Судоку

02 Dec 2013

Мне тут попалось судоку, которое я не мог решить без полного перебора вариантов. Когда было сделано несколько итераций, то стало понятно, что решения можно найти лишь перебирая все варианты. Нельзя было найти ни одной подсказки.

Вот оно:

  7         4   5
      6 8 9      
          4      
9   5   6       7
  2       7   3  
6 3              
3   1            
      3     8    
4     8     6    

Стало интересно: может быть где-то ошибка в поиске решения? Все же человек не машина, а следовательно ошибается чаще.

Что мы знаем про простейшие решения судоку?

Исходники

Знаем мы несколько паттернов:

Есть еще одно правило, которое можно пронаблюдать на примере.

Формулируется следующим образом:

Непонятно. Да. Непонятно. Но всегда можно посмотреть на примере.

Разместим все возможные варианты цифры 5

  7         4   5
5 5   6 8 9      
5 5   5 5 4      
9   5   6       7
  2   5 5 7 5 3  
6 3   5 5 5 5 5  
3 5 1 5 5 5 5 5  
5 5   3 5 5 8 5  
4 5   8 5 5 6 5  

Очевидно, что пятерки, которые помечены красным существовать не могут.

Что в итоге? Напишем программу, которая пользуется вышеприведенными аксиомами и увидим, что иногда без полного перепора судоку не решается. :)

Вывод приложения:

```text; gutter: true; first-line: 1; highlight: [] before scan [0, 7, 0, 0, 0, 0, 4, 0, 5] [0, 0, 0, 6, 8, 9, 0, 0, 0] [0, 0, 0, 0, 0, 4, 0, 0, 0] [9, 0, 5, 0, 6, 0, 0, 0, 7] [0, 2, 0, 0, 0, 7, 0, 3, 0] [6, 3, 0, 0, 0, 0, 0, 0, 0] [3, 0, 1, 0, 0, 0, 0, 0, 0] [0, 0, 0, 3, 0, 0, 8, 0, 0] [4, 0, 0, 8, 0, 0, 6, 0, 0] suggest for board [[1, 2, 8], 7, [2, 3, 6, 8, 9], [1, 2], [1, 2, 3], [1, 2], 4, [1, 2, 6, 8, 9], 5] [[1, 2, 5], [1, 4, 5], [2, 3, 4], 6, 8, 9, [1, 2, 3, 7], [1, 2, 7], [1, 2]] [[1, 2, 8], [1, 6, 9], [2, 3, 6, 8, 9], [1, 2, 5, 7], [1, 2, 3, 5, 7], 4, [1, 2, 3, 9], [1, 2, 6, 8, 9], [1, 2, 8, 9]] [9, [1, 4], 5, [1, 2, 4], 6, [1, 2, 3, 8], [1, 2], [1, 2, 4, 8], 7] [[1, 8], 2, [4, 8], [1, 4, 5, 9], [1, 4, 5, 9], 7, [1, 5, 9], 3, [1, 4, 6, 8, 9]] [6, 3, [4, 7, 8], [1, 2, 4, 5, 9], [1, 2, 4, 5, 9], [1, 2, 5, 8], [1, 2, 5, 9], [1, 2, 4, 5, 8, 9], [1, 2, 4, 8, 9]] [3, [5, 6, 8, 9], 1, [2, 4, 5, 7, 9], [2, 4, 5, 7, 9], [2, 5, 6], [2, 5, 7, 9], [2, 4, 5, 7, 9], [2, 4, 9]] [[2, 5, 7], [5, 6, 9], [2, 6, 9], 3, [1, 2, 4, 5, 9], [1, 2, 5, 6], 8, [1, 2, 4, 5, 9], [1, 2, 4, 9]] [4, [5, 9], [2, 9], 8, [1, 2, 5, 7, 9], [1, 2, 5], 6, [1, 2, 5, 7, 9], [1, 2, 3, 9]] description [((5, 2), 7), ((3, 5), 3), ((4, 8), 6), ((7, 0), 7), ((6, 1), 8), ((8, 8), 3)] after solve [0, 7, 0, 0, 0, 0, 4, 0, 5] [0, 0, 0, 6, 8, 9, 0, 0, 0] [0, 0, 0, 0, 0, 4, 0, 0, 0] [9, 0, 5, 0, 6, 3, 0, 0, 7] [0, 2, 0, 0, 0, 7, 0, 3, 6] [6, 3, 7, 0, 0, 0, 0, 0, 0] [3, 8, 1, 0, 0, 0, 0, 0, 0] [7, 0, 0, 3, 0, 0, 8, 0, 0] [4, 0, 0, 8, 0, 0, 6, 0, 3] before scan [0, 7, 0, 0, 0, 0, 4, 0, 5] [0, 0, 0, 6, 8, 9, 0, 0, 0] [0, 0, 0, 0, 0, 4, 0, 0, 0] [9, 0, 5, 0, 6, 3, 0, 0, 7] [0, 2, 0, 0, 0, 7, 0, 3, 6] [6, 3, 7, 0, 0, 0, 0, 0, 0] [3, 8, 1, 0, 0, 0, 0, 0, 0] [7, 0, 0, 3, 0, 0, 8, 0, 0] [4, 0, 0, 8, 0, 0, 6, 0, 3] suggest for board [[1, 2, 8], 7, [3, 6, 8, 9], [1, 2], [1, 2, 3], [1, 2], 4, [1, 2, 6, 9], 5] [[1, 2, 5], [1, 4], [3, 4], 6, 8, 9, [1, 2, 3, 7], [1, 2, 7], [1, 2]] [[1, 2, 8], [1, 6, 9], [3, 6, 8, 9], [1, 2, 5, 7], [1, 2, 3, 5, 7], 4, [1, 2, 3, 9], [1, 2, 6, 9], [1, 2, 8, 9]] [9, [1, 4], 5, [1, 2, 4], 6, 3, [1, 2], [1, 2, 4, 8], 7] [[1, 8], 2, [4, 8], [1, 4, 5, 9], [1, 4, 5, 9], 7, [1, 5, 9], 3, 6] [6, 3, 7, [1, 2, 4, 5, 9], [1, 2, 4, 5, 9], [1, 2, 5, 8], [1, 2, 5, 9], [1, 2, 4, 5, 9], [1, 2, 4, 9]] [3, 8, 1, [2, 4, 5, 7, 9], [2, 4, 5, 7, 9], [2, 5, 6], [2, 5, 7, 9], [2, 4, 5, 7, 9], [2, 4, 9]] [7, [5, 6, 9], [2, 6, 9], 3, [1, 2, 4, 5, 9], [1, 2, 5], 8, [1, 2, 4, 5, 9], [1, 2, 4, 9]] [4, [5, 9], [2, 9], 8, [1, 2, 5, 7, 9], [1, 2, 5], 6, [1, 2, 5, 7, 9], 3] description [((1, 0), 5), ((2, 8), 8), ((5, 5), 8), ((3, 7), 8), ((6, 5), 6)] after solve [0, 7, 0, 0, 0, 0, 4, 0, 5] [5, 0, 0, 6, 8, 9, 0, 0, 0] [0, 0, 0, 0, 0, 4, 0, 0, 8] [9, 0, 5, 0, 6, 3, 0, 8, 7] [0, 2, 0, 0, 0, 7, 0, 3, 6] [6, 3, 7, 0, 0, 8, 0, 0, 0] [3, 8, 1, 0, 0, 6, 0, 0, 0] [7, 0, 0, 3, 0, 0, 8, 0, 0] [4, 0, 0, 8, 0, 0, 6, 0, 3] before scan [0, 7, 0, 0, 0, 0, 4, 0, 5] [5, 0, 0, 6, 8, 9, 0, 0, 0] [0, 0, 0, 0, 0, 4, 0, 0, 8] [9, 0, 5, 0, 6, 3, 0, 8, 7] [0, 2, 0, 0, 0, 7, 0, 3, 6] [6, 3, 7, 0, 0, 8, 0, 0, 0] [3, 8, 1, 0, 0, 6, 0, 0, 0] [7, 0, 0, 3, 0, 0, 8, 0, 0] [4, 0, 0, 8, 0, 0, 6, 0, 3] suggest for board [[1, 2, 8], 7, [3, 6, 8, 9], [1, 2], [1, 2, 3], [1, 2], 4, [1, 2, 6, 9], 5] [5, [1, 4], [3, 4], 6, 8, 9, [1, 2, 3, 7], [1, 2, 7], [1, 2]] [[1, 2], [1, 6, 9], [3, 6, 9], [1, 2, 5, 7], [1, 2, 3, 5, 7], 4, [1, 2, 3, 9], [1, 2, 6, 9], 8] [9, [1, 4], 5, [1, 2, 4], 6, 3, [1, 2], 8, 7] [[1, 8], 2, [4, 8], [1, 4, 5, 9], [1, 4, 5, 9], 7, [1, 5, 9], 3, 6] [6, 3, 7, [1, 2, 5, 9], [1, 2, 5, 9], 8, [1, 2, 5, 9], [1, 2, 4, 5, 9], [1, 2, 4, 9]] [3, 8, 1, [2, 4, 5, 7, 9], [2, 4, 5, 7, 9], 6, [2, 5, 7, 9], [2, 4, 5, 7, 9], [2, 4, 9]] [7, [5, 6, 9], [2, 6, 9], 3, [1, 2, 4, 5, 9], [1, 2, 5], 8, [1, 2, 4, 5, 9], [1, 2, 4, 9]] [4, [5, 9], [2, 9], 8, [1, 2, 5, 7, 9], [1, 2, 5], 6, [1, 2, 5, 7, 9], 3] description [] Sorry. couldn’t solve this


На месте пустых (они же нулевые) значений печатается подсказка по тому, какой элемент может быть на этом месте.

Description - это список подстановок элемента с индексами в базовой матрице.

И конечно решение на другом примере.

```text; gutter: true; first-line: 1; highlight: []
before scan [0, 0, 5, 0, 6, 0, 0, 0, 0] [0, 0, 7, 0, 0, 0, 9, 1, 0] [2, 0, 0, 0, 5, 4, 0, 0, 0] [0, 8, 0, 0, 0, 0, 6, 3, 0] [0, 0, 0, 6, 0, 0, 1, 8, 0] [0, 2, 0, 3, 0, 0, 0, 0, 0] [6, 0, 0, 0, 8, 2, 0, 0, 0] [0, 0, 0, 0, 0, 0, 3, 0, 4] [7, 0, 9, 0, 0, 0, 0, 0, 0] suggest for board [[1, 3, 8, 9], [1, 3, 9], 5, [1, 2, 7, 8, 9], 6, [1, 3, 7, 9], [2, 4, 7, 8], [2, 4, 7], [3, 7, 8]] [[3, 4, 8], [3, 4, 6], 7, [2, 8], [2, 3], [3], 9, 1, [3, 5, 6, 8]] [2, [1, 3, 6, 9], [1, 3, 8], [1, 7, 8, 9], 5, 4, [7, 8], [6, 7], [3, 6, 7, 8]] [[1, 4, 5, 9], 8, [1, 4], [1, 2, 4, 5, 7, 9], [1, 2, 4, 7, 9], [1, 5, 7, 9], 6, 3, [2, 7, 9]] [[3, 4, 5, 9], [3, 4, 5, 7, 9], [3, 4], 6, [2, 4, 9], [5, 9], 1, 8, [2, 9]] [[1, 9], 2, [1, 6], 3, [1, 7, 9], [1, 7, 8, 9], [4, 5, 7], [4, 5, 7, 9], [7, 9]] [6, [1, 3, 4, 5], [1, 3, 4], [1, 4, 5, 7, 9], 8, 2, [5, 7], [5, 7, 9], [1, 7, 9]] [[1, 5, 8], [1, 5], [1, 2, 8], [1, 5, 7, 9], [1, 7, 9], [1, 5, 6, 7, 9], 3, [5, 6, 7, 9], 4] [7, [1, 4, 5], 9, [1, 4, 5], [1, 3, 4], [1, 3, 5, 6], [2, 5, 8], [2, 5, 6], [1, 6, 8]] description [((1, 5), 3), ((1, 8), 5), ((5, 2), 6), ((4, 1), 7), ((5, 5), 8), ((7, 2), 2)] after solve [0, 0, 5, 0, 6, 0, 0, 0, 0] [0, 0, 7, 0, 0, 3, 9, 1, 5] [2, 0, 0, 0, 5, 4, 0, 0, 0] [0, 8, 0, 0, 0, 0, 6, 3, 0] [0, 7, 0, 6, 0, 0, 1, 8, 0] [0, 2, 6, 3, 0, 8, 0, 0, 0] [6, 0, 0, 0, 8, 2, 0, 0, 0] [0, 0, 2, 0, 0, 0, 3, 0, 4] [7, 0, 9, 0, 0, 0, 0, 0, 0] before scan [0, 0, 5, 0, 6, 0, 0, 0, 0] [0, 0, 7, 0, 0, 3, 9, 1, 5] [2, 0, 0, 0, 5, 4, 0, 0, 0] [0, 8, 0, 0, 0, 0, 6, 3, 0] [0, 7, 0, 6, 0, 0, 1, 8, 0] [0, 2, 6, 3, 0, 8, 0, 0, 0] [6, 0, 0, 0, 8, 2, 0, 0, 0] [0, 0, 2, 0, 0, 0, 3, 0, 4] [7, 0, 9, 0, 0, 0, 0, 0, 0] suggest for board [[1, 3], [1, 3, 9], 5, [1, 7, 8, 9], 6, [1, 7, 9], [2, 4, 7, 8], [2, 4, 7], [3, 7, 8]] [[4], [4, 6], 7, [2, 8], [2], 3, 9, 1, 5] [2, [1, 3, 9], [1, 3, 8], [1, 7, 8, 9], 5, 4, [7, 8], [6, 7], [3, 6, 7, 8]] [[1, 4, 5, 9], 8, [1, 4], [1, 2, 4, 5, 7, 9], [1, 2, 4, 7, 9], [1, 5, 7, 9], 6, 3, [2, 7, 9]] [[3, 4, 5, 9], 7, [3, 4], 6, [2, 4, 9], [5, 9], 1, 8, [2, 9]] [[1, 9], 2, 6, 3, [1, 7, 9], 8, [4, 5, 7], [4, 5, 7, 9], [7, 9]] [6, [1, 3, 4, 5], [1, 3, 4], [1, 4, 5, 7, 9], 8, 2, [5, 7], [5, 7, 9], [1, 7, 9]] [[1, 8], [1, 5], 2, [1, 5, 7, 9], [1, 7, 9], [1, 5, 6, 7, 9], 3, [5, 6, 7, 9], 4] [7, [1, 4, 5], 9, [1, 4, 5], [1, 3, 4], [1, 5, 6], [2, 5, 8], [2, 5, 6], [1, 6, 8]] description [((1, 0), 4), ((1, 1), 6), ((2, 2), 8), ((1, 4), 2), ((7, 0), 8), ((8, 4), 3)] after solve [0, 0, 5, 0, 6, 0, 0, 0, 0] [4, 6, 7, 0, 2, 3, 9, 1, 5] [2, 0, 8, 0, 5, 4, 0, 0, 0] [0, 8, 0, 0, 0, 0, 6, 3, 0] [0, 7, 0, 6, 0, 0, 1, 8, 0] [0, 2, 6, 3, 0, 8, 0, 0, 0] [6, 0, 0, 0, 8, 2, 0, 0, 0] [8, 0, 2, 0, 0, 0, 3, 0, 4] [7, 0, 9, 0, 3, 0, 0, 0, 0] before scan [0, 0, 5, 0, 6, 0, 0, 0, 0] [4, 6, 7, 0, 2, 3, 9, 1, 5] [2, 0, 8, 0, 5, 4, 0, 0, 0] [0, 8, 0, 0, 0, 0, 6, 3, 0] [0, 7, 0, 6, 0, 0, 1, 8, 0] [0, 2, 6, 3, 0, 8, 0, 0, 0] [6, 0, 0, 0, 8, 2, 0, 0, 0] [8, 0, 2, 0, 0, 0, 3, 0, 4] [7, 0, 9, 0, 3, 0, 0, 0, 0] suggest for board [[1, 3], [1, 3, 9], 5, [1, 7, 9], 6, [1, 7, 9], [2, 4, 7, 8], [2, 4, 7], [3, 7, 8]] [4, 6, 7, [8], 2, 3, 9, 1, 5] [2, [1, 3, 9], 8, [1, 7, 9], 5, 4, [7], [6, 7], [3, 6, 7]] [[1, 5, 9], 8, [1, 4], [1, 2, 5, 7, 9], [1, 4, 7, 9], [1, 5, 7, 9], 6, 3, [7, 9]] [[3, 5, 9], 7, [3, 4], 6, [4, 9], [5, 9], 1, 8, [2, 9]] [[1, 9], 2, 6, 3, [1, 7, 9], 8, [4, 5, 7], [4, 5, 7, 9], [7, 9]] [6, [1, 3, 4, 5], [1, 3], [1, 4, 5, 7, 9], 8, 2, [5, 7], [5, 7, 9], [1, 7, 9]] [8, [1, 5], 2, [1, 5, 7, 9], [1, 7, 9], [1, 5, 6, 7, 9], 3, [5, 6, 7, 9], 4] [7, [1, 4, 5], 9, [1, 4, 5], 3, [1, 5, 6], [2, 5, 8], [2, 5, 6], [1, 6, 8]] description [((1, 3), 8), ((1, 3), 8), ((2, 6), 7), ((3, 3), 2), ((4, 8), 2)] after solve [0, 0, 5, 0, 6, 0, 0, 0, 0] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, 0, 8, 0, 5, 4, 7, 0, 0] [0, 8, 0, 2, 0, 0, 6, 3, 0] [0, 7, 0, 6, 0, 0, 1, 8, 2] [0, 2, 6, 3, 0, 8, 0, 0, 0] [6, 0, 0, 0, 8, 2, 0, 0, 0] [8, 0, 2, 0, 0, 0, 3, 0, 4] [7, 0, 9, 0, 3, 0, 0, 0, 0] before scan [0, 0, 5, 0, 6, 0, 0, 0, 0] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, 0, 8, 0, 5, 4, 7, 0, 0] [0, 8, 0, 2, 0, 0, 6, 3, 0] [0, 7, 0, 6, 0, 0, 1, 8, 2] [0, 2, 6, 3, 0, 8, 0, 0, 0] [6, 0, 0, 0, 8, 2, 0, 0, 0] [8, 0, 2, 0, 0, 0, 3, 0, 4] [7, 0, 9, 0, 3, 0, 0, 0, 0] suggest for board [[1, 3], [1, 3, 9], 5, [1, 7, 9], 6, [1, 7, 9], [2, 4, 8], [2, 4], [3, 8]] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, [1, 3, 9], 8, [1, 9], 5, 4, 7, [6], [3, 6]] [[1, 5, 9], 8, [1, 4], 2, [1, 4, 7, 9], [1, 5, 7, 9], 6, 3, [7, 9]] [[3, 5, 9], 7, [3, 4], 6, [4, 9], [5, 9], 1, 8, 2] [[1, 9], 2, 6, 3, [1, 7, 9], 8, [4, 5], [4, 5, 7, 9], [7, 9]] [6, [1, 3, 4, 5], [1, 3], [1, 4, 5, 7, 9], 8, 2, [5], [5, 7, 9], [1, 7, 9]] [8, [1, 5], 2, [1, 5, 7, 9], [1, 7, 9], [1, 6, 7, 9], 3, [5, 6, 7, 9], 4] [7, [1, 4, 5], 9, [1, 4, 5], 3, [1, 6], [2, 5, 8], [2, 5, 6], [1, 6, 8]] description [((2, 7), 6), ((6, 6), 5)] after solve [0, 0, 5, 0, 6, 0, 0, 0, 0] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, 0, 8, 0, 5, 4, 7, 6, 0] [0, 8, 0, 2, 0, 0, 6, 3, 0] [0, 7, 0, 6, 0, 0, 1, 8, 2] [0, 2, 6, 3, 0, 8, 0, 0, 0] [6, 0, 0, 0, 8, 2, 5, 0, 0] [8, 0, 2, 0, 0, 0, 3, 0, 4] [7, 0, 9, 0, 3, 0, 0, 0, 0] before scan [0, 0, 5, 0, 6, 0, 0, 0, 0] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, 0, 8, 0, 5, 4, 7, 6, 0] [0, 8, 0, 2, 0, 0, 6, 3, 0] [0, 7, 0, 6, 0, 0, 1, 8, 2] [0, 2, 6, 3, 0, 8, 0, 0, 0] [6, 0, 0, 0, 8, 2, 5, 0, 0] [8, 0, 2, 0, 0, 0, 3, 0, 4] [7, 0, 9, 0, 3, 0, 0, 0, 0] suggest for board [[1, 3], [1, 3, 9], 5, [1, 7, 9], 6, [1, 7, 9], [2, 4, 8], [2, 4], [3, 8]] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, [1, 3, 9], 8, [1, 9], 5, 4, 7, 6, [3]] [[1, 5, 9], 8, [1, 4], 2, [1, 4, 7, 9], [1, 5, 7, 9], 6, 3, [7, 9]] [[3, 5, 9], 7, [3, 4], 6, [4, 9], [5, 9], 1, 8, 2] [[1, 9], 2, 6, 3, [1, 7, 9], 8, [4], [4, 5, 7, 9], [7, 9]] [6, [1, 3, 4], [1, 3], [1, 4, 7, 9], 8, 2, 5, [7, 9], [1, 7, 9]] [8, [1, 5], 2, [1, 5, 7, 9], [1, 7, 9], [1, 6, 7, 9], 3, [7, 9], 4] [7, [1, 4, 5], 9, [1, 4, 5], 3, [1], [2, 8], [2], [1, 6, 8]] description [((2, 8), 3), ((5, 6), 4), ((5, 7), 5), ((8, 5), 1), ((7, 5), 6), ((8, 7), 2), ((8, 8), 6)] after solve [0, 0, 5, 0, 6, 0, 0, 0, 0] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, 0, 8, 0, 5, 4, 7, 6, 3] [0, 8, 0, 2, 0, 0, 6, 3, 0] [0, 7, 0, 6, 0, 0, 1, 8, 2] [0, 2, 6, 3, 0, 8, 4, 5, 0] [6, 0, 0, 0, 8, 2, 5, 0, 0] [8, 0, 2, 0, 0, 6, 3, 0, 4] [7, 0, 9, 0, 3, 1, 0, 2, 6] before scan [0, 0, 5, 0, 6, 0, 0, 0, 0] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, 0, 8, 0, 5, 4, 7, 6, 3] [0, 8, 0, 2, 0, 0, 6, 3, 0] [0, 7, 0, 6, 0, 0, 1, 8, 2] [0, 2, 6, 3, 0, 8, 4, 5, 0] [6, 0, 0, 0, 8, 2, 5, 0, 0] [8, 0, 2, 0, 0, 6, 3, 0, 4] [7, 0, 9, 0, 3, 1, 0, 2, 6] suggest for board [[1, 3], [1, 3, 9], 5, [1, 7, 9], 6, [7, 9], [2], [4], [8]] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, [1, 9], 8, [1, 9], 5, 4, 7, 6, 3] [[1, 5, 9], 8, [1, 4], 2, [1, 4, 7, 9], [5, 7, 9], 6, 3, [7, 9]] [[3, 5, 9], 7, [3, 4], 6, [4, 9], [5, 9], 1, 8, 2] [[1, 9], 2, 6, 3, [1, 7, 9], 8, 4, 5, [7, 9]] [6, [3, 4], [3], [4, 7, 9], 8, 2, 5, [7, 9], [1]] [8, [1, 5], 2, [5, 7, 9], [7, 9], 6, 3, [7, 9], 4] [7, [4, 5], 9, [4, 5], 3, 1, [8], 2, 6] description [((0, 8), 8), ((0, 6), 2), ((0, 7), 4), ((0, 8), 8), ((0, 6), 2), ((0, 7), 4), ((6, 2), 3), ((7, 1), 1), ((8, 6), 8), ((6, 8), 1), ((8, 6), 8), ((6, 8), 1)] after solve [0, 0, 5, 0, 6, 0, 2, 4, 8] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, 0, 8, 0, 5, 4, 7, 6, 3] [0, 8, 0, 2, 0, 0, 6, 3, 0] [0, 7, 0, 6, 0, 0, 1, 8, 2] [0, 2, 6, 3, 0, 8, 4, 5, 0] [6, 0, 3, 0, 8, 2, 5, 0, 1] [8, 1, 2, 0, 0, 6, 3, 0, 4] [7, 0, 9, 0, 3, 1, 8, 2, 6] before scan [0, 0, 5, 0, 6, 0, 2, 4, 8] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, 0, 8, 0, 5, 4, 7, 6, 3] [0, 8, 0, 2, 0, 0, 6, 3, 0] [0, 7, 0, 6, 0, 0, 1, 8, 2] [0, 2, 6, 3, 0, 8, 4, 5, 0] [6, 0, 3, 0, 8, 2, 5, 0, 1] [8, 1, 2, 0, 0, 6, 3, 0, 4] [7, 0, 9, 0, 3, 1, 8, 2, 6] suggest for board [[1], [3, 9], 5, [7, 9], 6, [7, 9], 2, 4, 8] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, [9], 8, [1, 9], 5, 4, 7, 6, 3] [[5, 9], 8, [1, 4], 2, [4, 7, 9], [5, 7, 9], 6, 3, [7, 9]] [[3, 5, 9], 7, [4], 6, [4, 9], [5, 9], 1, 8, 2] [[9], 2, 6, 3, [1, 7, 9], 8, 4, 5, [7, 9]] [6, [4], 3, [4, 7, 9], 8, 2, 5, [7, 9], 1] [8, 1, 2, [5, 7, 9], [7, 9], 6, 3, [7, 9], 4] [7, [4, 5], 9, [4], 3, 1, 8, 2, 6] description [((0, 0), 1), ((2, 1), 9), ((0, 0), 1), ((0, 1), 3), ((2, 3), 1), ((5, 0), 9), ((4, 2), 4), ((4, 0), 3), ((3, 2), 1), ((5, 4), 1), ((6, 1), 4), ((8, 1), 5), ((8, 3), 4), ((7, 3), 5)] after solve [1, 3, 5, 0, 6, 0, 2, 4, 8] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, 9, 8, 1, 5, 4, 7, 6, 3] [0, 8, 1, 2, 0, 0, 6, 3, 0] [3, 7, 4, 6, 0, 0, 1, 8, 2] [9, 2, 6, 3, 1, 8, 4, 5, 0] [6, 4, 3, 0, 8, 2, 5, 0, 1] [8, 1, 2, 5, 0, 6, 3, 0, 4] [7, 5, 9, 4, 3, 1, 8, 2, 6] before scan [1, 3, 5, 0, 6, 0, 2, 4, 8] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, 9, 8, 1, 5, 4, 7, 6, 3] [0, 8, 1, 2, 0, 0, 6, 3, 0] [3, 7, 4, 6, 0, 0, 1, 8, 2] [9, 2, 6, 3, 1, 8, 4, 5, 0] [6, 4, 3, 0, 8, 2, 5, 0, 1] [8, 1, 2, 5, 0, 6, 3, 0, 4] [7, 5, 9, 4, 3, 1, 8, 2, 6] suggest for board [1, 3, 5, [7, 9], 6, [7, 9], 2, 4, 8] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, 9, 8, 1, 5, 4, 7, 6, 3] [[5], 8, 1, 2, [4, 7], [7], 6, 3, [9]] [3, 7, 4, 6, [9], [5, 9], 1, 8, 2] [9, 2, 6, 3, 1, 8, 4, 5, [7]] [6, 4, 3, [7, 9], 8, 2, 5, [7, 9], 1] [8, 1, 2, 5, [7, 9], 6, 3, [7, 9], 4] [7, 5, 9, 4, 3, 1, 8, 2, 6] description [((3, 0), 5), ((3, 0), 5), ((4, 4), 9), ((3, 5), 7), ((3, 4), 4), ((4, 5), 5), ((3, 8), 9), ((5, 8), 7), ((3, 8), 9), ((5, 8), 7)] after solve [1, 3, 5, 0, 6, 0, 2, 4, 8] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, 9, 8, 1, 5, 4, 7, 6, 3] [5, 8, 1, 2, 4, 7, 6, 3, 9] [3, 7, 4, 6, 9, 5, 1, 8, 2] [9, 2, 6, 3, 1, 8, 4, 5, 7] [6, 4, 3, 0, 8, 2, 5, 0, 1] [8, 1, 2, 5, 0, 6, 3, 0, 4] [7, 5, 9, 4, 3, 1, 8, 2, 6] before scan [1, 3, 5, 0, 6, 0, 2, 4, 8] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, 9, 8, 1, 5, 4, 7, 6, 3] [5, 8, 1, 2, 4, 7, 6, 3, 9] [3, 7, 4, 6, 9, 5, 1, 8, 2] [9, 2, 6, 3, 1, 8, 4, 5, 7] [6, 4, 3, 0, 8, 2, 5, 0, 1] [8, 1, 2, 5, 0, 6, 3, 0, 4] [7, 5, 9, 4, 3, 1, 8, 2, 6] suggest for board [1, 3, 5, [7], 6, [9], 2, 4, 8] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, 9, 8, 1, 5, 4, 7, 6, 3] [5, 8, 1, 2, 4, 7, 6, 3, 9] [3, 7, 4, 6, 9, 5, 1, 8, 2] [9, 2, 6, 3, 1, 8, 4, 5, 7] [6, 4, 3, [9], 8, 2, 5, [7], 1] [8, 1, 2, 5, [7], 6, 3, [9], 4] [7, 5, 9, 4, 3, 1, 8, 2, 6] description [((0, 5), 9), ((0, 3), 7), ((0, 5), 9), ((0, 3), 7), ((6, 3), 9), ((7, 4), 7), ((6, 3), 9), ((7, 4), 7), ((7, 7), 9), ((6, 7), 7), ((7, 7), 9), ((6, 7), 7)] after solve [1, 3, 5, 7, 6, 9, 2, 4, 8] [4, 6, 7, 8, 2, 3, 9, 1, 5] [2, 9, 8, 1, 5, 4, 7, 6, 3] [5, 8, 1, 2, 4, 7, 6, 3, 9] [3, 7, 4, 6, 9, 5, 1, 8, 2] [9, 2, 6, 3, 1, 8, 4, 5, 7] [6, 4, 3, 9, 8, 2, 5, 7, 1] [8, 1, 2, 5, 7, 6, 3, 9, 4] [7, 5, 9, 4, 3, 1, 8, 2, 6]