Featured Research

from universities, journals, and other organizations

Solving Sudokus: Coloring By Numbers

Date:
June 8, 2007
Source:
American Mathematical Society
Summary:
In a recent article, mathematicians explain the use of tools from the branch of mathematics called graph theory to systematically analyze Sudoku puzzles. They also find that analyzing Sudokus leads to some unsolved problems in graph theory.

Have you ever been trying to solve a Sudoku puzzle and been gripped by a sinking feeling that maybe you were stuck with a lemon? That maybe the puzzle you are struggling with actually has no solution at all? And, if you do find a solution, how can you be sure it's the only one? What if half an hour ago you had written 5 instead of 3---would you then have gone down a path to a completely different solution?

These questions and others are explored in the article "Sudoku Squares and Chromatic Polynomials", by Agnes M. Herzberg and M. Ram Murty, which appears in the June/July 2007 issue of the Notices of the AMS. The authors use tools from the branch of mathematics called graph theory to systematically analyze Sudoku puzzles. They also find that analyzing Sudokus leads to some unsolved problems in graph theory.

In this context, a "graph" is a collection of nodes connected by line segments (this object is also sometimes called a "network"). We can think of the 81 squares in a Sudoku puzzle as 81 nodes in a graph. We will also represent each of the numbers one through nine by a different color. In a Sudoku graph, two nodes are connected by a line segment if the two squares they represent are in the same row, column, or 3 by 3 subgrid. Since no row, column, or 3 by 3 subgrid can contain more than one instance of each number, the graph will have no connected nodes of the same color. (For example, suppose we represent 1 with red. Two red nodes connected with a line segment would mean a row, column, or 3 by 3 subgrid had two 1's in it, which is forbidden in Sudoku.)

In the language of graph theory, a colored graph with no connections between same-colored nodes is a called a "proper coloring." What Sudoku solvers do every day is try to extend a partially-colored graph (the original puzzle with open squares means the graph representing it has yet-to-be-colored nodes) to a proper coloring.

With this analogy between Sudoku puzzles and graphs in place, Herzberg and Murty are able to use tools from graph theory to prove theorems about Sudokus. For example, they prove that the number of ways of extending a partial coloring of a graph is given by a polynomial. If the value of this polynomial is zero for a given Sudoku puzzle, then the puzzle has no solution; if the value is 1, then the puzzle has only one solution; and so forth. They also prove that, in order for any Sudoku puzzle to have only one solution, at least 8 of the 9 numbers must appear as given entries in the puzzle; if only 7 numbers are given, then the puzzle has at least two solutions. And this brings up an unsolved mathematical question: "It would be extremely interesting to determine under what conditions a partial coloring can be extended to a unique [proper] coloring," Herzberg and Murty write.

Some Sudokus are harder than others, with the really difficult ones containing very few given entries. What is the minimum number of given entries needed to ensure that a puzzle has a unique solution" Herzberg and Murty give an example of a Sudoku puzzle with just 17 given entries that has only one solution. So the minimum number is at least 17. But could it be 16---or something even smaller" No one knows. One might think that a puzzle with many given entries is likelier to have a unique solution, but this is not necessarily the case. The article gives an example of a puzzle with 29 given entries that actually has two different solutions.

And if you have been wondering when your newspaper will run out of Sudoku puzzles, the authors argue that the number of distinct Sudoku puzzles is somewhere around 5.5 billion---enough to keep Sudoku afficianados busy for a long time to come.

The article "Sudoku Squares and Chromatic Polynomials" is available on the web site of the AMS Notices at the URL http://www.ams.org/notices/200706/tx070600708p.pdf.


Story Source:

The above story is based on materials provided by American Mathematical Society. Note: Materials may be edited for content and length.


Cite This Page:

American Mathematical Society. "Solving Sudokus: Coloring By Numbers." ScienceDaily. ScienceDaily, 8 June 2007. <www.sciencedaily.com/releases/2007/06/070608093815.htm>.
American Mathematical Society. (2007, June 8). Solving Sudokus: Coloring By Numbers. ScienceDaily. Retrieved September 17, 2014 from www.sciencedaily.com/releases/2007/06/070608093815.htm
American Mathematical Society. "Solving Sudokus: Coloring By Numbers." ScienceDaily. www.sciencedaily.com/releases/2007/06/070608093815.htm (accessed September 17, 2014).

Share This



More Computers & Math News

Wednesday, September 17, 2014

Featured Research

from universities, journals, and other organizations


Featured Videos

from AP, Reuters, AFP, and other news services

2K Drafts Face-Mapping Tech for New Game

2K Drafts Face-Mapping Tech for New Game

AP (Sep. 17, 2014) "NBA 2K15" is angling for a slam dunk with an innovative new way to put players in the game. Gamers will be able to digitally graft lifelike 3D renditions of their faces onto virtual players using the PlayStation 4 and Xbox One cameras. (Sept. 17) Video provided by AP
Powered by NewsLook.com
FBI Finishes $1 Billion Facial Recognition System

FBI Finishes $1 Billion Facial Recognition System

Newsy (Sep. 15, 2014) The FBI announced it plans to make its Next Generation Identification System available to law enforcement, but some privacy advocates are worried. Video provided by Newsy
Powered by NewsLook.com
A+ for Apple iPhone Pre-Sales

A+ for Apple iPhone Pre-Sales

Reuters - Business Video Online (Sep. 15, 2014) Apple says it received a record 4 million first-day pre-orders for its new iPhone 6 and iPhone 6 Plus, pushing delivery dates into October. Bobbi Rebell reports. Video provided by Reuters
Powered by NewsLook.com
Microsoft to Buy 'Minecraft' Maker for $2.5B

Microsoft to Buy 'Minecraft' Maker for $2.5B

AP (Sep. 15, 2014) Microsoft will acquire the maker of the long-running hit game Minecraft for $2.5 billion as the company continues to invest in its Xbox gaming platform and looks to grab attention on mobile phones. (Sept. 15) Video provided by AP
Powered by NewsLook.com

Search ScienceDaily

Number of stories in archives: 140,361

Find with keyword(s):
Enter a keyword or phrase to search ScienceDaily for related topics and research stories.

Save/Print:
Share:

Breaking News:
from the past week

In Other News

... from NewsDaily.com

Science News

Health News

    Environment News

      Technology News



      Save/Print:
      Share:

      Free Subscriptions


      Get the latest science news with ScienceDaily's free email newsletters, updated daily and weekly. Or view hourly updated newsfeeds in your RSS reader:

      Get Social & Mobile


      Keep up to date with the latest news from ScienceDaily via social networks and mobile apps:

      Have Feedback?


      Tell us what you think of ScienceDaily -- we welcome both positive and negative comments. Have any problems using the site? Questions?
      Mobile: iPhone Android Web
      Follow: Facebook Twitter Google+
      Subscribe: RSS Feeds Email Newsletters
      Latest Headlines Health & Medicine Mind & Brain Space & Time Matter & Energy Computers & Math Plants & Animals Earth & Climate Fossils & Ruins