Is there a Sudoku Principle?
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From: Ludovicus on 21 Jun 2010 10:04 As there is a Pigeon-hole Principle, I think a new util principle can be based in the practice of Sudoku: "If n cells must be filled with n elements and n-1 cells are prohibited to one of those, then that element must fill the free cell." Ludovicus
From: Arturo Magidin on 21 Jun 2010 12:41 On Jun 21, 9:04 am, Ludovicus <luir...(a)yahoo.com> wrote: > As there is a Pigeon-hole Principle, I think a new util principle can > be based > in the practice of Sudoku: > "If n cells must be filled with n elements and n-1 cells are > prohibited to > one of those, then that element must fill the free cell." > Ludovicus "Once the impossible has been eliminated, whatever remains, however unlikely, must be the truth" seems to have anticipated you by 100+ years... -- Arturo Magidin
From: Herman Jurjus on 21 Jun 2010 14:56 On 6/21/2010 6:41 PM, Arturo Magidin wrote: > On Jun 21, 9:04 am, Ludovicus<luir...(a)yahoo.com> wrote: >> As there is a Pigeon-hole Principle, I think a new util principle can >> be based >> in the practice of Sudoku: >> "If n cells must be filled with n elements and n-1 cells are >> prohibited to >> one of those, then that element must fill the free cell." >> Ludovicus > > "Once the impossible has been eliminated, whatever remains, however > unlikely, must be the truth" seems to have anticipated you by 100+ > years... More like 2200+ years (Chrysippus' dog). -- Cheers, Herman Jurjus
From: Virgil on 21 Jun 2010 16:58 In article <b490a8f8-008e-40df-a322-9c7b64a0739f(a)i31g2000yqm.googlegroups.com>, Arturo Magidin <magidin(a)member.ams.org> wrote: > On Jun 21, 9:04�am, Ludovicus <luir...(a)yahoo.com> wrote: > > As there is a Pigeon-hole Principle, I think a new util principle can > > be based > > in the practice of Sudoku: > > "If n cells must be filled with n elements and n-1 cells are > > prohibited to > > one of those, then that element must fill the free cell." > > Ludovicus > > "Once the impossible has been eliminated, whatever remains, however > unlikely, must be the truth" seems to have anticipated you by 100+ > years... > > -- > Arturo Magidin But the quote did not appear in quite such a mathematical context.
From: Rock Brentwood on 21 Jun 2010 18:04
On Jun 21, 9:04 am, Ludovicus <luir...(a)yahoo.com> wrote: > As there is a Pigeon-hole Principle, I think a new util principle can > be based > in the practice of Sudoku: > "If n cells must be filled with n elements and n-1 cells are > prohibited to0 > one of those, then that element must fill the free cell." > Ludovicus Conjecture: If all the digits are used, but one, and are all correctly filled in, then filling in the last digit will lead to a correct solution. A Sudoku grid is a 4-D array that embodies the function S: (a,b,c,d) in U x V x V x U -> S(a,b,c,d) in U x V The version of the game normally seen has the sets U, V of sizes 3 and 3, so U x V can be expressed as 9 elements, normally taken as the 9 digits. In this representation (a,b) mark a row, (c,d) a column on the large grid; (a,c) the (row,column) in a cell. The criteria are that S_{ab} = S(a,b,_,_), S_{cd} = S(_,_,c,d), S_{ac} = S(a,_,c,_) each be one-to-one functions between U x V and U x V. So, if all the digits are used but one, and all are correctly filled out, then each of the above one-to-one functions S_{ab} for all (a,b) in UxV, S_{cd} for all (c,d) in VxU, and S_{ac} for all (a,c) in UxV will be correct in all but one place. The Pigeonhole Principle can then be used. This is all off the cuff. You may want to take the time to carefully verify these claims. So, regard my statement as a conjecture, until someone writes a formal proof. |