166 lines
4.5 KiB
Plaintext
166 lines
4.5 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "89053792",
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"metadata": {},
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"source": [
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"# [Arithmetic Expressions](https://projecteuler.net/problem=93)\n",
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"\n",
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"We can brute force this answer.\n",
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"\n",
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"There are five ways to place the parentheses in an expression with four numbers (I'm only using addition here, but obviously you could use any operators):\n",
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"* $((a+b)+c)+d$\n",
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"* $(a+(b+c))+d$\n",
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"* $a+((b+c)+d)$\n",
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"* $a+(b+(c+d))$\n",
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"* $(a+b)+(c+d)$\n",
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"\n",
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"(It turns out that the number of ways to place the parentheses around $n$ numbers is $C_{n-1}$, where $C_n$ is the $n$th [Catalan number](https://en.wikipedia.org/wiki/Catalan_number).)\n",
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"\n",
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"If we iterate through every possible permutation of four digits, along with every possible set of three operators, and evaluate using each of the five possible groupings, we can get every number that can be formed from four digits under the problem's rules.\n",
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"\n",
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"Here's a function to find the longest streak of positive integers you can form from four digits."
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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": "0d75c8ce",
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"metadata": {},
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"outputs": [],
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"source": [
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"from itertools import product, permutations, count\n",
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"from operator import add, sub, mul, truediv\n",
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"\n",
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"def streak(digits):\n",
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" targets = set()\n",
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" for (a, b, c, d) in permutations(digits):\n",
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" for (op1, op2, op3) in product((add, sub, mul, truediv), repeat=3):\n",
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" try:\n",
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" targets.add(op3(op2(op1(a, b), c), d))\n",
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" except ZeroDivisionError:\n",
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" pass\n",
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" \n",
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" try:\n",
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" targets.add(op3(op2(a, op1(b, c)), d))\n",
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" except ZeroDivisionError:\n",
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" pass\n",
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" \n",
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" try:\n",
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" targets.add(op3(a, op2(op1(b, c), d)))\n",
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" except ZeroDivisionError:\n",
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" pass\n",
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" \n",
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" try:\n",
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" targets.add(op3(a, op2(b, op1(c, d))))\n",
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" except ZeroDivisionError:\n",
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" pass\n",
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" \n",
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" try:\n",
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" targets.add(op3(op1(a, b), op2(c, d)))\n",
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" except ZeroDivisionError:\n",
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" pass\n",
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" \n",
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" for n in count(1):\n",
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" if n not in targets:\n",
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" return n - 1"
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]
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},
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{
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"cell_type": "markdown",
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"id": "7ef3f5d3",
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"metadata": {},
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"source": [
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"Now we just try every combination of four digits."
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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": "90d4d5dd",
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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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"(1, 2, 5, 8)"
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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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"from itertools import combinations\n",
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"\n",
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"max(combinations(range(0, 10), 4), key=streak)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "eda00caf",
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"metadata": {},
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"source": [
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"Under the problem's rules, these numbers can form all the positive integers up to 51."
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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": 3,
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"id": "735dab1c",
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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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"51"
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]
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},
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"execution_count": 3,
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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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"streak((1,2,5,8))"
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]
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},
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{
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"cell_type": "markdown",
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"id": "e56a1679",
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"metadata": {},
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"source": [
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"## Relevant sequences\n",
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"* Catalan numbers: [A000108](https://oeis.org/A000108)\n",
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"\n",
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"#### Copyright (C) 2025 filifa\n",
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"\n",
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"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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]
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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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