{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Triangular, pentagonal, and hexagonal (Euler Problem 45)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "[https://projecteuler.net/problem=45](https://projecteuler.net/problem=45)\n",
    "\n",
    "Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:\n",
    "\n",
    "Triangle\t \t$T_n=n(n+1)/2 = 1, 3, 6, 10, 15, ...$\n",
    "\n",
    "Pentagonal\t \t$P_n=n(3n−1)/2 = 1, 5, 12, 22, 35, ...$\n",
    "\n",
    "Hexagonal\t \t$H_n=n(2n−1) = 1, 6, 15, 28, 45, ...$\n",
    "\n",
    "It can be verified that T285 = P165 = H143 = 40755.\n",
    "\n",
    "Find the next triangle number that is also pentagonal and hexagonal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def T(n):\n",
    "    return n * (n + 1) // 2\n",
    "\n",
    "assert(T(3) == 6)\n",
    "\n",
    "def P(n):\n",
    "    return n * (3 * n - 1) // 2\n",
    "\n",
    "assert(P(3) == 12)\n",
    "\n",
    "def H(n):\n",
    "    return n * (2 * n - 1)\n",
    "\n",
    "assert(H(3) == 15)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n_max = 100000\n",
    "s = 0\n",
    "\n",
    "T_list = [T(n) for n in range(1, n_max + 1)]\n",
    "P_set = {P(n) for n in range(1, n_max + 1)}\n",
    "H_set = {H(n) for n in range(1, n_max + 1)}\n",
    "\n",
    "for n in range(1, n_max + 1):\n",
    "    if T_list[n - 1] in P_set and T_list[n - 1] in H_set:\n",
    "        t = T_list[n - 1]\n",
    "        if t > 40755:\n",
    "            s = t"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1533776805\n"
     ]
    }
   ],
   "source": [
    "print(s)\n",
    "assert(s == 1533776805)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "completion_date": "Sat, 22 Dec 2018, 23:17",
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  "tags": [
   "triangular",
   "pentagonal",
   "hexagonal"
  ]
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