add problem 68
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filifa
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@ -0,0 +1,183 @@
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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "371d0171",
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"metadata": {},
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"source": [
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"# [Magic 5-gon ring](https://projecteuler.net/problem=68)\n",
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"\n",
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"We can brute force this by considering all permutations of 1 through 10. For each permutation, we'll assign the first five values to the five external nodes, and the last five values will go in the internal ring. By assigning this way, we can easily obtain the indices of all the nodes along any given line in the 5-gon. The first node along a line will have index $i$, the second $i + 5$, and the third $i + 6$ (except if $i=4$, in which case the third index will be 5).\n",
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"\n",
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"Here's a function to handle the index logic (this and following methods are written to work for any $n$-gon)."
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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": "72c416e9",
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"metadata": {},
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"outputs": [],
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"source": [
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"def line_indices(i, n):\n",
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" j = i + n\n",
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" k = (i + 1) % n + n\n",
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" return (i, j, k)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "3510bee0",
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"metadata": {},
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"source": [
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"We'll have $10!=3628800$ permutations to check. If all the lines add to the same value as the first line, we have a magic ring."
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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": "37ea9e61",
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"metadata": {},
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"outputs": [],
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"source": [
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"from itertools import permutations\n",
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"\n",
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"def is_magic_ring(x, n):\n",
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" line_sum = x[0] + x[n] + x[n + 1]\n",
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" for i in range(1, n):\n",
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" i, j, k = line_indices(i, n)\n",
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" if x[i] + x[j] + x[k] != line_sum:\n",
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" return False\n",
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" \n",
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" return True\n",
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"\n",
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"\n",
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"def magic_rings(n):\n",
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" for x in permutations(range(1, 2*n + 1)):\n",
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" if is_magic_ring(x, n):\n",
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" yield x"
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]
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},
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{
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"cell_type": "markdown",
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"id": "515484ce",
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"metadata": {},
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"source": [
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"We'll need to reformat any magic rings we find in the form described in the problem. The solution string always starts with the lowest external node, so we use a [deque](https://docs.python.org/3/library/collections.html) to rotate."
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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": "669c130b",
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"metadata": {},
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"outputs": [],
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"source": [
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"from collections import deque\n",
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"\n",
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"def solution_string(x):\n",
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" lines = deque()\n",
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" n = len(x) // 2\n",
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" for i in range(0, n):\n",
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" i, j, k = line_indices(i, n)\n",
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" lines.append((x[i], x[j], x[k]))\n",
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" \n",
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" lowest_ext = min(range(0, n), key=lambda i: lines[i][0])\n",
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" lines.rotate(-lowest_ext)\n",
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" \n",
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" return \"\".join(str(m) for line in lines for m in line)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "6db5a5a6",
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"metadata": {},
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"source": [
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"Then we can just map this function over all the magic rings we find."
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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": 4,
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"id": "07928c04",
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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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"{'11069627285843410',\n",
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" '11085864693972710',\n",
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" '16103104548782926',\n",
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" '18102107379496568',\n",
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" '21049436378715110',\n",
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" '24105101817673934',\n",
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" '2594936378711015',\n",
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" '27858434106101917',\n",
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" '28797161103104548',\n",
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" '2951051817673439',\n",
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" '6357528249411013',\n",
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" '6531031914842725'}"
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]
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},
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"execution_count": 4,
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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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"sols = {solution_string(x) for x in magic_rings(5)}\n",
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"sols"
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]
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},
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{
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"cell_type": "markdown",
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"id": "b1f8a706",
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"metadata": {},
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"source": [
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"Note that we're only considering 16-digit solution strings."
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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": 5,
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"id": "9b0e2f19",
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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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"'6531031914842725'"
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]
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},
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"execution_count": 5,
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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((s for s in sols if len(s) == 16), key=int)"
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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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