143 lines
3.1 KiB
Plaintext
143 lines
3.1 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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"When we know the corner of a certain spiral we can calculate it's total like $f_n = 4 c_n + 6 (n - 1)$. We then only have to update the corner value for each spiral. "
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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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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"total = 1\n",
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"current_corner = 3\n",
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"\n",
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"for n in range(3, 1002, 2):\n",
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" total += 4 * current_corner + 6 * (n - 1)\n",
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" current_corner += 4 * n - 2\n",
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"\n",
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"s = total"
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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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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"669171001"
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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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"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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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The only missing piece is how could we calculate the current corner value for a certain n. The series for this is as follows:\n",
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"\n",
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"$c = 1, 3, 13, 31, 57, 91, 133, 183, 241, \\dots$\n",
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"\n",
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"With some experimenting it can be seen that\n",
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"\n",
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"$c_n = (n - 1)^2 - (n - 2) = n^2 - 2n + 1 - n + 2 = n^2 - 3n + 3$.\n",
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"\n",
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"Now, we can insert $c_n$ into $f_n$ which gives as the sum of corners for the nth spiral:\n",
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"\n",
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"$f_n = 4n^2 - 12n + 12 + 6n - 6 = 4n^2 - 6n + 6$"
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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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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"669171001"
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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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"s = 1 + sum([4 * n * n - 6 * n + 6 for n in range(3, 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": "3.5.4"
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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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