# [Maths-Education] Combinatorics and Sudoku

R.N.Goodman at brighton.ac.uk R.N.Goodman at brighton.ac.uk
Mon Dec 8 14:37:08 GMT 2008

```The claim that you never need to blind guess must depend on the quality
of the puzzle setter.  If I assume that harder puzzles have fewer
starting digits entered then an extreme puzzle could have too few digits
for a unique solution - and a logical trail of reasoning is unlikely?  I
have seen an apology in a newspaper following a published sudoku with
two solutions!

I have often wondered what algorithm is used to determine the level of
difficulty attached to the puzzles?  While I frequently get stumped on
solving the puzzles, creating them and ensuring uniqueness seems quite a
different problem?

Richard Goodman
University of Brighton

-----Original Message-----
From: maths-education-bounces at lists.nottingham.ac.uk
[mailto:maths-education-bounces at lists.nottingham.ac.uk] On Behalf Of
David H Kirshner
Sent: 08 December 2008 11:47
To: Mathematics Education discussion forum
Subject: RE: [Maths-Education] Combinatorics and Sudoku

************************************************************************
***********************************
This message has been generated through the Mathematics Education email
discussion list.
Hitting the REPLY key sends a message to all list members.
************************************************************************
***********************************
At the risk of distracting us further, here's a psychological question
related to Sudoku that has bugged me since I started doing the daily
puzzle in the newspaper a couple of years ago. Somewhere I read a CLAIM
to the effect that if you approach the puzzle effectively you'll NEVER
have to make a blind guess. For any configuration, there is ALWAYS a
valid reasoning path that enables you to definitively determine some
number location.

One of the joys of Sudoku is that with experience you do get better. You
come to recognize new possibilities for logical analysis you've not
previously thought of. What this has meant for me as I've matured as a
Sudokuist is the frequency with which I'm stumped--unable to develop a
logical argument that keeps me from having to make a blind guess--has
steadily declined. But I'm still stumped occasionally.

What is the analysis of the space of possible reasoning strategies, in
conjunction with the analysis of all possible configurations, that
enables someone to make the above CLAIM. Is it simply empirical--some
people who have worked on Sudoku for a long enough time have arrived at
a level of competence that enables them to always complete Sudoku's
without guessing? Or is there a theory of possible reasoning strategies
that proves the claim?

David

-----Original Message-----
From: maths-education-bounces at lists.nottingham.ac.uk
[mailto:maths-education-bounces at lists.nottingham.ac.uk] On Behalf Of
Alan Rogerson
Sent: Monday, December 08, 2008 4:29 AM
To: Mathematics Education discussion forum
Subject: [Maths-Education] Combinatorics and Sudoku

************************************************************************
***********************************
This message has been generated through the Mathematics Education email
discussion list.
Hitting the REPLY key sends a message to all list members.
************************************************************************
***********************************
It is almost trivial to write a little programme and let the computer
work out for us all the combinatorics to enumerate all the "possible"
sudoku grids, so in that sense the problem is neither difficult nor in a

way interesting (especially to sudoku widows and widowers), BUT the
complications come with the question of deciding what we regard as
"equivalent" solutions, and thank goodness someone else has actually
done all this hard work for us!
Neil Pateman was kind enough to send the webapge with the answer to all
the questions we have, but may have been afraid to ask, about sudoku
(thanks Peter Cave for confirming the spelling and the pronunciation!).
Here is Neil's email from Hawaii (hence the Aloha), enjoy and marvel at
the webpage, hard work by Ed Russell and Frazer Jarvis. And yes, ther
are no less than  5,472,730,538 "different", or as the authors put it
more subtly:: "essentially" different, solutions.
Hi Alan:
I found a website that may prove interesting to you.
http://www.afjarvis.staff.shef.ac.uk/sudoku/sudgroup.html
Aloha,
Neil

+++++++++++++++++++++++++++++++++++++++++++++++++++++++
An international directory of mathematics educators is available on the
web at www.nottingham.ac.uk/csme/directory/main.html
______________________________________________
Maths-Education mailing list
Maths-Education at lists.nottingham.ac.uk
http://lists.nottingham.ac.uk/mailman/listinfo/maths-education
+++++++++++++++++++++++++++++++++++++++++++++++++++++++
An international directory of mathematics educators is available on the
web at www.nottingham.ac.uk/csme/directory/main.html
______________________________________________
Maths-Education mailing list
Maths-Education at lists.nottingham.ac.uk
http://lists.nottingham.ac.uk/mailman/listinfo/maths-education
```