{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "43f43a42",
   "metadata": {},
   "source": [
    "# [Longest Collatz Sequence](https://projecteuler.net/problem=14)\n",
    "\n",
    "The [Collatz conjecture](https://en.wikipedia.org/wiki/Collatz_conjecture) is a famous unsolved problem, and a classic example of a seemingly simple question that has proven very difficult, if not impossible, to answer.\n",
    "\n",
    "It's easy enough to *define* a [recursive](https://en.wikipedia.org/wiki/Recursion_(computer_science%29) function to calculate the chain length for a starting number $n$.\n",
    "$$\n",
    "f(n) = \\begin{cases}\n",
    "1 & n = 1 \\\\\n",
    "1+f(n/2) & n \\equiv 0 \\pmod{2} \\\\\n",
    "1+f(3n+1) & n \\neq 1\\ \\text{and}\\ n \\equiv 1 \\pmod{2}\n",
    "\\end{cases}\n",
    "$$\n",
    "However, we want its *implementation* to be efficient. We can optimize greatly if we cache the outputs we compute (the computer science term for this is [memoization](https://en.wikipedia.org/wiki/Memoization)). For instance, if we store the fact that $f(4) = 3$ after we've computed it, when we later compute $f(8) = 1 + f(4)$, the program can use the stored value of 3 rather than recomputing $f(4)$. For large inputs, this will save us (or really the computer, I guess) from redoing work.\n",
    "\n",
    "Python has a nice decorator called [cache](https://docs.python.org/3/library/functools.html) that will automagically memoize our function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "d1017296",
   "metadata": {},
   "outputs": [],
   "source": [
    "from functools import cache\n",
    "\n",
    "@cache\n",
    "def collatz_length(n):\n",
    "    if n == 1:\n",
    "        return 1\n",
    "    elif n % 2 == 0:\n",
    "        return 1 + collatz_length(n // 2)\n",
    "    else:\n",
    "        return 1 + collatz_length(3 * n + 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "fd145a5b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "837799"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "max(range(1, 1000000), key=collatz_length)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bc0666ce",
   "metadata": {},
   "source": [
    "## Relevant sequences\n",
    "* Collatz chain lengths: [A008908](https://oeis.org/A008908)\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)."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 9.5",
   "language": "sage",
   "name": "sagemath"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}