95 lines
2.2 KiB
Plaintext
95 lines
2.2 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Euler Problem 28\n",
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"\n",
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"Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows:\n",
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"\n",
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"~~~\n",
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"21 22 23 24 25\n",
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"20 7 8 9 10\n",
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"19 6 1 2 11\n",
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"18 5 4 3 12\n",
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"17 16 15 14 13\n",
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"~~~\n",
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"\n",
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"$1 + 3 + 5 + 7 + 9 + 13 + 17 + 21 + 25 = 101$\n",
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"\n",
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"It can be verified that the sum of the numbers on the diagonals is 101.\n",
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"\n",
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"What is the sum of the numbers on the diagonals in a 1001 by 1001 spiral formed in the same way?"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"I would try to create a function $f(n)$ which yields the sum of the outmost ring of a n by n spiral.\n",
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"\n",
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"For example:\n",
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"\n",
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"$f(1) = 1$\n",
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"\n",
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"$f(3) = 3 + 5 + 7 + 9 = 24$\n",
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"\n",
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"$f(5) = 13 + 17 + 21 + 25 = 76$\n",
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"\n",
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"When we have this function we calculate the solution simply by\n",
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"\n",
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"~~~\n",
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"s = sum([f(n) for n in range(1, 1002, 2)])\n",
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"~~~\n",
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"\n",
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"For each outer ring there is an initial corner value c ($c_3 = 3, c_5 = 76$). Once we have this value we can caluclate f like $f(n) = c_{n} + (c_n + n - 1) + (c_n + 2(n-1)) + (c_n + 3(n-1)) = 4c_n + 6 (n-1)$"
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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": null,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"def f(n):\n",
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" if n == 1:\n",
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" return 1\n",
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" return 0\n",
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"\n",
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"s = sum([f(n) for n in range(1, 1002, 2)])\n",
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"assert(s == 669171001)\n",
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"s"
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]
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}
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],
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"metadata": {
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"completion_date": "Wed, 23 Aug 2017, 15:54",
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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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": "3.6.3"
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},
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"tags": [
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"spiral",
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"diagonals"
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]
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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