[Maths-Education] Didactics and Sudoku??
Alan Rogerson
alan at rogerson.pol.pl
Mon Dec 8 15:05:24 GMT 2008
Dear David,
I suspect you are right about the claim, it probably refers to an
extended guess and check strategy which eventually yields the first and
then other digits, rather than any blind guessing. Certainly it doesn't
look like there is any deterministic algorithm, so it must be something
else (thank goodness, publishers of the millions of sudoku books will say!!)
I consulted the oracle again on this and as you suggested she thinks
most (maybe all) advanced addicts do use a well developed strategy of
looking for the areas of maximum information (hence fewer choices)
pencilling in all the possibilities, scanning the nearby grid for
further foci of fewer choices, and thus step by step finally fixing on a
unique digit somewhere, this can take a lot of time and pencils but it
does (eventually) work, with patience and a clear head. Having seen this
process often I am sure there is an almost subconscious problem-solving
strategy at work which the solver may be using implicitly without
needing to even think about it (hence experience), such is the speed
sometimes of the pencil, and then moments of reflection, looking for the
next "useful" clues, but the uncertainty goes on a long time before the
first digit pops up at last - this is the fascination (and conversely
the repulsion) the game exerts on its addicts and non-addicts. It may be
other people guess a digit and then guess some others until they reach a
contradiction but this seems like a more wasteful strategy, better to
have many pencilled possible entries until that eureka moment reveals
the first unique digit...it seems. Players with very good memories may
not even need pencils...
Has anyone used sudoku in schools for the obvious merit it has in
practising arithmetic and reinforcing some basic logic and problem
solving? It would be interesting to know if it has any didactical use
and success?!
Some years ago we produced two booklets for the Maths in Society project
which were later published, one dealt with Magic and Numbers and
actually taught conjuring tricks as well as running through those
excellent number puzzles, for example the remarkable number 1089x9 =
9801 leads to an extraordinary depth of investigation which can make a
project for a week and ends up with the Fibonacci numbers and some nice
combinatorics on the way.
The other more relevant booklet introduced playing cards and after some
elementary discussion taught students to play Cribbage, an excellent and
popular card game for which success depends largely on probabilistic
thinking, which can be formalised and used to teach some introductory
probability - but (maybe like sudoku?) the motivation is also to learn
to play an exciting and skilful card game.
Alan
David H Kirshner wrote:
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> Alan and Dave,
> I guess I need to define what I mean by "blind guess." Of course, delimiting some possibilities and checking to see if you can eliminate all but one of them through logical analysis is a basic strategy. But by blind guessing I mean the strategy of logically delimiting some possibilities (usually two), substituting one of the possibilities and trying to do the remainder of the Sudoku puzzle to see if it leads to a contradiction. This is still a "logical approach," but requires only a very low level of reasoning, and is characterized by its tedium. I can't imagine becoming addicted to a game played like that. I assume the CLAIM that one need never resort to blind guessing refers to this procedure. But maybe addicts do submit themselves to just that mental abuse.
> 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 7:34 AM
> To: Mathematics Education discussion forum
> Subject: Re: [Maths-Education] Combinatorics and Sudoku
>
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> David H Kirshner wrote:
>
>> 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.
>>
>
> Dear David,
>
> The claim you mention is very interesting, if we forget about the very
> simple sudoku versions and concentrate on the very hard (the most recent
> we saw had NO numbers at all in it, but the clues were that the 9
> sub-squares, each contained the digits 1-9 AND there were inequality
> signs linking each small square to the next, quite fiendish even for
> Margaret my wife and Sudoku addict. For the hardest levels she, and
> everyone else we have seen, use a pencil to put in the invariable 2-3
> possibilities once they have started on the grid, and then move along by
> guess and check until they identify a definite digit, and then repeat
> the guess and check. There may be super-experts who can logically work
> out every digit one by one with no guessing, and maybe there is a neat
> proof that they CAN, assuming obviously that each problem has a unique
> solution (which I believe is one of the rules of the game, but see
> later), but life is too short I suspect for the addicts and they all
> seem to use guess and check, sometimes the whole grid can be covered
> with such pencilling provisional possibilities until a definite digit
> pops up! Experience in the puzzle obviously helps guide solvers as to
> where to start and what to do.
>
> The other thing is that the electronic versions Margaret has used also
> have this facility for guess and check so you can change previous
> selections, etc. My impression as a non-addict is that it would take
> maybe a long time and certainly a very clever brain indeed to work out
> the digits one by one without any guessing, unless we count some kind of
> subconscious guessing and checking, otherwise what are we doing (?), if
> it is possible to do the puzzle without guessing presumably *there is a
> deterministic algorithm to do it*, and that would immediately ruin the
> game and make it no fun at all?
>
> Maybe someone like Neil can helpfully point us to the webpage with all
> this worked out already to avoid any sleepless nights, of course the
> obvious thing to do is Google sudoku but I am too afraid of being sucked
> in to risk that...... wait...
>
> I have just checked Wikapedia and they give a very full and
> comprehensive discussion of the whole puzzle including this
>
> "The maximum number of givens provided while still not rendering a
> unique solution is four short of a full grid; if two instances of two
> numbers each are missing and the cells they are to occupy form the
> corners of an orthogonal rectangle, and exactly two of these cells are
> within one region, there are two ways the numbers can be assigned. Since
> this applies to Latin squares in general, most variants of /Sudoku/ have
> the same maximum. The inverse problem---the fewest givens that render a
> solution unique---is unsolved
> <http://en.wikipedia.org/wiki/Unsolved_problems_in_mathematics>,
> although the lowest number yet found for the standard variation without
> a symmetry constraint is 17, a number of which have been found by
> Japanese puzzle enthusiasts,^[11]
> <http://en.wikipedia.org/wiki/Sudoku#cite_note-seventeen1-10> ^[12]
> <http://en.wikipedia.org/wiki/Sudoku#cite_note-seventeen2-11> and 18
> with the givens in rotationally symmetric cells. Over 47,000 examples of
> Sudokus with 17 givens resulting in a unique solution are known."
>
> I couldn't find any mention of a way to *solve* a grid by deterministic
> logic or algorithm! But came across this frightening statistic related
> to the Number of grids:
> ^
> "The standard 3×3 calculation can be carried out in less than a second
> on a PC. The 3×4 (= 4×3) problem is much harder and took 2568 hours to
> solve, split over several computers. solution is
> "81171437193104932746936103027318645818654720000 = c. 8.1×10^46"
>
> Whow.
>
> Best wishes
> Alan
>
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