
The second costs one of the two stars available \textit{after} revealing the first hint.
\begin{hintlist}
Hint 1:
\hintcontent{2}{4}{This is a hint. Only instructors see this.}
Hint 2:
\hintcontent{1}{2}{This is another hint.}
\end{hintlist}
\vfill
You may notice that hints often cost a significant fraction of their problem's score. \par
This is intentional---don't ask for hints unless you're very, very stuck.
\vfill
\pagebreak
\input{parts/intro}
\section{Simple problems}
\input{parts/easy}
\section{Harder problems}
\section{Slightly harder problems}
\input{parts/medium}
\section{Very difficult problems}
\input{parts/hard}
%\section{Very difficult problems}
%\input{parts/hard}
\chessboard \hfill \chessboard \par
\chessboard \hfill \chessboard \par

154
Advanced/Retrograde Analysis/parts/easy.tex
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@ -1,6 +1,7 @@
% Sherlock, A little exercise
\problem{A little exercise}
\difficulty{1}{5}
%\difficulty{1}{5}
\starvalue{3}
Black has just moved in the game below. White started on the south side of the board.\par
What was Black's last move, and what was White's last move? \par
@ -11,6 +12,14 @@ What was Black's last move, and what was White's last move? \par
Bg1
}
\begin{hintlist}
Hint 1:
\hintcontent{1}{3}{What was Black's last move? What White do to make this happen?}
Hint 2:
\hintcontent{1}{2}{White uncovered a check. What piece did Black capture?}
\end{hintlist}
\begin{solution}
It's pretty clear that Black just moved out of check from A7.
@ -37,9 +46,15 @@ What was Black's last move, and what was White's last move? \par
% Sherlock, Which color?
\problem{Which color?}
\difficulty{2}{5}
%\difficulty{2}{5}
\starvalue{4}
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.
There is a pawn at G3. What color is it? \par
\hint{Again, White started on the bottom.}
@ -51,6 +66,16 @@ There is a pawn at G3. What color is it? \par
Pd2,Pf2
}
\begin{hintlist}
Hint 1:
\hintcontent{2}{4}{How did the white king get off E1? It must have castled!}
Hint 2:
\hintcontent{1}{2}{
It castled kingside (how do we know?). Now, how did it get off G1?
}
\end{hintlist}
\begin{solution}
The white king is the key to this solution. How did it get off of E1? \par
It must have castled kingside---castling queenside would move a rook from black to white.
@ -67,13 +92,23 @@ There is a pawn at G3. What color is it? \par
% Arabian Knights 2
\problem{Invisible, but not invincible}
\difficulty{2}{5}
%\difficulty{2}{5}
\starvalue{4}
Seeing that this battle was lost, the black king has turned himself invisible. \par
Unfortunately, his position is hopeless. Mate the king in one move. \par
\hint{You don't need to find the king, you only need a checkmate.} \par
The black king has turned himself invisible. Unfortunately, his position is hopeless. \par
Mate the king in one move. \par
\manyboards{
Ra8,rb8,Kf8,
@ -81,6 +116,14 @@ Unfortunately, his position is hopeless. Mate the king in one move. \par
Pa6,Rc6
}
\begin{hintlist}
Hint 1:
\hintcontent{1}{4}{Do you need to know \textit{exactly} where the black king is?}
Hint 2:
\hintcontent{1}{3}{Don't forget about promotion.}
\end{hintlist}
\begin{solution}
Since it is White's move, Black cannot be in check. \par
So, either White is in check or the black king is on C8. \par
@ -97,9 +140,14 @@ Unfortunately, his position is hopeless. Mate the king in one move. \par
\pagebreak
% Sherlock, a question of survival
\problem{An empty board}
\difficulty{2}{5}
%\difficulty{2}{5}
\starvalue{4}
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.
There is one more piece on the board, which isn't shown. What color square does it stand on? \par
@ -109,6 +157,11 @@ There is one more piece on the board, which isn't shown. What color square does
Ke1
}
\begin{hintlist}
Hint 1:
\hintcontent{1}{4}{Which piece performed the last capture?}
\end{hintlist}
\begin{solution}
Which piece performed the last capture on a black square? It couldn't have been a white pawn, which haven't moved.
@ -126,7 +179,8 @@ There is one more piece on the board, which isn't shown. What color square does
% Sherlock, another monochromatic
\problem{The knight's grave}
\difficulty{3}{5}
%\difficulty{3}{5}
\starvalue{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
@ -137,6 +191,20 @@ Use this information to show that a pawn was promoted. \par
Ke1
}
\begin{hintlist}
Hint 1:
\hintcontent{1}{5}{
\par
Who took the knights? Only one of them is interesting---most are easy to account for.
\vspace{2mm}
}
Hint 2:
\hintcontent{2}{4}{We can easily account for the white knights and the
black knight on G8. Which white pieces can move to capture the black knight on B8?}
\end{hintlist}
\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?
@ -156,9 +224,13 @@ Use this information to show that a pawn was promoted. \par
% Arabian Knights, intro (given with solution)
\problem{Promotion?}
\difficulty{2}{5}
%\difficulty{2}{5}
\starvalue{6}
It is White's move. Have there been any promotions this game? \par
@ -167,6 +239,14 @@ It is White's move. Have there been any promotions this game? \par
Bc1,Kd1,Rh1
}
\begin{hintlist}
Hint 1:
\hintcontent{3}{6}{
The black king must have moved from F1. (Why not G1?) \par
This would be impossible if something hadn't blocked check from the white rook.
}
\end{hintlist}
\begin{solution}
Since it is White's move, Black has just moved his king. Where did he move it from?
@ -192,9 +272,11 @@ It is White's move. Have there been any promotions this game? \par
% Sherlock Holmes, two bagatelles (1)
\problem{Whodunit}
\difficulty{2}{5}
%\difficulty{2}{5}
\starvalue{5}
It is Black's move. Can Black castle? \par
\hint{Remember the rules of chess: you may not castle if you've moved your rook.}
\manyboards{
ra8,bc8,ke8,rh8,
@ -205,6 +287,14 @@ It is Black's move. Can Black castle? \par
Bc1,Qd1,Ke1,Bf1
}
\begin{hintlist}
Hint 1:
\hintcontent{1}{5}{
Black captured a knight on his last move. \par
Why do we know this, and how did he do it?
}
\end{hintlist}
\begin{solution}
White's last move was with the pawn. \par
Black's last move must have been to capture the white piece which moved before that.
@ -232,10 +322,12 @@ It is Black's move. Can Black castle? \par
% Sherlock Holmes, two bagatelles (2)
\problem{Castle contradiction}
\difficulty{2}{5}
%\difficulty{2}{5}
\startimes{7}
Neither Black nor White captured a piece on their last move. \par
It is Black's move. Can he castle? \par
\hint{What was White's last move? Check the cases.}
\manyboards{
ke8,rh8,
@ -245,6 +337,12 @@ It is Black's move. Can he castle? \par
bd1,Rf1,Kg1
}
\begin{hintlist}
Hint 1:
\hintcontent{2}{7}{If White moved his king, Black cannot castle. Why? \par
That's the simple case. The other option: White castled. What did Black do before that?}
\end{hintlist}
\begin{solution}
If White's last move was with the king, then the black rook moved to check him and Black can't castle.
@ -270,10 +368,12 @@ It is Black's move. Can he castle? \par
% Arabian Knights, intro (given with solution)
\problem{A matter of order}
\difficulty{3}{5}
%\difficulty{3}{5}
\startimes{8}
A black bishop captured a White piece earlier in this game. \par
Which bishop was it, and what did it capture? \par
\hint{Black and White start with 16 pieces each.}
\manyboards{
ra8,qd8,ke8,
@ -286,16 +386,34 @@ Which bishop was it, and what did it capture? \par
Kc1,Rd1,Rh1
}
\begin{instructornote}
\textbf{Hints to give:} (these should be answered in order)
\begin{hintlist}
Hint 1:
\hintcontent{1}{8}{
\begin{itemize}
\item How many pieces does Black have? Where were the missing ones captured?
\item Which pieces is White missing? Where could they have been captured?
\item How did those white pieces get to the place they were captured?
\item How many pieces are missing? Where were the missing ones captured?
\item How did those pieces get to the place they were captured?
\end{itemize}
\vspace{2mm}
}
\end{instructornote}
Hint 2:
\hintcontent{2}{7}{
\begin{itemize}
\item The pawn on C3 came from D2, capturing a black rook.
\item The black rook it captured couldn't have moved there before...
\end{itemize}
\vspace{2mm}
}
Hint 3:
\hintcontent{1}{4}{
\begin{itemize}
\item ...the black pawn on G7 captured a white piece on H6.
\item What else is missing?
\end{itemize}
}
\end{hintlist}
\begin{solution}
First, notice that the pawn on C3 came from D2 by capturing a piece. \par

13
Advanced/Retrograde Analysis/parts/intro.tex
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@ -136,5 +136,18 @@ A king and rook can \textit{castle} under the following conditions:
\end{minipage}
\par
\vfill
\generic{General Strategies:}
Below is a list of questions you'll want to ask yourself if you get stuck. \par
\note[Note]{Yes, I know you don't have the problems yet. Patience.}
\begin{itemize}
\item What was the last move? (By black or white)
\item Were there any promotions? This may be indirect:\par
if White needs a third bishop, there was a promotion.\par
\note{Promotions often resolve seemingly impossible situations.}
\end{itemize}
\vfill
\pagebreak

92
Advanced/Retrograde Analysis/parts/medium.tex
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@ -1,7 +1,8 @@
% Sherlock, A matter of direction
\problem{A matter of direction}
\difficulty{3}{5}
%\difficulty{3}{5}
\starvalue{5}
The results of a game of chess are shown below. \par
Did White start on the north or south side of the board? \par
@ -14,6 +15,11 @@ Did White start on the north or south side of the board? \par
Bh1
}
\begin{hintlist}
Hint 1:
\hintcontent{2}{5}{The bishop on H1 is important. How did White deliver this check?}
\end{hintlist}
\begin{solution}
Let us first find White's last move. It wasn't with the pawns on D4 and E5, since Black wouldn't have a move before that.
(Note the double-check on A7).
@ -29,11 +35,15 @@ Did White start on the north or south side of the board? \par
% Arabian Knights 1
\problem{Where is the king?}
\difficulty{3}{5}
%\difficulty{3}{5}
\startimes{8}
The white king has turned himself invisible. Find him. \par
The white king has again become invisible. Find him. \par
\hint{White started on the bottom. En passant.} \par
\manyboards{
@ -42,6 +52,30 @@ The white king has turned himself invisible. Find him. \par
kd1
}
\begin{hintlist}
Hint 1:
\hintcontent{3}{8}{
Either the white king is on B3, or Black is in check. \par
First, show that the latter implies the former.
\vspace{2mm}
}
Hint 2:
\hintcontent{2}{5}{
Moving back in time, you'll need to add two pieces to the board (not counting the king). \par
They have been captured!
\vspace{2mm}
}
\end{hintlist}
\makeatletter
\if@solutions
\vfill
\pagebreak
\fi
\makeatother
\begin{solution}
Looking at the board, we see that the white king is on B3 or Black is in check.
@ -86,9 +120,22 @@ The white king has turned himself invisible. Find him. \par
\pagebreak
% Arabian Knights, intro (given with solution)
\problem{Double-checks}
\difficulty{3}{5}
%\difficulty{3}{5}
\startimes{10}
White to move. Which side of the board did each color start on? \par
\hint{What was Black's last move? }
@ -99,12 +146,14 @@ White to move. Which side of the board did each color start on? \par
Nd1,kf1,Kh1
}
\begin{instructornote}
\textbf{Hints to give:}\par
Clearly, Black just moved his king. From where? \par
All possible positions may seem impossible (thanks to double-checks), but E1 and F2 are a bit more reasonable than others.
Move the king to both and try to add (or un-promote) pieces to make the position make sense.
\end{instructornote}
\begin{hintlist}
Hint 1:
\hintcontent{4}{10}{
Double-checks make all positions seem impossible... \par
Try E1 and F2 anyway. Can you add pieces to make it make sense? \par
Don't forget about promotion.
}
\end{hintlist}
\begin{solution}
Black's last move was from F2, where his king was in double-check from both a rook and a knight.
@ -133,6 +182,17 @@ White to move. Which side of the board did each color start on? \par
% Arabian Knights 4
\problem{A missing piece}
\difficulty{4}{5}
@ -151,6 +211,16 @@ What is it, and what is its color? \par
Kc1,Rd1,nf1,Bh1
}
\begin{hintlist}
Hint 1:
\hintcontent{4}{10}{
Double-checks make all positions seem impossible... \par
Try E1 and F2 anyway. Can you add pieces to make it make sense? \par
Don't forget about promotion.
}
\end{hintlist}
\begin{instructornote}
\textbf{Hints to give:} (in this order)
\begin{itemize}
@ -173,7 +243,7 @@ What is it, and what is its color? \par
\begin{solution}
\textbf{Part 1:}
The black bishop on A2 cannot be original, since the white pawn on B3 would have prevented its getting there.
The black bishop on A2 cannot be original, since the white pawn on B3 would have prevented it from getting there.
That bishop is a promoted bishop. \par
The black pawn it was promoted from must have come from E7,