{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "
Peter Norvig
Oct. 2021
\n", "\n", "# Solving Sudoku at 200,000 puzzles per second\n", "\n", "The rules of [**Sudoku**](http://en.wikipedia.org/wiki/Sudoku) are [simple and finite](https://clip.cafe/legally-blonde-2001/the-rules-of-hair-care-are-simple-finite-s1/?srsltid=AfmBOoqLAWkreHwp-ZaP5JeYIlVYwljjgG-7aHmt-kLQ_D0q8zvxd4GG): fill in the empty squares in the puzzle so that each column, row, and 3x3 box in the solution contains all the digits from 1 to 9. For example:\n", "\n", "|Puzzle|Solution|\n", "|---|---|\n", "|![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Sudoku_Puzzle_by_L2G-20050714_standardized_layout.svg/250px-Sudoku_Puzzle_by_L2G-20050714_standardized_layout.svg.png)|![](https://upload.wikimedia.org/wikipedia/commons/thumb/1/12/Sudoku_Puzzle_by_L2G-20050714_solution_standardized_layout.svg/250px-Sudoku_Puzzle_by_L2G-20050714_solution_standardized_layout.svg.png)|\n", "\n", "\n", "In 2006, I wrote a [Python program](http://norvig.com/sudoku.html) to solve Sudoku. Soon after that, [Peter Seibel](https://gigamonkeys.com/) invited me to publish an article in [*Code Quarterly*](https://gigamonkeys.com/code-quarterly/) magazine which would go into more detail about backtracking search and constraint propagation in general, and would feature a more efficient program. Unfortunately, the magazine [folded](https://gigamonkeys.wordpress.com/2011/10/17/end-of-the-line-for-code-quarterly/) before I could finish the article. In 2021 I [updated the Python Sudoku program](Sudoku.ipynb) to Jupyter notebook form, with modern coding idioms, and now I do the same for the [more efficient Java Sudoku program](Sudoku.java). (I apologize for some remaining legacy Java idioms from decades ago).\n", "\n", "- [**Sudoku.java**](Sudoku.java) is the Java program discussed in this notebook.\n", "- [**Sudoku.ipynb**](Sudoku.ipynb) is the Python notebook." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Solving a Puzzle\n", "\n", "In the file [sudokus1.txt](sudokus1.txt) is the single puzzle shown at the top of the page, in this format (one puzzle per line):" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "53..7....6..195....98....6.8...6...34..8.3..17...2...6.6....28....419..5....8..79\n" ] } ], "source": [ "!cat sudokus1.txt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can solve the puzzle with the command `java Sudoku`, and display the solution with the `-grid` option:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Puzzle 1: Solution:\n", "5 3 . | . 7 . | . . . 5 3 4 | 6 7 8 | 9 1 2 \n", "6 . . | 1 9 5 | . . . 6 7 2 | 1 9 5 | 3 4 8 \n", ". 9 8 | . . . | . 6 . 1 9 8 | 3 4 2 | 5 6 7 \n", "------+-------+------ ------+-------+------\n", "8 . . | . 6 . | . . 3 8 5 9 | 7 6 1 | 4 2 3 \n", "4 . . | 8 . 3 | . . 1 4 2 6 | 8 5 3 | 7 9 1 \n", "7 . . | . 2 . | . . 6 7 1 3 | 9 2 4 | 8 5 6 \n", "------+-------+------ ------+-------+------\n", ". 6 . | . . . | 2 8 . 9 6 1 | 5 3 7 | 2 8 4 \n", ". . . | 4 1 9 | . . 5 2 8 7 | 4 1 9 | 6 3 5 \n", ". . . | . 8 . | . 7 9 3 4 5 | 2 8 6 | 1 7 9 \n" ] } ], "source": [ "!java Sudoku -grid -nosummary sudokus1.txt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Command Line Options\n", "\n", "Here are all the options for the `java Sudoku` command:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "usage: java Sudoku -(no)[fghnprstuv] | -[RT] | ...\n", "E.g., -v turns verify flag on, -nov turns it off. -R and -T require a number. The options:\n", "\n", " -g(rid) Print each puzzle grid and solution grid (default off)\n", " -h(elp) Print this usage message\n", " -n(aked) Run the naked pairs strategy (default on)\n", " -p(uzzle) Print summary stats for each puzzle (default off)\n", " -r(everse) Solve the reverse of each puzzle as well as each puzzle itself (default off)\n", " -s(ummary) Print per-file summary stats (default on)\n", " -t(hread) Print summary stats for each thread (default off)\n", " -u(nitTest) Run a suite of unit tests (default off)\n", " -v(erify) Verify each solution is valid (default on)\n", " -T Concurrently run threads (default 25)\n", " -R Repeat each puzzle times (default 1)\n", " Solve all puzzles in filename, which has one puzzle per line\n" ] } ], "source": [ "!java Sudoku -help" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Benchmark Puzzles\n", "\n", "I gathered some Sudoku puzzles from various sources:\n", "- [sudokus_hard.txt](sudokus_hard.txt): Allegedly hardest problems, form the [New Sudoku Players Forum](http://forum.enjoysudoku.com/the-hardest-sudokus-new-thread-t6539.html#p65791).\n", "- [sudokus_17.txt](sudokus_17.txt): Puzzles with exactly 17 filled-in squares, from [Gordon Royle at the University of Western Australia](https://web.archive.org/web/20131019184812if_/http://school.maths.uwa.edu.au/~gordon/sudokumin.php).\n", "- [sudokus.txt](sudokus.txt): My collection, from various sources, and some automatically generated (by permuting rows, or swapping digits).\n", "\n", "Hee are the number of puzzles in each file:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " 375 sudokus_hard.txt\n", " 49150 sudokus_17.txt\n", " 250000 sudokus.txt\n", " 299525 total\n" ] } ], "source": [ "!wc -l sudokus_hard.txt sudokus_17.txt sudokus.txt " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's run the `java Sudoku` program on all these puzzles. To get even more puzzles (more puzzles should lower variance), I use the `-r` option to include the reverse of each puzzle (the last square becomes the first, etc.) and `-R` to repeat puzzles." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Puzzles μsec KHz Threads Backtracks Name\n", "======= ====== ======= ======= ========== ====\n", " 3000 60.0 16.668 25 330.7 sudokus_hard.txt\n", " 196604 4.8 210.408 25 41.7 sudokus_17.txt\n", "1000000 3.2 311.832 25 32.5 sudokus.txt\n" ] } ], "source": [ "!java Sudoku -r -R4 sudokus_hard.txt -R2 sudokus_17.txt sudokus.txt " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The columns of the output are:\n", "\n", "- `Puzzles`: the total number of puzzles in each file.\n", "- `μsec`: the mean time to solve a puzzle in microseconds (millionths of a second).\n", "- `KHz`: the throughput expressed as thousands of puzzles solved per second (the inverse of the μsec column).\n", "- `Threads`: the number of concurrent threads used (the default is 25).\n", "- `Backtracks`: the average number of times per puzzle that the search guessed wrong and had to back up.\n", "- `Name`: here the name of the file; could also be a puzzle number.\n", "\n", "**Conclusion: The program solves about 16,000 puzzles per second with the hardest puzzles, and 200,000 to 300,000 per second with easier puzzles.**\n", "\n", "This is a pretty good performance for such a simple program! One of the fastest known Sudoku solvers, [**tdoku**](https://github.com/t-dillon/tdoku), is twice as fast on the hardest puzzles, and only about 15% faster for the 17-clue puzzles, but is a much more complex program.\n", "\n", "\n", "Note that the μsec and KHz columns are displaying **throughput**: the number of puzzles that can be solved in a given amount of time. This is different than **latency**: the time it takes to solve a single puzzle from start to end. Throughput and latency are different because there are multiple threads working in parallel. If, say, there are 25 threads working in parallel on 25 CPUs and each puzzle takes 1 second to solve, then latency would be 1 second but throughput would be 25 puzzles per second. I did some experiments, and on my machine (an M3 Ultra Mac Studio), the best performance is somewhere around 25 threads (although there is 5% to 10% variance from run to run, so I couldn't nail down an optimal number). \n", "\n", "To measure the latency, use the `-T1` option to request a single thread:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Puzzles μsec KHz Threads Backtracks Name\n", "======= ====== ======= ======= ========== ====\n", " 3000 250.5 3.992 1 292.2 sudokus_hard.txt\n", " 49151 65.1 15.363 1 40.2 sudokus_17.txt\n", " 250000 61.6 16.238 1 34.2 sudokus.txt\n" ] } ], "source": [ "!java Sudoku -T1 -R8 sudokus_hard.txt -R1 sudokus_17.txt sudokus.txt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The average latency is about 250 microseconds for the hard puzzles and 60 to 65 microseconds for the easier puzzles." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Effect of Naked Pairs Strategy\n", "\n", "The Java program explicitly uses the so-called [naked pairs strategy](https://www.learn-sudoku.com/naked-pairs.html) to eliminate some possible digits, whereas the Python program did not. We can check how much the naked pairs strategy helps by disabling it with the `-nonaked` option:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Puzzles μsec KHz Threads Backtracks Name\n", "======= ====== ======= ======= ========== ====\n", " 250000 10.7 93.850 25 34.2 sudokus.txt\n", " 250000 13.6 73.619 25 109.1 sudokus.txt\n" ] } ], "source": [ "!java Sudoku sudokus.txt -nonaked sudokus.txt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The program is three times as fast with naked pairs compared to without. Perhaps implementing [other strategies](https://bestofsudoku.com/sudoku-strategy) could give similar speedups. However, my feeling is that the pr ogram is already fast enough, and there would probably be diminishing returns from further strategies." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Distribution of Puzzle Difficulty\n", "\n", "What's the distribution of puzzle run times, or of backtracks, across the benchmark puzzles? \n", "\n", "I can get data for individual puzzles with the `-p` option, then eliminate the header lines with `tail`." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "!java Sudoku -T1 -p sudokus_17.txt sudokus_hard.txt | tail -n +3 > sudata.txt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "I can import the data into pandas:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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μsecKHzBacktracks
count49528.00000049528.00000049528.000000
mean74.74042621.11787242.138952
std135.9582449.120126187.339951
min11.8000000.1180000.000000
25%37.70000015.1520001.000000
50%45.20000022.1200005.000000
75%66.00000026.54900024.000000
max8487.50000085.10600013438.000000
\n", "
" ], "text/plain": [ " μsec KHz Backtracks\n", "count 49528.000000 49528.000000 49528.000000\n", "mean 74.740426 21.117872 42.138952\n", "std 135.958244 9.120126 187.339951\n", "min 11.800000 0.118000 0.000000\n", "25% 37.700000 15.152000 1.000000\n", "50% 45.200000 22.120000 5.000000\n", "75% 66.000000 26.549000 24.000000\n", "max 8487.500000 85.106000 13438.000000" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "columns = 'Puzzles μsec KHz Threads Backtracks Puzzle Number'.split()\n", "sudata = pd.read_csv('sudata.txt', sep='\\s+', names=columns).drop(columns=['Puzzles', 'Puzzle', 'Threads', 'Number'])\n", "sudata.describe()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The mean latency is 74 microseconds, the median is 45 microseconds, and the maximum is about 8 milliseconds. For backtracks, the mean is 42, the median is only 5, and the maximum is over 13,000. \n", "\n", "Here is a scatterplot:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sudata.plot.scatter(x='Backtracks', y='μsec', marker='.');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "That makes it clear that there is a strong linear relation between number of backtracks and run time (with a couple of outliers).\n", "\n", "One weakness of a scatter plot as a visualization tool is that there can be lots of points plotted on top of each other, and you can't tell the density of such points. A histogram portrays density better (but on only one attribute at a time):" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sudata.hist(rwidth=0.9, column='Backtracks', log=True, bins=20);\n", "sudata.hist(rwidth=0.9, column='μsec', log=True, bins=20);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For both attributes, almost all the puzzles are in the leftmost bin (in both plots the leftmost bar is two horizontal lines above the next bin; thus more than 100 times more frequent). " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Comparing my Python and Java Sudoku Programs\n", "\n", "There are many similarities and some differences between my Java and Python Sudoku programs:\n", "- Both programs represent a puzzle as a **grid** of 81 squares, with 27 **units** (9 rows, 9 columns, 9 boxes).\n", "- They both represent uncertainty about a square's contents with a **set** of the possible digits from 1 to 9.\n", "- They both do a depth-first [search](http://en.wikipedia.org/wiki/Search_algorithm) for a solution.\n", " - They find an unfilled square with the fewest remaining possible digits, guess a digit to fill the square, and try to solve the rest of the puzzle from there. If that fails, back up and try a different guess for the square.\n", " - Each guess is placed on a new *copy* of the board. If the guess turns out to be wrong, revert to the old board.\n", "- They both use [constraint propagation](http://en.wikipedia.org/wiki/Constraint_satisfaction) to limit the search:\n", " - *The one rule:* After filling a square with a digit, eliminate the digit from all of the square's [peers](https://www.sudocue.net/guide.php#peers).\n", " - *Arc consistency:* after eliminating a possible digit from a square, check that it still has some other possible digit.\n", " - *Dual consistency:* after eliminating a possible digit from a square, check that another square in each of the square's 3 units could hold that digit.\n", " - These changes do not require a new copy of the board, because they are logical consequences, not guesses.\n", " - If a check fails, back up a level in the search.\n", "\n", "\n", "The [Python program](http://norvig.com/sudoku.html) was written for clarity and brevity. The [Java program](Sudoku.java) for efficiency. It uses these tricks:\n", "- Primitive Java data types are used:\n", " - A grid is an `int[81]` array, not a hash table. \n", " - A square is an int (like `8`), not a string (like `'A9'`).\n", " - A set of possible digits is an int bitset (like `0b110010100`) not a string of digits (like `'9853'`). \n", "- Multiple puzzles are solved in parallel, in different threads.\n", "- Rather than allocating (and then garbage collecting) a new copy of the grid at each step in the depth-first search, instead we pre-allocate a `gridpool` array of grids once and for all (at the start of each thread), and then every recursive call to `search` re-uses `gridpool[level]`. \n", "- The specific Sudoku strategy of [naked pairs](https://www.learn-sudoku.com/naked-pairs.html) is (optionally) implemented.\n", "- The Java compiler produces code that is inherently faster than the Python interpreter.\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Verifying Correctness\n", "\n", "How do we know this program (or any program) is correct? Traditionally, there are four kinds of evidence:\n", "- A large number of example input/output pairs that give the right answer.\n", "- Manual inspection of a smaller number of example input/output pairs.\n", "- A suite of unit tests to verify that components function properly (at least on some inputs).\n", "- A formal proof that the code is correct (or at least an outline of an argument for a partial proof).\n", "\n", "For this program:\n", "- Unless `-noverify` is given, every puzzle/solution pair is verified with the `verify` method to make sure:\n", " - Each square in the solution contains a single digit. \n", " - Each unit in the solution contains all nine digits.\n", " - Squares in the puzzle that are filled with a digit keep that same digit in the solution.\n", "- I have looked at some example solutions and double-checked some with an [online Sudoku](https://sudokuspoiler.azurewebsites.net/) program. [LGTM](https://www.dictionary.com/e/acronyms/lgtm/).\n", "- The `-u` option runs a suite of unit tests.\n", "- I have looked at the code carefully, but I am nowhere near a proof of correctness.\n" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Unit tests pass.\n" ] } ], "source": [ "!java Sudoku -u " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You can manually inspect these 10 \"hard\" puzzle/solution pairs:" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Puzzle 1: Solution:\n", ". . . | . . . | . . 8 6 2 1 | 9 4 3 | 7 5 8 \n", ". . 3 | . . . | 4 . . 7 8 3 | 6 1 5 | 4 9 2 \n", ". 9 . | . 2 . | . 6 . 5 9 4 | 7 2 8 | 3 6 1 \n", "------+-------+------ ------+-------+------\n", ". . . | . 7 9 | . . . 1 4 2 | 8 7 9 | 6 3 5 \n", ". . . | . 6 1 | 2 . . 3 5 7 | 4 6 1 | 2 8 9 \n", ". 6 . | 5 . 2 | . 7 . 8 6 9 | 5 3 2 | 1 7 4 \n", "------+-------+------ ------+-------+------\n", ". . 8 | . . . | 5 . . 2 3 8 | 1 9 7 | 5 4 6 \n", ". 1 . | . . . | . 2 . 9 1 6 | 3 5 4 | 8 2 7 \n", "4 . 5 | . . . | . . 3 4 7 5 | 2 8 6 | 9 1 3 \n", "\n", "Puzzle 2: Solution:\n", ". . . | . . . | . . 2 6 3 9 | 8 4 7 | 5 1 2 \n", ". . 8 | . 1 . | 9 . . 4 7 8 | 5 1 2 | 9 6 3 \n", "5 . . | . . 3 | . 4 . 5 1 2 | 6 9 3 | 7 4 8 \n", "------+-------+------ ------+-------+------\n", ". . . | 1 . 9 | 3 . . 7 2 4 | 1 8 9 | 3 5 6 \n", ". 6 . | . 3 . | . 8 . 9 6 5 | 2 3 4 | 1 8 7 \n", ". . 3 | 7 . . | . . . 1 8 3 | 7 6 5 | 2 9 4 \n", "------+-------+------ ------+-------+------\n", ". 4 . | . . . | . . 5 8 4 7 | 9 2 1 | 6 3 5 \n", "3 . 1 | . 7 . | 8 . . 3 5 1 | 4 7 6 | 8 2 9 \n", "2 . . | . . . | . . . 2 9 6 | 3 5 8 | 4 7 1 \n", "\n", "Puzzle 3: Solution:\n", ". . 2 | . . . | 7 . . 8 3 2 | 4 1 6 | 7 9 5 \n", ". 1 . | . . . | . 6 . 4 1 7 | 9 8 5 | 2 6 3 \n", "5 . . | . . . | . 1 8 5 9 6 | 2 7 3 | 4 1 8 \n", "------+-------+------ ------+-------+------\n", ". . . | . 3 7 | . . . 9 5 1 | 8 3 7 | 6 2 4 \n", ". . . | . 4 9 | . . . 3 2 8 | 6 4 9 | 5 7 1 \n", ". . 4 | 1 . 2 | 3 . . 7 6 4 | 1 5 2 | 3 8 9 \n", "------+-------+------ ------+-------+------\n", ". . 3 | . 2 . | 9 . . 1 7 3 | 5 2 8 | 9 4 6 \n", ". 8 . | . . . | . 5 . 2 8 9 | 3 6 4 | 1 5 7 \n", "6 . . | . . . | . . 2 6 4 5 | 7 9 1 | 8 3 2 \n", "\n", "Puzzle 4: Solution:\n", ". . . | . . . | . . 7 9 6 3 | 8 1 4 | 5 2 7 \n", ". . 4 | . 2 . | 6 . . 1 5 4 | 3 2 7 | 6 8 9 \n", "8 . . | . . . | 3 1 . 8 2 7 | 9 6 5 | 3 1 4 \n", "------+-------+------ ------+-------+------\n", ". . . | . . 2 | 9 . . 3 7 1 | 4 8 2 | 9 5 6 \n", ". 4 . | . 9 . | . 3 . 6 4 5 | 7 9 1 | 8 3 2 \n", ". . 9 | 5 . 6 | . . . 2 8 9 | 5 3 6 | 7 4 1 \n", "------+-------+------ ------+-------+------\n", ". 1 . | . . . | . . 8 5 1 2 | 6 7 3 | 4 9 8 \n", ". . 6 | . 5 . | 2 . . 4 9 6 | 1 5 8 | 2 7 3 \n", "7 . . | . . . | . 6 . 7 3 8 | 2 4 9 | 1 6 5 \n", "\n", "Puzzle 5: Solution:\n", ". . 4 | . . 3 | . . . 5 9 4 | 2 6 3 | 8 7 1 \n", ". 7 . | . 8 . | . . . 6 7 1 | 5 8 4 | 3 2 9 \n", "2 . 8 | 1 . . | . . 6 2 3 8 | 1 9 7 | 5 4 6 \n", "------+-------+------ ------+-------+------\n", ". . 3 | . . . | . 9 . 7 6 3 | 4 1 8 | 2 9 5 \n", ". 8 . | . 2 . | . . . 9 8 5 | 3 2 6 | 7 1 4 \n", "1 . . | 7 . . | . . 3 1 4 2 | 7 5 9 | 6 8 3 \n", "------+-------+------ ------+-------+------\n", ". . . | . . . | 4 5 . 8 1 6 | 9 3 2 | 4 5 7 \n", ". . . | 8 . . | 9 . . 3 5 7 | 8 4 1 | 9 6 2 \n", ". . 9 | . . 5 | . . 8 4 2 9 | 6 7 5 | 1 3 8 \n", "\n", "Puzzle 6: Solution:\n", ". . 6 | . . 1 | . . . 8 2 6 | 9 7 1 | 3 5 4 \n", ". 5 . | . 3 . | . . . 7 5 4 | 8 3 6 | 1 9 2 \n", "9 . . | 4 . . | . . 7 9 1 3 | 4 2 5 | 8 6 7 \n", "------+-------+------ ------+-------+------\n", ". . 1 | . . . | . 2 . 5 7 1 | 6 4 3 | 9 2 8 \n", ". 3 . | . 9 . | . . . 2 3 8 | 1 9 7 | 5 4 6 \n", "4 . . | 5 . . | . . 1 4 6 9 | 5 8 2 | 7 3 1 \n", "------+-------+------ ------+-------+------\n", "3 . . | . . . | 6 8 . 3 4 7 | 2 1 9 | 6 8 5 \n", ". . . | 3 . . | 2 . . 1 8 5 | 3 6 4 | 2 7 9 \n", ". . 2 | . . 8 | . . 3 6 9 2 | 7 5 8 | 4 1 3 \n", "\n", "Puzzle 7: Solution:\n", ". . . | . . . | . . 3 8 6 2 | 7 1 4 | 9 5 3 \n", ". . 1 | . . 9 | . 6 . 7 4 1 | 3 5 9 | 8 6 2 \n", ". 5 . | . 8 . | 4 . . 9 5 3 | 2 8 6 | 4 7 1 \n", "------+-------+------ ------+-------+------\n", ". . . | 9 . . | . 8 . 3 7 4 | 9 2 1 | 6 8 5 \n", ". . 8 | 6 7 . | . . . 2 9 8 | 6 7 5 | 1 3 4 \n", ". 1 . | . . . | 2 . . 6 1 5 | 4 3 8 | 2 9 7 \n", "------+-------+------ ------+-------+------\n", ". . 6 | . . 7 | . 2 . 5 8 6 | 1 4 7 | 3 2 9 \n", ". 3 . | 8 . . | 5 . . 1 3 7 | 8 9 2 | 5 4 6 \n", "4 . . | . . . | . . 8 4 2 9 | 5 6 3 | 7 1 8 \n", "\n", "Puzzle 8: Solution:\n", ". . . | . . . | . . 5 7 1 4 | 9 6 3 | 2 8 5 \n", ". . 6 | . . 8 | 7 . . 9 2 6 | 5 1 8 | 7 3 4 \n", "3 . . | . . . | . 9 . 3 8 5 | 2 7 4 | 6 9 1 \n", "------+-------+------ ------+-------+------\n", ". . . | 1 . 7 | . 4 . 2 3 8 | 1 9 7 | 5 4 6 \n", ". . 7 | . . . | 8 . . 6 5 7 | 3 4 2 | 8 1 9 \n", ". 4 . | . . 6 | . . . 1 4 9 | 8 5 6 | 3 2 7 \n", "------+-------+------ ------+-------+------\n", ". 9 . | . 8 . | . . 3 4 9 2 | 7 8 5 | 1 6 3 \n", ". . 1 | 6 . . | 4 . . 8 7 1 | 6 3 9 | 4 5 2 \n", "5 . . | . 2 . | . . . 5 6 3 | 4 2 1 | 9 7 8 \n", "\n", "Puzzle 9: Solution:\n", ". . . | . . 5 | . . 3 4 2 7 | 1 6 5 | 8 9 3 \n", ". . 9 | . . . | . 4 . 5 3 9 | 2 7 8 | 6 4 1 \n", ". 8 1 | . 4 . | . . . 6 8 1 | 3 4 9 | 7 2 5 \n", "------+-------+------ ------+-------+------\n", ". . . | 7 . . | . . . 2 1 6 | 7 9 3 | 4 5 8 \n", ". . 4 | . . 2 | . . 6 3 9 4 | 5 8 2 | 1 7 6 \n", "8 . . | . 1 4 | . 3 . 8 7 5 | 6 1 4 | 9 3 2 \n", "------+-------+------ ------+-------+------\n", ". . . | . . . | 2 . . 7 5 8 | 4 3 1 | 2 6 9 \n", ". 4 . | . . 6 | . . 7 1 4 3 | 9 2 6 | 5 8 7 \n", "9 . . | . 5 . | . 1 . 9 6 2 | 8 5 7 | 3 1 4 \n", "\n", "Puzzle 10: Solution:\n", ". . . | . . 5 | . . 4 2 8 7 | 9 3 5 | 1 6 4 \n", ". 9 . | . . . | . 2 . 3 9 4 | 1 6 8 | 5 2 7 \n", ". . 6 | . 7 . | 3 . . 5 1 6 | 4 7 2 | 3 8 9 \n", "------+-------+------ ------+-------+------\n", ". . . | 7 . . | 8 . . 6 4 5 | 7 9 1 | 8 3 2 \n", ". . 8 | 6 . . | . . . 9 7 8 | 6 2 3 | 4 5 1 \n", "1 3 . | . 8 . | . . . 1 3 2 | 5 8 4 | 9 7 6 \n", "------+-------+------ ------+-------+------\n", ". . 3 | . 1 . | 6 . . 7 5 3 | 2 1 9 | 6 4 8 \n", ". 2 . | . . . | . . 5 8 2 9 | 3 4 6 | 7 1 5 \n", "4 . . | . . . | . 9 . 4 6 1 | 8 5 7 | 2 9 3 \n" ] } ], "source": [ "!head -10 sudokus_hard.txt > sudokus_hard10.txt\n", "\n", "!java Sudoku -grid -nosummary sudokus_hard10.txt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# What's Next?\n", "\n", "There are a few things I didn't get a chance to explore; maybe you can:\n", "\n", "- Would the program be even faster in Golang or C++ or Rust? Or with some more optimizations?\n", "- The depth-first search makes a choice to *fill* some square with a digit. What if instead the choice was to *eliminate* a digit from the square? At first glance it seems that would be slower, because more choices would be required, but would constraint propagation make it work well?\n", "- On each recursive call, the depth-first search copies the current grid into a new grid (re-using the one in `gridpool[level]`). That requires copying an array of 81 ints. Would it be faster to instead make changes directly to the current grid, and then undo the changes when failure is detected? You would need to keep track of the changes made so that they can be undone. My guess is that this would make the code more complex and not much faster (if any), but you might want to try it.\n", "- In the theory of constraint propagation, [shaving](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.175.7143&rep=rep1&type=pdf) means guessing a value for some variable, detecting a contradiction, and then keeping track of the fact that the value is not possible. But in our program, when we guess wrong we don't keep track of anything. Can the program be made faster by incorporating shaving?\n", "- Can you create an adversarial puzzle, where the program guesses wrong the maximal number of times? In other words, if you draw a tree of choices, what's a puzzle with a tree that has the solution in the bottom-right corner? How much time and how many backtracks does that puzzle take to solve?\n", "- What [other Sudoku strategies](https://bestofsudoku.com/sudoku-strategy) can be implemented? Can you find a suite of strategies that will solve all the puzzles with no search? \n", "- Can you develop a system to rank the difficulty of puzzles, based on the complexity of the strategies needed to solve it?\n", "\n", "One final word: I got around to writing this up after a friend mentioned to me \"*Hey, a while back didn't you do a Sudoku program that could solve like **a dozen puzzles a second** or something?*\" I felt like Robert Wagner when he had to explain to Dr. Evil that \"*a million dollars isn't exactly a lot of money these days.*\" So this is for all of you who were cryogenically frozen in 1967 (and for all of you who weren't).\n", "\n", "| | |\n", "|---|---|\n", "|![](https://upload.wikimedia.org/wikipedia/en/1/16/Drevil_million_dollars.jpg)|![](Number_2.webp)|\n", "|\"*One **dozen** puzzles per second!*|*A dozen isn't exactly a lot of throughput these days...*|\n", "" ] } ], "metadata": { "kernelspec": { "display_name": "Python [conda env:base] *", "language": "python", "name": "conda-base-py" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.13.9" } }, "nbformat": 4, "nbformat_minor": 4 }