diff --git a/ipython/EulerProblem028.ipynb b/ipython/EulerProblem028.ipynb new file mode 100755 index 0000000..e71f4cf --- /dev/null +++ b/ipython/EulerProblem028.ipynb @@ -0,0 +1,94 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Euler Problem 28\n", + "\n", + "Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows:\n", + "\n", + "~~~\n", + "21 22 23 24 25\n", + "20 7 8 9 10\n", + "19 6 1 2 11\n", + "18 5 4 3 12\n", + "17 16 15 14 13\n", + "~~~\n", + "\n", + "$1 + 3 + 5 + 7 + 9 + 13 + 17 + 21 + 25 = 101$\n", + "\n", + "It can be verified that the sum of the numbers on the diagonals is 101.\n", + "\n", + "What is the sum of the numbers on the diagonals in a 1001 by 1001 spiral formed in the same way?" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "I would try to create a function $f(n)$ which yields the sum of the outmost ring of a n by n spiral.\n", + "\n", + "For example:\n", + "\n", + "$f(1) = 1$\n", + "\n", + "$f(3) = 3 + 5 + 7 + 9 = 24$\n", + "\n", + "$f(5) = 13 + 17 + 21 + 25 = 76$\n", + "\n", + "When we have this function we calculate the solution simply by\n", + "\n", + "~~~\n", + "s = sum([f(n) for n in range(1, 1002, 2)])\n", + "~~~\n", + "\n", + "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 $f(n) = c_{n} + (c_n + n - 1) + (c_n + 2(n-1)) + (c_n + 3(n-1)) = 4c_n + 6 (n-1)$" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "def f(n):\n", + " if n == 1:\n", + " return 1\n", + " return 0\n", + "\n", + "s = sum([f(n) for n in range(1, 1002, 2)])\n", + "assert(s == 669171001)\n", + "s" + ] + } + ], + "metadata": { + "completion_date": "Wed, 23 Aug 2017, 15:54", + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.6.3" + }, + "tags": [ + "spiral", + "diagonals" + ] + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/ipython/EulerProblem029.ipynb b/ipython/EulerProblem029.ipynb new file mode 100755 index 0000000..0062700 --- /dev/null +++ b/ipython/EulerProblem029.ipynb @@ -0,0 +1,59 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Euler Problem 29\n", + "\n", + "Consider all integer combinations of ab for 2 ≤ a ≤ 5 and 2 ≤ b ≤ 5:\n", + "\n", + "$2^2=4, 2^3=8, 2^4=16, 2^5=32$\n", + "\n", + "32=9, 33=27, 34=81, 35=243\n", + "\n", + "42=16, 43=64, 44=256, 45=1024\n", + "\n", + "52=25, 53=125, 54=625, 55=3125\n", + "\n", + "If they are then placed in numerical order, with any repeats removed, we get the following sequence of 15 distinct terms:\n", + "\n", + "4, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125\n", + "\n", + "How many distinct terms are in the sequence generated by ab for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + } + ], + "metadata": { + "comopletion_date": "", + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.6.3" + }, + "tags": [] + }, + "nbformat": 4, + "nbformat_minor": 2 +}