Look at the 8x8 grid below at left. In the rows and columns there are repeated numbers. Erasing 19 of them, we achieve the grid at right, that has no repeated numbers in any row, in any column.
+---+---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+
| 5 | 7 | 1 | 2 | 5 | 4 | 4 | 3 | | | 7 | 1 | | 5 | | 4 | 3 |
+---+---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+
| 4 | 3 | 1 | 2 | 7 | 5 | 6 | 3 | | 4 | 3 | | 2 | 7 | 5 | 6 | |
+---+---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+
| 5 | 5 | 3 | 4 | 2 | 1 | 7 | 8 | | | 5 | 3 | | 2 | | 7 | 8 |
+---+---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+
| 6 | 6 | 2 | 7 | 3 | 3 | 3 | 1 | | 6 | | 2 | 7 | | 3 | | 1 |
+---+---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+
| 3 | 2 | 5 | 6 | 9 | 1 | 8 | 6 | | 3 | 2 | 5 | | 9 | 1 | 8 | 6 |
+---+---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+
| 2 | 1 | 3 | 4 | 6 | 2 | 5 | 2 | | | 1 | | 4 | 6 | | 5 | 2 |
+---+---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+
| 9 | 8 | 4 | 1 | 4 | 6 | 2 | 3 | | 9 | 8 | 4 | 1 | | 6 | 2 | |
+---+---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+
| 7 | 5 | 6 | 5 | 8 | 5 | 1 | 4 | | 7 | | 6 | 5 | 8 | | 1 | 4 |
+---+---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+
Do the same with this 8x8 grid, erasing the minimum number of squares.
+---+---+---+---+---+---+---+---+
| 8 | 4 | 6 | 5 | 3 | 5 | 7 | 4 |
+---+---+---+---+---+---+---+---+
| 6 | 5 | 5 | 4 | 7 | 8 | 3 | 1 |
+---+---+---+---+---+---+---+---+
| 5 | 7 | 2 | 5 | 5 | 4 | 8 | 7 |
+---+---+---+---+---+---+---+---+
| 8 | 6 | 5 | 3 | 2 | 5 | 4 | 4 |
+---+---+---+---+---+---+---+---+
| 3 | 8 | 1 | 4 | 8 | 6 | 5 | 2 |
+---+---+---+---+---+---+---+---+
| 5 | 3 | 7 | 6 | 4 | 2 | 2 | 2 |
+---+---+---+---+---+---+---+---+
| 5 | 8 | 7 | 7 | 6 | 2 | 1 | 3 |
+---+---+---+---+---+---+---+---+
| 1 | 1 | 3 | 7 | 6 | 4 | 6 | 8 |
+---+---+---+---+---+---+---+---+
(In reply to
A solution by Dej Mar)
Dej Mar and I have proposed nearly identical solutions -- since there are obviously a number of places in which the same overall result could be obtained by selecting a different one of duplicated cells to exclude. Perhaps someone will find fewer.
I think we have all proceeded by "trial and error" (i.e. no exhaustive computer search). Does anyone (including the proposer) have a proof mechanism in mind for determining the minimum erasures for any given initial diagram? Can a given problem be "built" by starting some given set of consistent cells?
Now, can we prove this is minimal?
