129 lines
3.5 KiB
Plaintext
129 lines
3.5 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "c102f0bb",
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"metadata": {},
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"source": [
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"# [Cuboid Route](https://projecteuler.net/problem=86)\n",
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"\n",
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"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",
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"\n",
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"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",
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"\n",
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"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",
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"* $1 \\leq a$\n",
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"* $s - M \\leq a$\n",
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"* $a \\leq \\frac{s}{2}$\n",
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"* $a \\leq M$\n",
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"\n",
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"From these bounds, we can get the number of cuboids that can be constructed from an $(s, M)$ pair."
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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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"id": "6ce96a6e",
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"metadata": {},
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"outputs": [],
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"source": [
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"def leg_splits(s, M):\n",
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" max_a = min(M, s // 2 + 1)\n",
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" min_a = max(s - M, 1)\n",
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" return max_a - min_a"
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]
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},
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{
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"cell_type": "markdown",
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"id": "00e62dfc",
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"metadata": {},
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"source": [
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"Then we can write a function to find the number of cuboids with at least one edge equaling $M$."
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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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"id": "caf59499",
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"metadata": {},
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"outputs": [],
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"source": [
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"def count_cuboids(M):\n",
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" return sum(leg_splits(s, M) for s in range(1, 2 * M + 1) if is_square(s^2 + M^2))"
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]
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},
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{
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"cell_type": "markdown",
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"id": "d110565c",
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"metadata": {},
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"source": [
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"To get our answer, we just compute a running total and stop when it exceeds one million."
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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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"id": "984b7665",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"1818"
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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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"from itertools import count\n",
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"\n",
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"total = 0\n",
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"for M in count(1):\n",
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" total += count_cuboids(M)\n",
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" if total > 1000000:\n",
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" break\n",
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" \n",
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"M"
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]
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},
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{
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"cell_type": "markdown",
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"id": "488ec2af",
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"metadata": {},
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"source": [
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"## Relevant sequences\n",
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"* Number of pairs $a,b$ such that $(a+b)^2 + n^2$ is square: [A143714](https://oeis.org/A143714)\n",
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"* Partial sums of A143714: [A143715](https://oeis.org/A143715)\n",
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"\n",
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"#### Copyright (C) 2025 filifa\n",
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"\n",
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"This work is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International license](https://creativecommons.org/licenses/by-sa/4.0/) and the [BSD Zero Clause license](https://spdx.org/licenses/0BSD.html)."
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "SageMath 9.5",
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"language": "sage",
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"name": "sagemath"
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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": "3.11.2"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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