From 830ca940de901ee579020eeb2255a8cb3cabf327 Mon Sep 17 00:00:00 2001
From: filifa <filifa.breath652@8alias.com>
Date: Sun, 29 Jun 2025 22:16:01 -0400
Subject: [PATCH] add problem 86

---
 notebooks/problem0086.ipynb | 124 ++++++++++++++++++++++++++++++++++++
 1 file changed, 124 insertions(+)
 create mode 100644 notebooks/problem0086.ipynb

diff --git a/notebooks/problem0086.ipynb b/notebooks/problem0086.ipynb
new file mode 100644
index 0000000..259a943
--- /dev/null
+++ b/notebooks/problem0086.ipynb
@@ -0,0 +1,124 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "c102f0bb",
+   "metadata": {},
+   "source": [
+    "# [Cuboid Route](https://projecteuler.net/problem=86)\n",
+    "\n",
+    "Suppose you have a cuboid with side lengths $a \\leq b \\leq M$. Then the shortest route will be $\\sqrt{(a + b)^2 + M^2}$. We're interested in when this distance is an integer.\n",
+    "\n",
+    "However, rather than iterate through values of $a$, $b$, and $M$, we can be more efficient by iterating through values of $M$, then values of $s$, where $s \\leq 2M$. If $s^2 + M^2$ is a square number, then that means any $a,b$ such that $s = a+b$ and $a \\leq b \\leq M$ will correspond to an $a \\times b \\times M$ cuboid with integer shortest route.\n",
+    "\n",
+    "So, if $s = a + b$, naturally $b = s - a$, and we want to know how many values of $a$ satisfy $1 \\leq a \\leq s - a \\leq M$. We can derive four bounds on $a$ from this.\n",
+    "* $1 \\leq a$\n",
+    "* $s - M \\leq a$\n",
+    "* $a \\leq \\frac{s}{2}$\n",
+    "* $a \\leq M$\n",
+    "\n",
+    "From these bounds, we can get the number of cuboids that can be constructed from an $(s, M)$ pair."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "6ce96a6e",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def leg_splits(s, M):\n",
+    "    max_a = min(M, s // 2 + 1)\n",
+    "    min_a = max(s - M, 1)\n",
+    "    return max_a - min_a"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "00e62dfc",
+   "metadata": {},
+   "source": [
+    "Then we can write a function to find the number of cuboids with at least one edge equaling $M$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "caf59499",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def count_cuboids(M):\n",
+    "    return sum(leg_splits(s, M) for s in range(1, 2 * M + 1) if is_square(s^2 + M^2))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d110565c",
+   "metadata": {},
+   "source": [
+    "To get our answer, we just compute a running total and stop when it exceeds one million."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "984b7665",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "1818"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from itertools import count\n",
+    "\n",
+    "total = 0\n",
+    "for M in count(1):\n",
+    "    total += count_cuboids(M)\n",
+    "    if total > 1000000:\n",
+    "        break\n",
+    "        \n",
+    "M"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "488ec2af",
+   "metadata": {},
+   "source": [
+    "## Relevant sequences\n",
+    "* Number of pairs $a,b$ such that $(a+b)^2 + n^2$ is square: [A143714](https://oeis.org/A143714)\n",
+    "* Partial sums of A143714: [A143715](https://oeis.org/A143715)"
+   ]
+  }
+ ],
+ "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
+}