add problem 71

notebooks/problem0071.ipynb

@@ -0,0 +1,50 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "e7323b89",
+   "metadata": {},
+   "source": [
+    "# [Ordered Fractions](https://projecteuler.net/problem=71)\n",
+    "\n",
+    "This problem can be solved by hand.\n",
+    "\n",
+    "The concept the problem is describing is called a [Farey sequence](https://en.wikipedia.org/wiki/Farey_sequence). The example given in the problem is $F_8$, and we are tasked with finding the numerator of the left neighbor of $\\frac{3}{7}$ in $F_{1000000}$.\n",
+    "\n",
+    "It turns out there is a very simple method for determining this. Whenever you have two neighbors $\\frac{a}{b}$ and $\\frac{c}{d}$ in a Farey sequence, the next term that will appear between them in a subsequent Farey sequence is simply $\\frac{a+c}{b+d}$, called the [mediant](https://en.wikipedia.org/wiki/Mediant_(mathematics)) of the two neighbors. For example, since we're given that the left neighbor of $\\frac{3}{7}$ in $F_8$ is $\\frac{2}{5}$, the next fraction to appear between the two will be\n",
+    "$$\\frac{2+3}{5+7} = \\frac{5}{12}$$\n",
+    "Naturally, this fraction will first appear in $F_{12}$, meaning $\\frac{5}{12}$ is the left neighbor of $\\frac{3}{7}$ in that Farey sequence. We could then, in turn, find the mediant of $\\frac{5}{12}$ and $\\frac{3}{7}$ to find the next left neighbor of $\\frac{3}{7}$ ($\\frac{8}{19}$, appearing in $F_{19}$).\n",
+    "\n",
+    "We can repeat this process until we find a mediant with a denominator greater than 1000000. At that point, we know that mediant will *not* be in $F_{1000000}$, so whatever left neighbor we just used to calculate it must be the left neighbor of $\\frac{3}{7}$ in $F_{1000000}$.\n",
+    "\n",
+    "To compute this by hand, observe that the $n$th mediant computed in this manner is simply\n",
+    "$$\\frac{2 + 3n}{5 + 7n}$$\n",
+    "Therefore, we just need to find the largest $n$ such that $5 + 7n \\leq 1000000$, which is $n=142856$. This gives us a numerator of 428570.\n",
+    "\n",
+    "## Relevant sequences\n",
+    "* Numerators of Farey sequences: [A007305](https://oeis.org/A007305)"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "SageMath 9.5",
+   "language": "sage",
+   "name": "sagemath"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.11.2"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}