In the example below, three rectangular pieces of dimensions 2x1, with the numbers (0,5), (2,3) and (2,3) are put into the 3x3 box so that all the 6 sums (3 rows and 3 columns) are the same (=5).
+---------------------+
| +=====++=====+|
| | 2 || 3 ||
|+=====+|-----||-----||
|| 0 || 3 || 2 ||
||-----|+=====++=====+|
|| 5 | |
|+=====+ |
+---------------------+I put these six similar pieces - (0,2), (0,6), (1,1), (1,5), (2,4) and (2,4) - with the numbers upwards, in a 4x4 box and showed it to my next door neighbour. He noticed that all 8 sums (4 rows and 4 columns) added up to the same number. How did I do it?
+=====+=====+ +=====+=====+ +=====+=====+
| 0 | 2 | | 0 | 6 | | 1 | 1 |
+=====+=====+ +=====+=====+ +=====+=====+
+=====+=====+ +=====+=====+ +=====+=====+
| 1 | 5 | | 2 | 4 | | 2 | 4 |
+=====+=====+ +=====+=====+ +=====+=====+Note: the "6" shown is still a "6" even when you put that piece upside down.
The sum of all the numbers on the blocks is 28, so if all 8 sums in the grid are equal, they must all = 7. Since there are only four odd entries in the pieces (1,1,1,5), that means exactly one odd entry must lie in each row and each column. This means no two odd entried can lie next to each other. But the odd entries each lie on the same piece as another odd entry, so it is impossible to generate a solution with exactly one odd entry in each row an column.
What am I missing?
impossible?
