Open Problems in Sudoku Game pierre.douillet@ensait.fr cole - - PowerPoint PPT Presentation

Douillet Luminy 2006

✬ ✫ ✩ ✪ Luminy 2006

Open Problems in Sudoku Game

pierre.douillet@ensait.fr

École Nationale Supérieure des Arts et Industries Textiles Roubaix, France

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⇒ • deterministic solutions . . . . . . . . . . . . . 3

elementary definitions rules for solving taboo letter, taboo move totem cell, totem move minimal game

• probabilistic solutions . . . . . . . . . . . . .

9

• computer science issues

. . . . . . . . . . . . 14

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deterministic solutions

elementary definitions

4 5 6 3 2 6 5 2 4 6 2

• N = m × n. Here N = 3 × 2
• N letters and a N × N grid
• N rows, N columns
• N = n × m blocks, each

block being N = m × n sized

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rules for solving

• a group is either a row, a column or a block
• in a complete grid :

each group contains one letter of each kind and each letter appears once and only once in each group

• solving is not only completing, but also proving the unicity
• f the obtained completion

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taboo letter, taboo move

4 5 6 3 2 6 5 2 X 4 6 2

• a neighbor of cell x is any other

cell sharing a group with x

• letter λ is taboo for cell x when

λ is affected to a neighbor of x

• a taboo move affects a letter λ

to a cell x when all other letters are taboo for x

• here, all letters but 1 are taboo for cell X
• a taboo move like 1 → (4, 5) is not a guess, but a proof

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totem cell, totem move

4 6 5 Y 4 1 6 3 2 6 3 5 2 1 4 6 2 3 5 6

• cell x ∈ G is totem for the letter

λ and the group G when λ is taboo for all cells of G but x

• a totem move affects letter λ to

cell x knowing that λ is taboo for all the cells of G \ x

• here, cell Y is totem for letter 3 and block 1 (upper-left)
• a taboo move like 3 → (2, 1) is not a guess, but a proof

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minimal game

• a game is taboo-solvable when it can be proved by a

sequence of taboo moves

• a game is totem-solvable when it can be proved by a mix of

totem and taboo moves

• the size of a game is the number of published cells
• a minimal game is a game of a given kind with minimal

size, i.e. such that erasing any cell leads to what is no more a game of the given kind

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√ • deterministic solutions . . . . . . . . . . . . . 3 ⇒ • probabilistic solutions . . . . . . . . . . . . . 9

pivot and attempt proven wrong pick move

• pen mathematical questions (1)
• pen mathematical questions (2)
• computer science issues

. . . . . . . . . . . . 14

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probabilistic solutions

pivot and attempt

• assume that no more taboo nor totem moves are possible
• choose an empty cell (pivot) and a non taboo letter

(attempt) for that cell and then see what happen when applying all available totem/taboo moves

• it can happen that all letters become taboo for a given

empty cell (taboo error) or that a given letter becomes taboo for all the cells of a group (totem error).

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proven wrong

• a totem/taboo error is not a "bad thing" but a proof that

another attempt must be done for that pivot cell

• when all available totem/taboo moves don’t lead to

completion, another pivot cell is chosen, and the process is iterated

• if all non-taboo letters for a given pivot become proven

wrong (dead end), another attempt must be done for the previous pivot

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pick move

• dont forget : a (fair) game has one and only one

completion, and solving is proving the unicity

• a pick move consists into chosing an empty cell x (the

pivot), proving wrong all but one of the non taboo letters for that cell and finally affecting the remaining letter to the pivot cell

• any game is pick solvable, i.e. solvable by a mix of

totem/taboo and pick moves

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• pen mathematical questions (1)
• determine the number of (complete) square grids having a

given dimension N.

• given a grid, what can be said about all the associated

minimal taboo-solvable games ?

• given N, what is the minimal size of a taboo-solvable

game ?

• change taboo into totem in previous questions
• are there other powerful deterministic moves ?

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• pen mathematical questions (2)
• given N, what is the minimal size of a pick solvable game ?
• given a grid, what can be said about all the associated

minimal taboo-solvable games ?

• given a game, examine if all solutions contain the same

number of pick moves

• are there minimal totem/taboo solvable games that are also

minimal pick-solvable games ?

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computer science issues

• how to obtain other strategies ?

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