97 lines
2.7 KiB
Plaintext
97 lines
2.7 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "dcfe1d77",
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"metadata": {},
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"source": [
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"# [Right Triangles with Integer Coordinates](https://projecteuler.net/problem=91)\n",
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"\n",
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"We'll start by generating all the possible values of $P$ and $Q$."
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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": "cba05ca7",
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"metadata": {},
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"outputs": [],
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"source": [
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"limit = 50\n",
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"points = ((x, y) for x in range(0, limit + 1) for y in range(0, limit + 1) if (x, y) != (0, 0))"
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]
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},
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{
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"cell_type": "markdown",
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"id": "90cb43dd",
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"metadata": {},
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"source": [
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"If we think about $P$ and $Q$ as vectors instead of points, we can solve this problem with [dot products](https://en.wikipedia.org/wiki/Dot_product). Since the dot product of orthogonal vectors is 0, we can check for a right angle in the triangle by seeing if $\\vec{P} \\cdot \\vec{Q} = 0$, $\\vec{P} \\cdot (\\vec{Q} - \\vec{P}) = 0$, or $\\vec{Q} \\cdot (\\vec{Q} - \\vec{P}) = 0$. By distributing in the last two equations, we can simply check if $\\vec{P} \\cdot \\vec{Q}$ equals 0, $\\vec{P} \\cdot \\vec{P}$, or $\\vec{Q} \\cdot \\vec{Q}$."
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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": "18e334eb",
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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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"14234"
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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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"from itertools import combinations\n",
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"\n",
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"triangles = set()\n",
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"for ((x1, y1), (x2, y2)) in combinations(points, 2):\n",
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" d = x1 * x2 + y1 * y2\n",
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" if d == 0 or d == x1^2 + y1^2 or d == x2^2 + y2^2:\n",
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" triangles.add(((x1, y1), (x2, y2)))\n",
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" \n",
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"len(triangles)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "35a4a234",
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"metadata": {},
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"source": [
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"## Relevant sequences\n",
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"* Answers for limits of 0, 1, 2, ...: [A155154](https://oeis.org/A155154)\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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