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   "source": [
    "If we take each intersection as a node and each direction as an edge, then we can see that this is just a [DAG](https://en.wikipedia.org/wiki/Directed_acyclic_graph).\n",
    "To count the possible paths from start to end we can observe than the number of paths of a node is the sum of the number of paths of its children. We do a [topological sort](https://en.wikipedia.org/wiki/Topological_sorting) of the graph and starting from the bottom right corner we go back calculating the number of paths. Since for each $v$ the paths of its children have already been calculated, we need to just sum them."
   ]
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    {
     "data": {
      "text/plain": [
       "137846528820"
      ]
     },
     "execution_count": 3,
     "metadata": {},
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    }
   ],
   "source": [
    "import networkx as nx\n",
    "\n",
    "\n",
    "def build_graph(n):\n",
    "    g = nx.DiGraph()\n",
    "\n",
    "    def i(col, row):\n",
    "        return n * (row - 1) + col\n",
    "\n",
    "    for c in range(1, n + 1):\n",
    "        for r in range(1, n + 1):\n",
    "            if c != n:\n",
    "                g.add_edge(i(c, r), i(c + 1, r))\n",
    "            if r != n:\n",
    "                g.add_edge(i(c, r), i(c, r + 1))\n",
    "    return g\n",
    "\n",
    "\n",
    "# Kahn's algorithm\n",
    "def topological_sort(g):\n",
    "    g = g.copy()\n",
    "    sorted_nodes = []\n",
    "    without_incoming_edges = [1]\n",
    "    while without_incoming_edges:\n",
    "        n = without_incoming_edges.pop()\n",
    "        sorted_nodes.append(n)\n",
    "        edges_to_remove = []\n",
    "        for s in g.successors(n):\n",
    "            edges_to_remove.append((n, s))\n",
    "            if g.in_degree(s) == 1:  # the edge n -> s will be removed, so we check for 1 instead of 0\n",
    "                without_incoming_edges.append(s)\n",
    "        for (n, s) in edges_to_remove:\n",
    "            g.remove_edge(n, s)\n",
    "    return sorted_nodes\n",
    "\n",
    "\n",
    "def count_paths(g, f, t):\n",
    "    sorted_nodes = topological_sort(g)\n",
    "    cache = {t: 1}\n",
    "    for n in reversed(sorted_nodes[:-1]):\n",
    "        count = 0\n",
    "        for s in g.successors(n):\n",
    "            count += cache[s]\n",
    "        cache[n] = count\n",
    "    return cache[f]\n",
    "\n",
    "\n",
    "nodes = 20 + 1\n",
    "g = build_graph(nodes)\n",
    "count_paths(g, 1, nodes * nodes)"
   ]
  }
 ],
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