+-----+-----+-----+
|     |     |     |
|  1  |  2  |  x  |
|     |     |     |
+-----+-----+-----+
|     |     |     |
|  4  |  6  |  y  |
|     |     |     |
+-----+-----+-----+

With the numbers chosen, four of the values could be obtained in two different ways each, and the rest could be obtained in only one way each.

Your job is to figure out the numeric values of x and y.

Note that the square portion already given, containing 1, 2, 4 and 6, would, by itself satisfy the mentioned criterion: all the values from 1 to 13 can be achieved as the total of orthogonally connected stamps, such as 1, 2 and 4 to make 7 (the 6 and the 1 are not orthogonally connected, so that would not be allowed).

Bravo.

Was it analytically solved or comp.-aided?

How do we know that no better , that is higher reaching pairs exist?

Please advise.