Estoy intentando crear una cuadrícula de camino que tenga agujeros usando el triángulo de Pascal en látex. También la brújula

Question

La siguiente imagen da la idea de lo que estoy tratando de hacer.

Además, no estoy seguro de qué etiqueta usar para esta pregunta específica.

El código que estoy usando:

 \documentclass[28pt]{article}

 \usepackage{fancyhdr}

 \usepackage[includeheadfoot,margin=1.0cm]{geometry} 
 \usepackage{amsmath,amsthm,amssymb}
 \usepackage{enumitem}
 \usepackage{mathtools}
 \usepackage{framed}
 \usepackage{chessfss} %chess figure for HW #2
 \usepackage[english]{babel} %table for problem A.43
 \usepackage{multirow} %table for problem A.43
%\usepackage[table]{xcolor} color certain blocks in a table
%\usepackage[pass,showframe]{geometry}  just to show the margins
 \usepackage[makeroom]{cancel}
 \usepackage{array}  %BETWEEN TWO 2-DIGIT NUMBERS

 \newcommand{\N}{\mathbb{N}}
 \newcommand{\Z}{\mathbb{Z}}
 \newcommand{\thedate}{\today}

 \newtheoremstyle{case}{}{}{}{}{}{:}{ }{}
 \theoremstyle{case}
 \newtheorem{case}{Case}


 \newenvironment{theorem}[2][Theorem]{\begin{trivlist}
 \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
 \newenvironment{lemma}[2][Lemma]{\begin{trivlist}
 \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
 \newenvironment{exercise}[2][Exercise]{\begin{trivlist}
 \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
 \newenvironment{problem}[2][Problem]{\begin{trivlist}
 \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
 \newenvironment{question}[2][Question]{\begin{trivlist}
 \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
 \newenvironment{corollary}[2][Corollary]{\begin{trivlist}
 \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}

 \begin{document}

 \end{document}

Answer 1

Creo que esto lo hace:

\documentclass[tikz,border=5]{standalone}
\usetikzlibrary{math}
\begin{document}
\begin{tikzpicture}[x=2cm,y=-2cm, node 0/.style={fill=red!20}]
\tikzmath{%
  int \i, \j, \m, \n, \t;
  \m = 6; \n = 6;
  % Initialise board.
  for \i in {0,...,\m}{
    for \j in {0,...,\n}{
      \t{\i,\j} = 0;
    };
  };
  % Create holes.
  \t{0,3} = -1;
  \t{4,4} = -1;
  \t{5,1} = -1;
  \t{2,5} = -1;
  % Perform calculations.
  for \i1 in {0,...,\m}{
    for \j1 in {0,...,\n}{
      if (\t{\i1,\j1} == -1) then {
          \t{\i1,\j1} = 0;
      } else {
        if (\i1 == 0 || \j1 == 0) then  {
          \t{\i1,\j1} = 1;            
        } else {
          \i2 = \i1 - 1;
          \j2 = \j1 - 1;
          \t{\i1,\j1} = \t{\i2,\j1} + \t{\i1,\j2}; 
        };
      };
    };
  };
  % Draw nodes.
  for \i1 in {0,...,\m}{
    for \j1 in {0,...,\n}{
    { \node [circle, fill=blue!20, minimum size=1cm, node \t{\i1,\j1}/.try] 
        (n-\i1-\j1) at (\j1, \i1) {\t{\i1,\j1}}; };
    };   
  };
  % Draw edges.
  for \i1 in {0,...,\m}{
    for \j1 in {0,...,\n}{
      \i2 = \i1 + 1;
      \j2 = \j1 + 1;
      if (\i1 < \m) then {
        if (\t{\i2,\j1} > 0) then { 
              { \draw [thick, -stealth] (n-\i1-\j1) -- (n-\i2-\j1); };
          }; 
        };
        if (\j1 < \n) then {
        if (\t{\i1,\j2} > 0) then { 
              { \draw [thick, -stealth] (n-\i1-\j1) -- (n-\i1-\j2); };
        }; 
      };
    };
  };
}
\end{tikzpicture}
\end{document}

Sin 'agujeros' se puede hacer de forma mucho más sencilla:

\documentclass[tikz,border=5]{standalone}
\usepackage{xintexpr}
\begin{document}
\begin{tikzpicture}[x=2cm,y=2cm]
\draw [help lines] grid [step=1] (6,-6);
\foreach \x in {0,...,6}
  \foreach \y in {0,...,6}
    \node [circle, fill=blue!20, minimum size=1cm]
      at (\x, -\y) {\xinttheiiexpr binomial(\x+\y,\x)\relax};
\end{tikzpicture}
\end{document}