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   "source": [
    "# [Cube Digit Pairs](https://projecteuler.net/problem=90)\n",
    "\n",
    "There's only ${10 \\choose 6} = 210$ distinct cubes, making this problem easy to brute force. We first make a generator for those cubes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "06b0661f",
   "metadata": {},
   "outputs": [],
   "source": [
    "from itertools import combinations\n",
    "\n",
    "def cubes():\n",
    "    yield from combinations(range(0, 10), 6)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "87ba05f8",
   "metadata": {},
   "source": [
    "We also write a simple function for handling the fact that we can flip 6s and 9s."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "15dab5dd",
   "metadata": {},
   "outputs": [],
   "source": [
    "def extended_set(s):\n",
    "    s = set(s)\n",
    "    if 6 in s:\n",
    "        s.add(9)\n",
    "    elif 9 in s:\n",
    "        s.add(6)\n",
    "        \n",
    "    return s"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "42f875d0",
   "metadata": {},
   "source": [
    "Another simple function tests if we can display all the square numbers with a given pair of cubes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "7396b4e8",
   "metadata": {},
   "outputs": [],
   "source": [
    "def all_squares_displayable(s, t):\n",
    "    es = extended_set(s)\n",
    "    et = extended_set(t)\n",
    "    \n",
    "    square_digits = ((0, 1), (0, 4), (0, 9), (1, 6), (2, 5), (3, 6), (4, 9), (6, 4), (8, 1))\n",
    "    for (m, n) in square_digits:\n",
    "        if not ((m in es and n in et) or (n in es and m in et)):\n",
    "            return False\n",
    "    \n",
    "    return True"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9efa8055",
   "metadata": {},
   "source": [
    "Then we just check all the cube pairs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "2d11c86f",
   "metadata": {},
   "outputs": [],
   "source": [
    "arrangements = set()\n",
    "for s, t in combinations(cubes(), 2):\n",
    "    if all_squares_displayable(s, t):\n",
    "        arrangements.add((s, t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "8ce1fab4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1217"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(arrangements)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7229398d",
   "metadata": {},
   "source": [
    "#### 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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