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ipynb/CrossProduct.ipynb
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407
ipynb/CrossProduct.ipynb
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<div align=\"right\" style=\"text-align: right\"><i>Peter Norvig, Feb 2020</i></div>\n",
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"\n",
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"# CrossProduct Puzzle\n",
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"\n",
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"The 538 Riddler [poses a puzzle](https://fivethirtyeight.com/features/can-you-cross-like-a-boss/) called ***CrossProduct***, which works like this:\n",
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"> *Fill in each empty cell of the table with a single digit, 1—9, so that the product of the digits in each row equals the number to the right of the row, and the product of the digits in each column equals the number below the column.*\n",
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"\n",
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"<table class=\"viz full\" data-carpenter-slug=\"wissner-gross.riddler.7\"><thead style=\"border-bottom: none\"><tr><th class=\"number\" style=\"text-align: center;border-right: none;width: 25%;border-left: none\" colspan=\"3\"></th>\n",
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"<th class=\"number\" style=\"text-align: center;border-right: none;width: 25%;border-left: none\" colspan=\"1\"></th>\n",
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"</tr></thead><tbody><tr style=\"border-top: 1px solid #cdcdcd\"><td class=\"number\" style=\"text-align: center;border-right: 1px solid #cdcdcd;width: 25%;border-left: 1px solid #cdcdcd\"></td>\n",
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"<td class=\"number\" style=\"text-align: center;border-right: 1px solid #cdcdcd;width: 25%\"></td>\n",
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"<td class=\"number\" style=\"text-align: center;border-right: 2px solid;width: 25%\"></td>\n",
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"<td class=\"number\" style=\"border-right: 1px solid #cdcdcd;width: 25%\">210</td>\n",
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"</tr><tr><td class=\"number\" style=\"text-align: center;border-right: 1px solid #cdcdcd;width: 25%;border-left: 1px solid #cdcdcd\"></td>\n",
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"<td class=\"number\" style=\"text-align: center;border-right: 1px solid #cdcdcd;width: 25%\"></td>\n",
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"<td class=\"number\" style=\"text-align: center;border-right: 2px solid;width: 25%\"></td>\n",
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"<td class=\"number\" style=\"border-right: 1px solid #cdcdcd;width: 25%\">144</td>\n",
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"</tr><tr><td class=\"number\" style=\"text-align: center;border-right: 1px solid #cdcdcd;width: 25%;border-left: 1px solid #cdcdcd\"></td>\n",
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"<td class=\"number\" style=\"text-align: center;border-right: 1px solid #cdcdcd;width: 25%\"></td>\n",
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"<td class=\"number\" style=\"text-align: center;border-right: 2px solid;width: 25%\"></td>\n",
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"<td class=\"number\" style=\"border-right: 1px solid #cdcdcd;width: 25%\">54</td>\n",
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"</tr><tr><td class=\"number\" style=\"text-align: center;border-right: 1px solid #cdcdcd;width: 25%;border-left: 1px solid #cdcdcd\"></td>\n",
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"<td class=\"number\" style=\"text-align: center;border-right: 1px solid #cdcdcd;width: 25%\"></td>\n",
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"<td class=\"number\" style=\"text-align: center;border-right: 2px solid;width: 25%\"></td>\n",
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"<td class=\"number\" style=\"border-right: 1px solid #cdcdcd;width: 25%\">135</td>\n",
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"</tr><tr><td class=\"number\" style=\"text-align: center;border-right: 1px solid #cdcdcd;width: 25%;border-left: 1px solid #cdcdcd\"></td>\n",
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"<td class=\"number\" style=\"text-align: center;border-right: 1px solid #cdcdcd;width: 25%\"></td>\n",
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"<td class=\"number\" style=\"text-align: center;border-right: 2px solid;width: 25%\"></td>\n",
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"<td class=\"number\" style=\"border-right: 1px solid #cdcdcd;width: 25%\">4</td>\n",
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"</tr><tr><td class=\"number\" style=\"text-align: center;border-right: 1px solid #cdcdcd;width: 25%;border-left: 1px solid #cdcdcd\"></td>\n",
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"<td class=\"number\" style=\"text-align: center;border-right: 1px solid #cdcdcd;width: 25%\"></td>\n",
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"<td class=\"number\" style=\"text-align: center;border-right: 2px solid;width: 25%\"></td>\n",
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"<td class=\"number\" style=\"border-right: 1px solid #cdcdcd;width: 25%\">49</td>\n",
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"</tr><tr style=\"border-top: 2px solid;border-bottom: 1px solid #cdcdcd\"><td class=\"number\" style=\"text-align: center;border-right: 1px solid #cdcdcd;width: 25%;border-left: 1px solid #cdcdcd\">6,615</td>\n",
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"<td class=\"number\" style=\"text-align: center;border-right: 1px solid #cdcdcd;width: 25%\">15,552</td>\n",
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"<td class=\"number\" style=\"text-align: center;border-right: 2px solid;width: 25%\">420</td>\n",
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"<td class=\"number\" style=\"border-right: 1px solid #cdcdcd;width: 25%\"></td>\n",
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"</tr></tbody></table>\n",
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"\n",
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" \n",
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"# Data Type definitions\n",
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" \n",
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"Here are the data types we will use in trying to solve CrossProduct puzzles: \n",
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"- `Row`: a sequence of digits that forms a row in the table, e.g. `(7, 6, 5)`.\n",
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"- `Table`: a filled-in table (but not the row and column products): a list of rows, e.g. `[(7, 6, 5), (9, 8, 2), ...]`.\n",
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"- `Products`: a list of the numbers that corresponding digits must multiply to, e.g. in the puzzle above, `[210, 144, 54, 135, 4, 49]` for the row products, and `[6615, 15552, 420]` for the column products."
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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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"metadata": {},
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"outputs": [],
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"source": [
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"from typing import Tuple, List, Set, Iterable, Optional\n",
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"\n",
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"Row = Tuple[int, ...] \n",
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"Table = List[Row] \n",
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"Products = List[int] "
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Filling in one row\n",
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"\n",
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"A first step in solving the puzzle is filling in a single row of the table.\n",
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"<br>`fill_row(n, k)` will return the set of all `k`-digit tuples whose product is `n`.\n",
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"<br>By default there are 3 digits in a row, but any number of digits is allowed."
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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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"metadata": {},
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"outputs": [],
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"source": [
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"def fill_row(n, k=3) -> Set[Row]:\n",
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" \"All permutations of k digits that multiply to n.\"\n",
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" if k == 0:\n",
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" return {()} if n == 1 else set()\n",
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" else:\n",
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" return {(d, *rest) for d in range(1, 10)\n",
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" if (n / d).is_integer()\n",
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" for rest in fill_row(n // d, k - 1)}"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"For example:"
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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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"metadata": {},
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"outputs": [],
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"source": [
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"fill_row(210)"
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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": 4,
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"metadata": {},
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"outputs": [],
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"source": [
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"fill_row(729)"
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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": 5,
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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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"{(1, 9, 9, 9),\n",
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" (3, 3, 9, 9),\n",
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" (3, 9, 3, 9),\n",
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" (3, 9, 9, 3),\n",
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" (9, 1, 9, 9),\n",
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" (9, 3, 3, 9),\n",
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" (9, 3, 9, 3),\n",
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" (9, 9, 1, 9),\n",
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" (9, 9, 3, 3),\n",
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" (9, 9, 9, 1)}"
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]
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},
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"execution_count": 5,
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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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"fill_row(729, 4)"
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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": 6,
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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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"{(1, 5, 5, 5, 5, 5),\n",
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" (5, 1, 5, 5, 5, 5),\n",
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" (5, 5, 1, 5, 5, 5),\n",
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" (5, 5, 5, 1, 5, 5),\n",
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" (5, 5, 5, 5, 1, 5),\n",
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" (5, 5, 5, 5, 5, 1)}"
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]
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},
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"execution_count": 6,
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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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"fill_row(5**5, 6)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Solving the whole puzzle\n",
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"\n",
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"We can now solve the whole puzzle with a simple brute-force strategy: for every possible way of filling the first row, try every way of recursively solving the rest of the puzzle, and yield a table for each way that works. We'll define the function `solutions` to yield all possible solutions, and `solve` to find just the first solution. (A well-formed puzzle will have exactly one solution, but some puzzles might have no solution, or multiple solutions.)\n",
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"\n",
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"The function `solutions` has three main cases to consider:\n",
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"- Any puzzle with a non-integer column product is unsolvable. Return without yielding anything.\n",
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"- The empty puzzle (with no rows) has the empty table, `[]`, as a solution, as long as the column products are all 1. \n",
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"- In the general case, call `fill_row` to get all possible ways to fill the first row, and recursively call `solutions` to get all the possible ways of filling the rest of the rows (making sure to pass in an altered `col_prods` that accounts for the first row). "
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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": 7,
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"metadata": {},
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"outputs": [],
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"source": [
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"def solutions(row_prods: Products, col_prods: Products) -> Iterable[Table]:\n",
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" \"\"\"Yield table(s) that solve the puzzle.\n",
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" The product of the digits in row r must equal row_prods[r], for all r.\n",
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" The product of the digits in column c must equal col_prods[c], for all c.\"\"\"\n",
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" if not all(c == int(c) for c in col_prods):\n",
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" return\n",
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" if not row_prods and all(c == 1 for c in col_prods):\n",
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" yield []\n",
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" elif row_prods:\n",
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" for row1 in fill_row(row_prods[0], len(col_prods)):\n",
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" for rows in solutions(row_prods[1:], divide(col_prods, row1)):\n",
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" yield [row1, *rows]\n",
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" \n",
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"def solve(r, c) -> Optional[Table]: return next(solutions(r, c), None)\n",
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" \n",
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"def divide(A, B) -> List[float]: return [a / b for a, b in zip(A, B)]"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Solutions\n",
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"\n",
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"Here are solutions to the puzzles posed by *The Riddler*:"
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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": 8,
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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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"[(3, 9, 5), (5, 9, 1), (8, 1, 8), (5, 7, 8), (5, 7, 2)]"
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]
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},
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"execution_count": 8,
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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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"solve([135, 45, 64, 280, 70], [3000, 3969, 640])"
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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": 9,
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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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"[(7, 6, 5), (9, 8, 2), (3, 9, 2), (5, 9, 3), (1, 4, 1), (7, 1, 7)]"
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]
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},
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"execution_count": 9,
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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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"# The puzzle that appears at the top of this notebook\n",
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"solve([210, 144, 54, 135, 4, 49], [6615, 15552, 420])"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Tests\n",
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"\n",
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"A suite of unit tests:"
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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": 10,
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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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"True"
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]
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},
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"execution_count": 10,
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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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"def test():\n",
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" assert fill_row(1, 0) == {()}\n",
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" assert fill_row(2, 0) == set()\n",
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" assert fill_row(9, 1) == {(9,)}\n",
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" assert fill_row(10, 1) == set()\n",
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" assert fill_row(73, 3) == set()\n",
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" \n",
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" assert solve([], []) == []\n",
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" assert solve([], [1]) == []\n",
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" assert solve([], [2]) == None\n",
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" assert solve([5], [5]) == [(5,)]\n",
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" assert solve([0], [0]) == None # Maybe should allow zero as a digit?\n",
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" \n",
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" assert fill_row(729, 3) == {(9, 9, 9)} # Unique fill\n",
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" \n",
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" assert fill_row(729, 4) == {\n",
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" (1, 9, 9, 9),\n",
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" (3, 3, 9, 9),\n",
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" (3, 9, 3, 9),\n",
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" (3, 9, 9, 3),\n",
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" (9, 1, 9, 9),\n",
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" (9, 3, 3, 9),\n",
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" (9, 3, 9, 3),\n",
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" (9, 9, 1, 9),\n",
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" (9, 9, 3, 3),\n",
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" (9, 9, 9, 1)}\n",
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" \n",
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" assert max(range(1, 9*9*9 + 1), key=lambda n: len(fill_row(n, 3))) == 72\n",
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" assert fill_row(72, 3) == { # 72 has the most ways to fill a 3-digit row\n",
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" (1, 8, 9),\n",
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" (1, 9, 8),\n",
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" (2, 4, 9),\n",
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" (2, 6, 6),\n",
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" (2, 9, 4),\n",
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" (3, 3, 8),\n",
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" (3, 4, 6),\n",
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" (3, 6, 4),\n",
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" (3, 8, 3),\n",
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" (4, 2, 9),\n",
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" (4, 3, 6),\n",
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" (4, 6, 3),\n",
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" (4, 9, 2),\n",
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" (6, 2, 6),\n",
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" (6, 3, 4),\n",
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" (6, 4, 3),\n",
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" (6, 6, 2),\n",
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" (8, 1, 9),\n",
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" (8, 3, 3),\n",
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" (8, 9, 1),\n",
|
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" (9, 1, 8),\n",
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" (9, 2, 4),\n",
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" (9, 4, 2),\n",
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" (9, 8, 1)}\n",
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" \n",
|
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" assert solve([6, 120, 504], [28, 80, 162]) == [\n",
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" (1, 2, 3), \n",
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" (4, 5, 6), \n",
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" (7, 8, 9)]\n",
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" \n",
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" assert solve([210, 144, 54, 135, 4, 49], [6615, 15552, 420]) == [\n",
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" (7, 6, 5), \n",
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" (9, 8, 2), \n",
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" (3, 9, 2), \n",
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" (5, 9, 3), \n",
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" (1, 4, 1), \n",
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" (7, 1, 7)]\n",
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" \n",
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" assert sorted(solutions([8, 8, 1], [8, 8, 1])) == [ # Multi-solution puzzle\n",
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" [(1, 8, 1), \n",
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" (8, 1, 1), \n",
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" (1, 1, 1)],\n",
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" [(2, 4, 1), \n",
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" (4, 2, 1), \n",
|
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" (1, 1, 1)],\n",
|
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" [(4, 2, 1), \n",
|
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" (2, 4, 1), \n",
|
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" (1, 1, 1)],\n",
|
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" [(8, 1, 1), \n",
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" (1, 8, 1), \n",
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" (1, 1, 1)]]\n",
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" \n",
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" assert not list(solutions([8, 8, 1], [8, 8, 2])) # Unsolvable puzzle\n",
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" \n",
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" assert solve([1470, 720, 270, 945, 12, 343], \n",
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" [6615, 15552, 420, 25725]) == [ # 4 column puzzle\n",
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" (7, 6, 5, 7),\n",
|
||||
" (9, 8, 2, 5),\n",
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" (3, 9, 2, 5),\n",
|
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" (5, 9, 3, 7),\n",
|
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" (1, 4, 1, 3),\n",
|
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" (7, 1, 7, 7)]\n",
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||||
" \n",
|
||||
" return True\n",
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||||
" \n",
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"test()"
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]
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||||
}
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],
|
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"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.7.7"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 4
|
||||
}
|
||||
695
ipynb/Life.ipynb
695
ipynb/Life.ipynb
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Reference in New Issue
Block a user