\documentclass[11pt]{amsart}
\usepackage{graphicx}
\usepackage{amssymb}


\textwidth = 6.5 in
\textheight = 8.3 in
\oddsidemargin = 0.0 in
\evensidemargin = 0.0 in
\topmargin = 0.0 in
\headheight = 0.0 in
%\headsep = 0.1 in
\parskip = 0.2in
\parindent = 0.0in

\newcommand{\pchoose}[2]{\begin{pmatrix}#1\\ #2\end{pmatrix}}
\newcommand{\bchoose}[2]{\begin{bmatrix}#1\\ #2\end{bmatrix}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%% Young diagrams, grace a Francois Bergeron

\newdimen\squaresize \squaresize=12pt
\newdimen\thickness \thickness=0.4pt


\def\square#1{\hbox{\vrule width \thickness
     \vbox to \squaresize{\hrule height \thickness\vss
        \hbox to \squaresize{\hss#1\hss}
     \vss\hrule height\thickness}
\unskip\vrule width \thickness}
\kern-\thickness}

\def\vsquare#1{\vbox{\square{$#1$}}\kern-\thickness}
\def\blk{\omit\hskip\squaresize}
\def\noir{\vrule height\squaresize width\squaresize}%
\def\noir{\gray{\vrule height\squaresize width\squaresize}}%
\def\blkk{\omit}

\def\young#1{
\vbox{\smallskip\offinterlineskip
\halign{&\vsquare{##}\cr #1}}}

\def\thisbox#1{\kern-.09ex\fbox{#1}}
\def\downbox#1{\lower1.200em\hbox{#1}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\title{Homework \#2}
\author{Assigned: January 15, 2020; Due: January 29, 2020
}
\begin{document}
\maketitle

Create a rectangle of width $m$ and of height $n$ of squares and the total
number of smaller squares in the rectangle is equal to $n\cdot m$.  For example
in the picture below, the grid is $7 \times 4$.

$$\young{&&&&&&\cr&&&&&&\cr&&&&&&\cr&&&&&&\cr}$$

The rectangle above also contains 18 $~~~2\times 2$ squares, 10 $~~~3 \times 3$ squares and 4
$~~~4 \times 4$ squares.\\\\

Question 1: For each $k$ that is between $1$ and the minimum of $m$ and $n$,
how many $k \times k$ squares are in the rectangle?\\

Question 2: What is the total number of squares that one can find in a $n \times m$ rectangle?\\\\

A full answer to this question will have a formula which one can calculate quickly given $n$ and $m$ (even for $n$ and $m$ very large).
\end{document}