??????

????? ???????

?????? ????? ??????

????????? (? ????? : ?? , ???? ????) ??? ???? ?????? ??? ???? ???? ????? ?? ??? ????? ?????? (????) 9×9, ?????? ?-9 ??????? (???? ??? ??????? , ?? ?? ????) ??? 9 ?????? ?? ???. ???? ????? - ???? ?? 9 ?????? (???? ?????? 1 ?? 9) ?? ??? ??? ????? ?? ???? ???, ??? ????, ???? ????? ????? ?? ??? ????? ??? ???.

???? ????? ?????? ??? ?????, ??? ??????? ????? ??? ????? ?????? ????? ?????.

????? ??? ?????????? ? ??? ???? 1986 ?? ??????? , ? ???? ?? ????? ???? 2005 ?????? ?????? ????????.

?? ?????? ???????? ?????? ????? ?????? ?????? ?? ????? ?? ????? ??????? ?????? ^[1].

???????? [ ????? ??? ???? | ????? ]

? ??? ?-18 ??? ????????? ? ???????? ?????? ????? ??????? ???? ???? ??????? ???? ??? ???? ???? ?????. ??? ??? ???????? ???? ?????? ??? , ?? ????? ?????? ???? ?? ?????? ???????? ??????? ???????, ????? ???? ?? ??? ??????? ??????? . ???? ?????? ?? ??????? ??????? ?? ????? ??????? ? ????? ??? ????? ??? " Math Puzzles and Logic Problems " ("????? ??????? ?????? ??????") ? ???? ?-70 ?? ???? ?-20 .

???? ?????? ????? ??????? ?????? ?? 6 ????? 1895 ? ????? ? ????? La France ^[2]. ????? ???????? ??????? ????? ??????? ?- 1979 ???? ???????? ?? ?? ?????? ( Dell Magazines ) . ???? ?? ?????? ?? ??? ?????? ????? ( Howard Garns ) , ??????? ??????? ?? 74 ? ??? ??????? . ????? ?? ??? ???? ?? ??? ?????? ?????? , ??????? ?? ??????? ??? ?????? ??????? ????? ??? ???? ????????.

???? ???????, ??? ???? ?????? ????? ??? ???? ????, ????? ??????? ???? ???? 1984 ?????: "?????'? ??? ??????? ?? ??????" (?字は?身に限る) ?????? - " ????? ????? ???? " (=????? ???? ????)". ????? ???? ??? ?"??????", ?????? ?????? (???? ?? ????? ???? ????? ?? ?????? ????) "???? ????" (??).

??? ?????? ??? ???? ???? , ????? ???? ? ???? ???? , ???? ?????? ???? ??????? ?????? ???, ????? 2004 ??? ????? ? ????? " ?????? " ??????, ??? ??? ?????? ????? ???? ????.

???? ?????? ???????? ???? ????????? ?? ???????? ??????, ??? ???????? " New York Times " ?-" New York Post " ? ??? ???? . ?? ???????? ?????????? "??????????" ??"????? ??????? ????" ???? ?????? ?????? ??????.

???? 2005 ?? ??????? ??????? ???? ????? ?????? ?? ???? ????. ?????? ????? ?????? ?????? ??????? " ????? ??? ??? ", ??? ???? ????? ?????? ????????, ??? ??? "??? ?????". ?????? ?? ?????? ?? ???????? ?????? ??? ??? "?????? ????????".

?????? ????? [ ????? ??? ???? | ????? ]

???? ??? ????? ????? ?? ???? ?? 9×9 ???? (?? ?? ????? ?????? ?????), ???????? ???? ????? ?? 3×3 ????. ???? ?????? ?????? ?????, ??? ?????? ????? ?? ????? ?????? ?????? 1?9, ?? ???? ???, ??? ????, ???? ????, ?????? ?? ??????.

????? ????? [ ????? ??? ???? | ????? ]

????? ?? 9 ??????, ????? ???? ?????, ??????, ???? ????? ?? ????? ????? 5. ??? ????? ????? ??????, ????? 5 ??????? ????? ??????? ??? ?? ????? 5 ??????? ????? ?????? ?????? ???? ?? ????? 5 ??????? ???? ????? ????? ?????? ??? ?? ?? 7 ???? 9 ?????? ????? ??? ?????? ????? ?? ????? 5, ??? ??? ?????? ??? ?? ?? ???? ?? ??? ????? ????? ????? ??????. ???? ??? ??????? ??????? ??? ??? ?????, ??? ?? ????? 5 ???? ????? ?????? ?????? ??????, ?????? ????????.

??? ????? [ ????? ??? ???? | ????? ]

???? ?????? ??????? ??? ????? ?? ?????? ?????? ??????? (????=3X3 ???????) ??????? ??? ?????? ?????? ?????. ?????? ????? ????? ????? ???? ???? ????? ?? ????? 5 ????? ????? ??????. ???? ???? ????????? ?? ???????? ??? ????? ?? ???? ??????.

??? ??? [ ????? ??? ???? | ????? ]

???? ???? ?? ?? ???????, ?????? ???????? ???????? ????, ????? ?? ??????? ?????? ???? ????? ???? ?????? ?????? ????? ???. ????? ???? ????? ?? ??????, ????? ?? ?????? ???? ????? ??? (2 ?? 3) ??????? ?????, ?????? ???? ?????? ????? ??? ?????? ???????. ?? ???? ?????? ??? ?????? ???????? ?????? ??? ???????.

??? ????? [ ????? ??? ???? | ????? ]

??? ?????? ??? ????, ?? ???? ??????? ???? ??? ???? ???? ????? ?????? ??????. ????? ?? ???? ??? ???? ???? ??? ??????? ?? ???? ??????? ?????????? ?? ????? ????? ?????? ?? ????? ?????? . ?? ????? ???? ????? ????? ?????? ?? ??? ??????????. ???? ????? - ????? ?? ?? ??? (???? 2) ?????? ???????, ?????? ???? ??? ???? ??????? ?????? ???????? ?????. ?? ????? ?? ?????? ?? ??????? ?????? (????? ??????? ?????, ???? ?? ?????), ?? ????? ???? ????? ?? ?????. ????? ??? ????? ???? ?????? ????? ?????? ??????. ???? ?? ??? ?? ???? ????? ????.

????? ?????? [ ????? ??? ???? | ????? ]

???? ?????? ????? ????? ????? ?????? ??????? ???? ??? ?? ??? ???? ?? ?? ?????????, ?????? ?? ??? ?? ? ????? ???? . ??????? ?? ?????? ?????? ???????, ???? ?? ?? ????? 1, ??????? ??? ?????? ????, ????? ???? ????? ???????. ?? ??, ??????? ?????? ?????? ???? ??????? ?? ???? ?????; ?? ?? - ?? ??? ????????: ?? ????? ??????? ???? ?-9, ?????? ???? ?-1 ?????? ?????? ???????; ???? ?????? ?? ????? ???????, ?????? ???? ????? ??? ?? ????? ???? ?-9, ?????? ???? ?-1 ?????? ?????? ???????; ???? ?????? ?? ?????, ?????? ???? ?????, ??? ????. ???? ?? ????? ???????? ??? ????? ????, ????? ????? ?????? ???? ????? ?? ??????. ?? ?? ??? ???????? ????????? ??????? ?? ????? ????? ????? ????? ?????? ????, ?????? ?????? ?????? ???? ?? ?????? ????? (??? ?????????? ?????? ?????? ( ??' ) ).

????? ????? ??????? ???? ????? ????? ( $n^{2}$ ????? ??????? ??????? ????? $n\times n$ ) ????? ????? NP-???? .

??????? [ ????? ??? ???? | ????? ]

???? 2006 ??????? ???? ???? , ? ?????? ?????? ????? ??????? ?????? ??????. ???????? ??? ??? ????? ?????, ???? 15 (?????? ? ?????? ) ??? ??? 61 (?????? ???????). ??????? ?????? 85 ??? ?-22 ??????. ??????? ? ???? ????? ?'??? ?? 31 ??? ???? ??????? (Jana Tylova) ???? ???????; ??? ?????? ?????? ?? ????? ??????, ???? ?????????? ??????? ?? 26, ????? ????? ????, ??? ????-??-???? ?? 30, ????? ????? ? ???? , ????????? , ????? ????? ??????. ?? ????? ????? ?????? ????? ???? ????. ?????? ????? ?????? ??????? ???? 2007 ? ???? , ????? ??? ????? ?????? (Thomas Snyder, ????? ????? ???? ?-2006). ?????? ????? ??????? ??????? ?????? 2008 ? ???? , ???? ??? ????? ??????.

?????? ????? [ ????? ??? ???? | ????? ]

???? ?? 9×9 ??? ????? ?????, ???? ??????? ?? ????? ?? 4 ?????? ?? 2×2, ??? ????? ?? 4 ?????? ?? 5×5. ???? ??? ?????? ??? ?? 16×16 ?????? ?? 16 ?????? ?? 4×4. ?????? ?? ????? ???? ???????? ?? 6 ?????? ?? 2×3 ??? ????? ?? 8 ?????? ?? 2×4 ??? ????? ?? 12 ?????? ?? 3×4.

????? ?????? ????? ????? ????? ?? 9*9, ??????? ???? ??? ?????? ??????? ?? ??? ??????? (Sub Doku). ????? ?????? ????? ????? ?????? ?? ??? ???????? (Super Doku).

??? ?? ????? ?????? ??????? ???? ?? ???????:

?????? ??????? , ??? ??? ????????? ??? 9 ??????? ???? ???????. ???????? ?????? ????? ???? ???? ?????? ????????, ?? ??? ?? ?????.
?????? ??????? ?? ??????? ??? - ?????? ???????, ??? ???? ????? ?????? ?????? ?????? ?? ??? ????? ?? ???? ?? ??? ?????? ?????. ????? ?? ??? ???? ?? ???"? , ?????? ?? ??????? ?? ?????? .
?????? ???? , ????? ??????? ???????, ????? ???? ????? ?? ????? ?? 9 ??????, ?? ?? ??? ?? 9 ?????? ??????? ??? ^[3].
?????? ????? , ??? ??????? ????????, ?? ?? ????????? ??????? ?????. ??????, ??????? ??????? ???????? ?????? ?????, ???? ?????? ????? ?? ????.
?????? ???????? , ??? ?? ??? ????????? ??????? ?????? ?? ?????? 1 ?? 9.
?????? ??????? ?? ???????? - ????? ?? ?????? ??????? ?? ?????? ????????.
?????? ?????? , ???, ????? ????? ???????? ??????? 9 ?????? ?? ???, ?? ??????? ?????? ??? 9 ??????, ??? ?? ????, ??????? ??????? ??? ???? ???? ??????? ????? ?? ????? ????? ??????. ??? ??????? ?????? ?? ?????? ?????? ??? ???? ?????? ????? ????? ?????? ???????.
?????? ???? ??? , ??? ??????? ??????? ????? ???? ?? ??????? ??? ??????? ???????? ?????? ?????? ???? ??????? ???????? ?????? ?? ??????. ????? ???? ???? ?????? ??? ?????? ????? ???? ?????? ?????? ???? ^[4].
?????? ?????? ??????? ??? ????? ?? ??? ????? ?????? ???? ???? ?????, ????? ????? ???? ???????? ??? ?????? ??????? . ???? ?????? ???? ???? ???? ?????? ??? 54 ??????. ???? ?? ??????? ?? ????? ????? ?????? ??????? ???????, ????? ?????, ????? ????? ?????. ?????? ???? ????? ?? ?? ???? ?????? ?? ??? ?????.
?????? ?????? , ???? ????? ?? ????? ????? ??????, ???????? ??????. ??? ???? ??????? ?????? ?? ????? ????? ?? ?????? ???? ? ?????? .
?????? ????? , ??? ?? ????? ?? ?????? ?? ????? ??????. ??? ????, ??? ?????? 1 ?-9 ?????? ????? ??? ????? ??????, ???? ???? ???? ??????? ????? ????. ????? 1 ?? ????? ?? ?? ????? 2 ?????? 9 ?? ????? ?? ?? ????? 8.
?????? ??? ????? , ??? ?????? ?????? ? ??? ??????? - ???????? ?? ?? ????? ??????? ?????, ?????? ?? ??????, ?????? ?????? ?? ??? ???????.
?????? ?? ????? ????? ??? ?????? ????, ?? ?????? ??????? ?? ???? ?? ????? ?? ???? ?? ??? ?????? ??????, ????? ?????? ?? ???? ???????.
?????? ??????? , ??? ?? ????? ????? ???? ??? ????? ??????? ??????? ????? - ???? ?? ??? ???? ?????.
???"? , ???? ????? ?? ?????? ? ?????? . ???? "??????" ?? ??????, ???? ?? ???? ????? ?? ???? ?????? ??????? ????? ????. ???? ?? ??? ???? ????? ??????? ????? ?? ???????. ?? ?????? ??????? ?? ??????? ??? (??? ????) ??? ??? ?? ???"?.
???"? ???????? , ???? ??????? ?? ???"? ??? ????? ??????? ???????? ?????? ?????? ????????? ?? ??????? ???????? ????????????.
????? , ?????? ?? ?? ?????? , ???? ??????? ???? ?? ???"? ??? ?????? ?? ?????? ??????? ???? ?? ???? ?????.

?????? ?? ?????? ????? , ??? ?????? ?????? ??????? ?????? ?? ?? ????? 3 ?? 3 ?????? ??????? ?????? ??? ?? ?????? ???? ???? ???? ????? ????? ?? 5 ?? 5 ??????, ???? ??????.
?????? ????????? , ??? ??????, ???? ???????? ?? 4 ??????, ????? ?? 4 ?????? ????? ???? ????? ??????, ?? ?? ????? ????? ???? ???? ????? ????? ????? ?????.
?????? ??? ?????? ???? 9 ?? 9 ?? ????? ???? ????? 3 ?? 3 ?????? ????? ???, ?? ???? ?????? ??? ????, ??? ?????? ???.

?????? ???????? [ ????? ??? ???? | ????? ]

????? ?????? ??? ??? ?? ????? ????? , ???? ??? ???? ????. ?????? ???? ???? ???????? ??????? ???? ??????? ???????, ???? ?????? ?????? ???? ????? ????, ????? (???????). ???? ????????? ??????? ?? 9X9 ??? 6,670,903,752,021,072,936,960 ^[5]. ???? ?? ???? ? $\ 9!\times 72^{2}\times 2^{7}\times 27,704,267,971$ ???? ????? ?????? ??? ??????. ????? ?? ????? ?????? ???????????? ?????? ?????? " ??? ?? " ??????? ????.

?? ???? ???????? ????? ????? (?? ????) ?? ? ?????? $\ S_{9}$ ( ????? ?? ??????), $\ (S_{3}\cdot (S_{3})^{3})^{2}$ (????? ??? ?????? ?? ??????? ??? ????, ?????? ???????) ?- $\ D_{4}$ (???????? ?? ??????) ^[6]. ???? ?????? ??????? ???? ?????? ???, ??? 5,472,730,538. ???, ?? ??, ???? ???????? ?????? ??????? ?? ???.

????? ???????? ?? ?????? ????? ???? ???? ??????? ???? ??????? ??? ???? ???? ???? ???? 4. ????: ?? ????? ??? ?????? ????? ??? ????? ?? ?????? ???????, ??? ?????? ??? ????? ?????. ????? ???????? ?? ?????? ??? ???? ?? ??? ??????? ???? ????? ?????? ???? ???? ?????? ??? 17, ??? ?? ???? ?-2012 ?? ??? ????? ????? ???? ? ????? ??????? ???? . ?????? ??????? ?? ???? ?????? ??.

??? ?? [ ????? ??? ???? | ????? ]

???????

?????? ????? [ ????? ??? ???? | ????? ]

???? ???????, ?????? ?????? ????? , ????? ???? , 2006

Jason Rosenhouse and Laura Taalman, Taking Sudoku Seriously , Oxford University Press , 2011.

??????? ???????? [ ????? ??? ???? | ????? ]

???? ??????? ????????
???????? ??????? ?????? ????????? - ????? ?????
???????? ?? ?????? ???? ?? ????? ???? .
?????? ???? - ?????? ???????? ?????? ???? ??? ??? ??? ??????
?????? , ???? MathWorld (???????)
??????? ?????? ???? ????? (???????)
?????? , ???? ???'?, "???? ???????", ?? ???? 69 , ???? ???? , 17 ??????? 2021
?????? ??????? - ???? ????? ?????? ?????
?????? , ???? ??????????? ???????? (???????)

Jean-Paul Delahaye, "The Science Behind Sudoku" , Scientific American , June 2006.
J. Scott Provan, Sudoku: Strategy Versus Structure , American Mathematical Monthly , October 2009
Sudoku Solver

????? ?????? [ ????? ??? ???? | ????? ]

^ ?????? ??????? ?????? ?????? ????? ????? ????? (???????)
^ Sudoku's French ancestors
^ ?????? ???????, ?????? ???? ???? ???? ?? ????? ????
^ ?????? ???? ??? ???? ?? ????? ????
^ Bertram Felgenhauer and Frazier Jarvis, "Mathematics of Sudoku, I", Mathematical Spectrum, Vol. 39(1), 2006-2007, pp. 15-22.
^ ????? ?? ???? ????? ???? ?????? ???? ????? ?? ???? ?????? ?????? ????? ?? ??????.

[1] ?????? ??????? ?????? ?????? ????? ????? ????? (???????)

[2] Sudoku's French ancestors

[3] ?????? ???????, ?????? ???? ???? ???? ?? ????? ????

[4] ?????? ???? ??? ???? ?? ????? ????

[5] Bertram Felgenhauer and Frazier Jarvis, "Mathematics of Sudoku, I", Mathematical Spectrum, Vol. 39(1), 2006-2007, pp. 15-22.

[6] ????? ?? ???? ????? ???? ?????? ???? ????? ?? ???? ?????? ?????? ????? ?? ??????.

[1]

[2]

[3]

[4]

[5]

[6]