A page of postage stamps in a booklet contained 6 stamps from which it was possible to obtain any total value from 1 to the total value of all the stamps on the page by the appropriate selection of one or more orthogonally connected stamps from those laid out in the following manner:

+-----+-----+-----+
| | | |
| 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).*

Didn't we have this, or an almost identical, puzzle not long ago? It is not hard to find a solution, using the minimum values, i.e. x=5 and y=3. As I read the puzzle we need only be sure that all stamps used are attached to one another; the only ones which are excluded are those which attach to one of the others ONLY orthogonally (as would 1 and 6 alone, to represent 7).

1 (1) 2 (2) 3(3) 4(4) 5(5) 6(6) 7(1,2) 8(35) 9(36) 10 (46) 11(146 ) 12(246) 13(1246) 14(1346) 15(2346) 16(2356) 17(12356) 18(3456) 19 (13456) 20 (23456) 21 (123456)

Four alternatives: 8(26) 9(126) 10(235) 13(256).

However, this does not satisfy the condition that ONLY four numbers could be met in more than one way. (e.g. 14(356) 15(12345).