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 ✫ ✪ Ensait - Roubaix 1

✬ ✩ Douillet Luminy 2006 ⇒ • 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 ✫ ✪ Ensait - Roubaix 2

✬ ✩ Douillet Luminy 2006 deterministic solutions elementary definitions 4 5 • N = m × n . Here N = 3 × 2 • N letters and a N × N grid 6 3 2 • N rows, N columns 6 5 2 4 • N = n × m blocks, each 6 2 block being N = m × n sized ✫ ✪ Ensait - Roubaix 3

✬ ✩ Douillet Luminy 2006 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 of the obtained completion ✫ ✪ Ensait - Roubaix 4

✬ ✩ Douillet Luminy 2006 taboo letter, taboo move • a neighbor of cell x is any other 4 5 cell sharing a group with x • letter λ is taboo for cell x when 6 3 2 λ is affected to a neighbor of x 6 5 2 X 4 • a taboo move affects a letter λ 6 2 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 ✫ ✪ Ensait - Roubaix 5

✬ ✩ Douillet Luminy 2006 totem cell, totem move 4 6 5 • cell x ∈ G is totem for the letter Y 4 1 λ and the group G when λ is taboo for all cells of G but x 6 3 2 • a totem move affects letter λ to 6 3 5 2 1 4 cell x knowing that λ is taboo 6 2 3 for all the cells of G \ x 5 6 • 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 ✫ ✪ Ensait - Roubaix 6

✬ ✩ Douillet Luminy 2006 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 ✫ ✪ Ensait - Roubaix 7

✬ ✩ Douillet Luminy 2006 √ • deterministic solutions . . . . . . . . . . . . . 3 ⇒ • probabilistic solutions . . . . . . . . . . . . . 9 pivot and attempt proven wrong pick move open mathematical questions (1) open mathematical questions (2) • computer science issues . . . . . . . . . . . . 14 ✫ ✪ Ensait - Roubaix 8

✬ ✩ Douillet Luminy 2006 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). ✫ ✪ Ensait - Roubaix 9

✬ ✩ Douillet Luminy 2006 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 ✫ ✪ Ensait - Roubaix 10

✬ ✩ Douillet Luminy 2006 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 ✫ ✪ Ensait - Roubaix 11

✬ ✩ Douillet Luminy 2006 open 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 ? ✫ ✪ Ensait - Roubaix 12

✬ ✩ Douillet Luminy 2006 open 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 ? ✫ ✪ Ensait - Roubaix 13

✬ ✩ Douillet Luminy 2006 computer science issues • how to obtain other strategies ? ✫ ✪ Ensait - Roubaix 14