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    "# [Cuboid Route](https://projecteuler.net/problem=86)\n",
    "\n",
    "Suppose you have a cuboid with side lengths $a \\leq b \\leq M$. Then the shortest route will be $\\sqrt{(a + b)^2 + M^2}$. We're interested in when this distance is an integer.\n",
    "\n",
    "However, rather than iterate through values of $a$, $b$, and $M$, we can be more efficient by iterating through values of $M$, then values of $s$, where $s \\leq 2M$. If $s^2 + M^2$ is a square number, then that means any $a,b$ such that $s = a+b$ and $a \\leq b \\leq M$ will correspond to an $a \\times b \\times M$ cuboid with integer shortest route.\n",
    "\n",
    "So, if $s = a + b$, naturally $b = s - a$, and we want to know how many values of $a$ satisfy $1 \\leq a \\leq s - a \\leq M$. We can derive four bounds on $a$ from this.\n",
    "* $1 \\leq a$\n",
    "* $s - M \\leq a$\n",
    "* $a \\leq \\frac{s}{2}$\n",
    "* $a \\leq M$\n",
    "\n",
    "From these bounds, we can get the number of cuboids that can be constructed from an $(s, M)$ pair."
   ]
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    "def leg_splits(s, M):\n",
    "    max_a = min(M, s // 2 + 1)\n",
    "    min_a = max(s - M, 1)\n",
    "    return max_a - min_a"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "00e62dfc",
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   "source": [
    "Then we can write a function to find the number of cuboids with at least one edge equaling $M$."
   ]
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   "source": [
    "def count_cuboids(M):\n",
    "    return sum(leg_splits(s, M) for s in range(1, 2 * M + 1) if is_square(s^2 + M^2))"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "To get our answer, we just compute a running total and stop when it exceeds one million."
   ]
  },
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   "execution_count": 3,
   "id": "984b7665",
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    {
     "data": {
      "text/plain": [
       "1818"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from itertools import count\n",
    "\n",
    "total = 0\n",
    "for M in count(1):\n",
    "    total += count_cuboids(M)\n",
    "    if total > 1000000:\n",
    "        break\n",
    "        \n",
    "M"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "488ec2af",
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   "source": [
    "## Relevant sequences\n",
    "* Number of pairs $a,b$ such that $(a+b)^2 + n^2$ is square: [A143714](https://oeis.org/A143714)\n",
    "* Partial sums of A143714: [A143715](https://oeis.org/A143715)\n",
    "\n",
    "#### Copyright (C) 2025 filifa\n",
    "\n",
    "This work is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International license](https://creativecommons.org/licenses/by-sa/4.0/) and the [BSD Zero Clause license](https://spdx.org/licenses/0BSD.html)."
   ]
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