97 lines
2.8 KiB
Plaintext
97 lines
2.8 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "43f43a42",
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"metadata": {},
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"source": [
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"# [Longest Collatz Sequence](https://projecteuler.net/problem=14)\n",
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"\n",
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"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",
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"\n",
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"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",
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"$$\n",
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"f(n) = \\begin{cases}\n",
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"1 & n = 1 \\\\\n",
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"1+f(n/2) & n \\equiv 0 \\pmod{2} \\\\\n",
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"1+f(3n+1) & n \\neq 1\\ \\text{and}\\ n \\equiv 1 \\pmod{2}\n",
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"\\end{cases}\n",
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"$$\n",
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"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",
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"\n",
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"Python has a nice decorator called [cache](https://docs.python.org/3/library/functools.html) that will automagically memoize our function."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "d1017296",
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"metadata": {},
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"outputs": [],
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"source": [
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"from functools import cache\n",
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"\n",
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"@cache\n",
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"def collatz_length(n):\n",
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" if n == 1:\n",
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" return 1\n",
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" elif n % 2 == 0:\n",
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" return 1 + collatz_length(n // 2)\n",
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" else:\n",
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" return 1 + collatz_length(3 * n + 1)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "fd145a5b",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"837799"
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]
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},
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"execution_count": 2,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"max(range(1, 1000000), key=collatz_length)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "bc0666ce",
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"metadata": {},
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"source": [
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"## Relevant sequences\n",
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"* Collatz chain lengths: [A008908](https://oeis.org/A008908)"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "SageMath 9.5",
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"language": "sage",
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"name": "sagemath"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.11.2"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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