From 0b3aa2c3c24d0af1318dcbbe6f092adf6ded229e Mon Sep 17 00:00:00 2001
From: penguinn <michellewerth@hotmail.com>
Date: Mon, 6 Jan 2020 18:40:06 +0100
Subject: [PATCH] Add Euler Problem
---
Euler_Problem.ipynb | 182 +++++++++++++++++++++++++++++++++
resources/triangle_Euler67.def | 103 +++++++++++++++++++
resources/triangle_Euler67.txt | 100 ++++++++++++++++++
3 files changed, 385 insertions(+)
create mode 100644 Euler_Problem.ipynb
create mode 100644 resources/triangle_Euler67.def
create mode 100644 resources/triangle_Euler67.txt
diff --git a/Euler_Problem.ipynb b/Euler_Problem.ipynb
new file mode 100644
index 0000000..a2a0372
--- /dev/null
+++ b/Euler_Problem.ipynb
@@ -0,0 +1,182 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Euler Problem 67 - Maximum Path Sum II\n",
+ "\n",
+ "In the following we will give a solution for the Euler Problem 67. For convenience, the description of the problem can be found on the official site for the [Euler Problem 67](https://projecteuler.net/problem=67).\n",
+ "\n",
+ "By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.\n",
+ "\n",
+ "```\n",
+ "3\n",
+ "7 4\n",
+ "2 4 6\n",
+ "8 5 9 3\n",
+ "```\n",
+ "\n",
+ "That is, 3 + 7 + 4 + 9 = 23.\n",
+ "\n",
+ "In this notebook we are going to find the maximum total from top to bottom in triangle.txt, a 15K text file containing a triangle with one-hundred rows.\n",
+ "\n",
+ "Please note, that this is a much more difficult version of Problem 18. It is not possible to try every route to solve this problem, as there are 299 altogether! If you could check one trillion (1012) routes every second it would take over twenty billion years to check them all. There is an efficient algorithm to solve it."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Loaded machine: Euler_Problem_067"
+ ]
+ },
+ "execution_count": 2,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "::load\n",
+ "MACHINE Euler_Problem_067\n",
+ "DEFINITIONS\n",
+ " \"resources/triangle_Euler67.def\";\n",
+ " SET_PREF_TIME_OUT == 5000\n",
+ "CONSTANTS n, Sol, OptimalSolution, TriangleC\n",
+ "PROPERTIES TriangleC = Triangle &\n",
+ "n = size (TriangleC) &\n",
+ "\n",
+ "Sol : 1..n --> seq(INTEGER) &\n",
+ "Sol(n) = TriangleC(n) /* Initialisation */\n",
+ "&\n",
+ "! i. (i:1..(n-1) =>\n",
+ " (size(Sol(i)) = size(TriangleC(i))\n",
+ " & !j.(j: 1..size(TriangleC(i)) => Sol(i)(j) = TriangleC(i)(j)+max({Sol(i+1)(j),Sol(i+1)(j+1)}))\n",
+ " )\n",
+ " )\n",
+ " & OptimalSolution = Sol(1)(1)\n",
+ "END"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "An interesting fact is the that dynamic programming is done automatically by ProB’s constraint solving.\n",
+ "\n",
+ "For a better overview we are showing the optimal solution in this notebook with the `:eval` command.\n",
+ "\n",
+ "```\n",
+ "ASSERTIONS\n",
+ " OptimalSolution = 7273;\n",
+ " n = 100\n",
+ "```"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Machine constants set up using operation 0: $setup_constants()"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":constants"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Machine initialised using operation 1: $initialise_machine()"
+ ]
+ },
+ "execution_count": 4,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":init"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$7273$"
+ ],
+ "text/plain": [
+ "7273"
+ ]
+ },
+ "execution_count": 5,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":eval OptimalSolution"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/markdown": [
+ "$100$"
+ ],
+ "text/plain": [
+ "100"
+ ]
+ },
+ "execution_count": 6,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ ":eval n"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "ProB 2",
+ "language": "prob",
+ "name": "prob2"
+ },
+ "language_info": {
+ "codemirror_mode": "prob2_jupyter_repl",
+ "file_extension": ".prob",
+ "mimetype": "text/x-prob2-jupyter-repl",
+ "name": "prob"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/resources/triangle_Euler67.def b/resources/triangle_Euler67.def
new file mode 100644
index 0000000..0498646
--- /dev/null
+++ b/resources/triangle_Euler67.def
@@ -0,0 +1,103 @@
+DEFINITIONS
+Triangle == [
+[59],
+[73, 41],
+[52, 40, 09],
+[26, 53, 06, 34],
+[10, 51, 87, 86, 81],
+[61, 95, 66, 57, 25, 68],
+[90, 81, 80, 38, 92, 67, 73],
+[30, 28, 51, 76, 81, 18, 75, 44],
+[84, 14, 95, 87, 62, 81, 17, 78, 58],
+[21, 46, 71, 58, 02, 79, 62, 39, 31, 09],
+[56, 34, 35, 53, 78, 31, 81, 18, 90, 93, 15],
+[78, 53, 04, 21, 84, 93, 32, 13, 97, 11, 37, 51],
+[45, 03, 81, 79, 05, 18, 78, 86, 13, 30, 63, 99, 95],
+[39, 87, 96, 28, 03, 38, 42, 17, 82, 87, 58, 07, 22, 57],
+[06, 17, 51, 17, 07, 93, 09, 07, 75, 97, 95, 78, 87, 08, 53],
+[67, 66, 59, 60, 88, 99, 94, 65, 55, 77, 55, 34, 27, 53, 78, 28],
+[76, 40, 41, 04, 87, 16, 09, 42, 75, 69, 23, 97, 30, 60, 10, 79, 87],
+[12, 10, 44, 26, 21, 36, 32, 84, 98, 60, 13, 12, 36, 16, 63, 31, 91, 35],
+[70, 39, 06, 05, 55, 27, 38, 48, 28, 22, 34, 35, 62, 62, 15, 14, 94, 89, 86],
+[66, 56, 68, 84, 96, 21, 34, 34, 34, 81, 62, 40, 65, 54, 62, 05, 98, 03, 02, 60],
+[38, 89, 46, 37, 99, 54, 34, 53, 36, 14, 70, 26, 02, 90, 45, 13, 31, 61, 83, 73, 47],
+[36, 10, 63, 96, 60, 49, 41, 05, 37, 42, 14, 58, 84, 93, 96, 17, 09, 43, 05, 43, 06, 59],
+[66, 57, 87, 57, 61, 28, 37, 51, 84, 73, 79, 15, 39, 95, 88, 87, 43, 39, 11, 86, 77, 74, 18],
+[54, 42, 05, 79, 30, 49, 99, 73, 46, 37, 50, 02, 45, 09, 54, 52, 27, 95, 27, 65, 19, 45, 26, 45],
+[71, 39, 17, 78, 76, 29, 52, 90, 18, 99, 78, 19, 35, 62, 71, 19, 23, 65, 93, 85, 49, 33, 75, 09, 02],
+[33, 24, 47, 61, 60, 55, 32, 88, 57, 55, 91, 54, 46, 57, 07, 77, 98, 52, 80, 99, 24, 25, 46, 78, 79, 05],
+[92, 09, 13, 55, 10, 67, 26, 78, 76, 82, 63, 49, 51, 31, 24, 68, 05, 57, 07, 54, 69, 21, 67, 43, 17, 63, 12],
+[24, 59, 06, 08, 98, 74, 66, 26, 61, 60, 13, 03, 09, 09, 24, 30, 71, 08, 88, 70, 72, 70, 29, 90, 11, 82, 41, 34],
+[66, 82, 67, 04, 36, 60, 92, 77, 91, 85, 62, 49, 59, 61, 30, 90, 29, 94, 26, 41, 89, 04, 53, 22, 83, 41, 09, 74, 90],
+[48, 28, 26, 37, 28, 52, 77, 26, 51, 32, 18, 98, 79, 36, 62, 13, 17, 08, 19, 54, 89, 29, 73, 68, 42, 14, 08, 16, 70, 37],
+[37, 60, 69, 70, 72, 71, 09, 59, 13, 60, 38, 13, 57, 36, 09, 30, 43, 89, 30, 39, 15, 02, 44, 73, 05, 73, 26, 63, 56, 86, 12],
+[55, 55, 85, 50, 62, 99, 84, 77, 28, 85, 03, 21, 27, 22, 19, 26, 82, 69, 54, 04, 13, 07, 85, 14, 01, 15, 70, 59, 89, 95, 10, 19],
+[04, 09, 31, 92, 91, 38, 92, 86, 98, 75, 21, 05, 64, 42, 62, 84, 36, 20, 73, 42, 21, 23, 22, 51, 51, 79, 25, 45, 85, 53, 03, 43, 22],
+[75, 63, 02, 49, 14, 12, 89, 14, 60, 78, 92, 16, 44, 82, 38, 30, 72, 11, 46, 52, 90, 27, 08, 65, 78, 03, 85, 41, 57, 79, 39, 52, 33, 48],
+[78, 27, 56, 56, 39, 13, 19, 43, 86, 72, 58, 95, 39, 07, 04, 34, 21, 98, 39, 15, 39, 84, 89, 69, 84, 46, 37, 57, 59, 35, 59, 50, 26, 15, 93],
+[42, 89, 36, 27, 78, 91, 24, 11, 17, 41, 05, 94, 07, 69, 51, 96, 03, 96, 47, 90, 90, 45, 91, 20, 50, 56, 10, 32, 36, 49, 04, 53, 85, 92, 25, 65],
+[52, 09, 61, 30, 61, 97, 66, 21, 96, 92, 98, 90, 06, 34, 96, 60, 32, 69, 68, 33, 75, 84, 18, 31, 71, 50, 84, 63, 03, 03, 19, 11, 28, 42, 75, 45, 45],
+[61, 31, 61, 68, 96, 34, 49, 39, 05, 71, 76, 59, 62, 67, 06, 47, 96, 99, 34, 21, 32, 47, 52, 07, 71, 60, 42, 72, 94, 56, 82, 83, 84, 40, 94, 87, 82, 46],
+[01, 20, 60, 14, 17, 38, 26, 78, 66, 81, 45, 95, 18, 51, 98, 81, 48, 16, 53, 88, 37, 52, 69, 95, 72, 93, 22, 34, 98, 20, 54, 27, 73, 61, 56, 63, 60, 34, 63],
+[93, 42, 94, 83, 47, 61, 27, 51, 79, 79, 45, 01, 44, 73, 31, 70, 83, 42, 88, 25, 53, 51, 30, 15, 65, 94, 80, 44, 61, 84, 12, 77, 02, 62, 02, 65, 94, 42, 14, 94],
+[32, 73, 09, 67, 68, 29, 74, 98, 10, 19, 85, 48, 38, 31, 85, 67, 53, 93, 93, 77, 47, 67, 39, 72, 94, 53, 18, 43, 77, 40, 78, 32, 29, 59, 24, 06, 02, 83, 50, 60, 66],
+[32, 01, 44, 30, 16, 51, 15, 81, 98, 15, 10, 62, 86, 79, 50, 62, 45, 60, 70, 38, 31, 85, 65, 61, 64, 06, 69, 84, 14, 22, 56, 43, 09, 48, 66, 69, 83, 91, 60, 40, 36, 61],
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+]
\ No newline at end of file
diff --git a/resources/triangle_Euler67.txt b/resources/triangle_Euler67.txt
new file mode 100644
index 0000000..00aa2bc
--- /dev/null
+++ b/resources/triangle_Euler67.txt
@@ -0,0 +1,100 @@
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GitLab