Is there a way to have different behavior for even pages and odd pages

Question

I would like to modify the below template so that it prints out 1 problem per sheet. Example: Problem 1 on pages 1-3, problem 2 on 5-6 and nothing on page 4. Another Example: Problem 1 on Pages 1-2, problem 2 on 3-5, nothing on p6, and problem 3 on p7.

The template is

\documentclass[letterpaper,12pt,twoside]{article}

\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[letterpaper, top = 1.5in, left = 1in, right = 1in, bottom = 1.5in]  
\usepackage{fancyhdr}       
\usepackage{amsfonts,amsmath,amsthm,amssymb}
\usepackage{parskip}

\title{}
\author{Author}     
\date{\today}               

\makeatletter               
\let\Author\@author         
\let\Date\@date             
\makeatother

\newcommand{\Problem}{\relax}
\newcounter{problemcount}
\newcommand{\nextproblem}[1]{\renewcommand{\Problem}{#1}\setcounter{equation {0}\setcounter{page}{0}}

\pagestyle{fancy}
\lhead{\Problem}
\chead{}
\rhead{
\Author
\\
Class
\\
Section
\\
\Date
}

\renewcommand{\headrulewidth}{0.4pt}
\setlength{\headheight}{56.2pt}
\setlength{\parindent}{0pt}

\makeatother

\begin{document}
\nextproblem{Problem 1}
\begin{proof}\begin{align*}1+1&=2\\2&=2\end{align*}\end{proof}
\newpage

\nextproblem{Problem 2}\begin{proof}\begin{align*}1+1&=2\\2&=2\end{align*}\end{proof}

\end{document}

Answer 1

This fixes your example, cleans up the preamble a bit and implements the \cleardoublepage:

\documentclass[letterpaper,12pt,twoside]{article}

\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[top = 1.5in, left = 1in, right = 1in, bottom = 1.5in, headheight=56.2pt]{geometry}% tell geometry about the head height
\usepackage{fancyhdr}
\usepackage{amsfonts,amsmath,amsthm,amssymb}
\usepackage{parskip}% already sets parindent to zero - no need to do it yourself later as well
\usepackage{kantlipsum}

\title{}
\author{Author}
\date{\today}

\makeatletter
\let\Author\@author% consider using the titling package if you need commands like this
\let\Date\@date
\makeatother

\newcommand{\Problem}{\relax}
\newcounter{problemcount}
\newcommand{\nextproblem}[1]{%
  \cleardoublepage
  \renewcommand{\Problem}{#1}%
  \setcounter{equation}{0}}% if you want problem 2 on pages 5-6, better not reset the page count to zero!

\pagestyle{fancy}
\lhead{\Problem}
\chead{}
\rhead{
  \Author
  \\
  Class
  \\
  Section
  \\
  \Date
}

\makeatother

\begin{document}
  \nextproblem{Problem 1}
  \begin{proof}\begin{align*}1+1&=2\\2&=2\end{align*}\end{proof}
  \kant[1-8]

  \nextproblem{Problem 2}\begin{proof}\begin{align*}1+1&=2\\2&=2\end{align*}\end{proof}
  \kant[9-14]

\end{document}

Update

Spurred by Christopher Hupfer's suggestion, this is an attempt to clean up the code a bit more. Since it is not entirely clear what you are trying to do, this may not be exactly what you want. However, it should at least give you some ideas.

I use \newtheorem as Christopher Hupfer suggested. Rather than create an additional \nextproblem macro, I have used etoolbox to append commands to the start and end of the problem environment. The header then uses the problem number, which amsthm manages automatically. In this version, page 4 is entirely empty - without even header or footer. If you do not want this, remove \thispagestyle{empty}. I have also used titling to ensure that the values from \author etc. are available throughout, and adapted the header to use the macros provided by that package.

\documentclass[letterpaper,12pt,twoside]{article}

\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[top = 1.5in, left = 1in, right = 1in, bottom = 1.5in, headheight=56.2pt]{geometry}% tell geometry about the head height
\usepackage{fancyhdr}
\usepackage{amsfonts,amsmath,amsthm,amssymb}
\usepackage{parskip}% already sets parindent to zero - no need to do it yourself later as well
\usepackage{etoolbox,titling}
\usepackage{kantlipsum}

\title{}
\author{Author}
\date{\today}

\newtheorem{problem}{Problem}
\newcommand{\nextproblem}[1]{%
  \cleardoublepage\thispagestyle{fancy}%
  \setcounter{equation}{0}}% if you want problem 2 on pages 5-6, better not reset the page count to zero!

\AtBeginEnvironment{problem}{%
  \cleardoublepage
  \setcounter{equation}{0}}
\AtEndEnvironment{problem}{%
  \clearpage
  \thispagestyle{empty}}

\pagestyle{fancy}
\lhead{Problem \theproblem}
\chead{}
\rhead{
  \theauthor
  \\
  Class
  \\
  Section
  \\
  \thedate
}

\begin{document}
  \begin{problem}
    \begin{proof}\begin{align*}1+1&=2\\2&=2\end{align*}\end{proof}
    \kant[1-8]
  \end{problem}

  \begin{problem}
    \begin{proof}\begin{align*}1+1&=2\\2&=2\end{align*}\end{proof}
    \kant[9-14]
  \end{problem}
\end{document}